# Measure-valued affine and polynomial diffusions

Christa Cuchiero Francesco Guida Luca di Persio
Sara Svaluto-Ferro Vienna University, Department of Statistics and Operations
Research, Data Science @ Uni Vienna, Kolingasse 14-16, 1090 Wien, Austria,
<EMAIL_ADDRESS>University of Trento and University of Verona,
Department of Mathematics, Via Sommarive 14, 38123 Povo, Italy,
<EMAIL_ADDRESS>University of Verona, Department of Computer Science,
Strada le Grazie 15, 37134 Verona, Italy<EMAIL_ADDRESS>of
Verona, Department of Economics, Via Cantarane 24, 37129 Verona, Italy,
<EMAIL_ADDRESS>
The first and the fourth author gratefully acknowledge financial support
through grant Y 1235 of the START-program.

###### Abstract

We introduce a class of measure-valued processes, which – in analogy to their
finite dimensional counterparts – will be called measure-valued polynomial
diffusions. We show the so-called moment formula, i.e. a representation of the
conditional marginal moments via a system of finite dimensional linear PDEs.
Furthermore, we characterize the corresponding infinitesimal generators and
obtain a representation analogous to polynomial diffusions on
$\mathbb{R}^{m}_{+}$, in cases where their domain is large enough. In general
the infinite dimensional setting allows for richer specifications strictly
beyond this representation. As a special case we recover measure-valued affine
diffusions, sometimes also called Dawson-Watanabe superprocesses.
From a mathematical finance point of view the polynomial framework is
especially attractive as it allows to transfer the most famous finite
dimensional models, such as the Black-Scholes model, to an infinite
dimensional measure-valued setting. We outline in particular the applicability
of our approach for term structure modeling in energy markets.

Keywords: measure-valued processes; polynomial and affine diffusions; Dawson-
Watanabe type superprocesses; martingale problem; maximum principle; HJM term
structure modeling; energy markets

AMS MSC 2020: 60J68, 60G57

###### Contents

  1. 1 Introduction
    1. 1.1 Notation and basic definitions
  2. 2 Polynomials and cylindrical functions of measure arguments
    1. 2.1 Monomials and polynomials
    2. 2.2 Directional derivatives
    3. 2.3 Cylindrical polynomials with regular coefficients
    4. 2.4 Cylindrical functions
  3. 3 Optimality conditions
  4. 4 Polynomial operators
    1. 4.1 Characterization of polynomial operators in the diffusion setting
    2. 4.2 Dual operators
  5. 5 Moment formula and existence of polynomial diffusions on $M_{+}(E)$
    1. 5.1 Moment formula
    2. 5.2 Existence
    3. 5.3 Uniqueness in law
  6. 6 Affine diffusions on $M_{+}(E)$
    1. 6.1 Operators of affine type and their characterization in the diffusion setting
    2. 6.2 Laplace transform and Riccati (partial) differential equations
    3. 6.3 Existence and uniqueness in law
  7. 7 Examples and applications
    1. 7.1 Finite underlying space
    2. 7.2 Underlying space $E\subseteq\mathbb{R}$
    3. 7.3 Measure-valued Pearson diffusions
  8. A Necessary conditions for the positive maximum principle
  9. B Existence for martingale problems
  10. C Generators of strongly continuous positive groups
    1. C.1 Case $E\subseteq\mathbb{R}$

##  1 Introduction

We introduce a class of (non-negative) measure-valued processes, which we call
in analogy to their finite dimensional counterparts _measure-valued polynomial
diffusions_. In spirit of Cuchiero et al. (2019), where the focus was on
probability measures, we now transfer the defining property of finite
dimensional polynomial processes considered e.g. in Cuchiero et al. (2012);
Filipović and Larsson (2016); Larsson and Pulido (2017); Cuchiero et al.
(2018) to the state space of (non-negative) measures.

This state space has high relevance for applications since stochastic modeling
of infinite dimensional non-negative quantities occurs in many areas. This
applies especially to biology and population genetics where measure-valued
processes have played a crucial role since the 1970’s, when branching Brownian
motion was introduced by Henry McKean, see McKean (1975). Also the polynomial
property occurs in this field often naturally, see, e.g., Blath et al. (2016)
for an example of a recently investigated two-dimensional polynomial process.

Similarly to finite dimensional polynomial processes on non-negative state
spaces which have been applied in many areas of finance, like credit risk,
stochastic volatility, or life insurance liability modeling (see, e.g.,
Ackerer and Filipović (2020); Ackerer et al. (2018); Arrouy et al. (2020);
Biagini and Zhang (2016)), measure-valued polynomial processes are also very
well suited for modeling in mathematical finance. Indeed, infinite dimensional
non-negative processes are of particular interest in view of term structures
as we shall illustrate below by analyzing forward price models in energy
markets. Another instance of measure-valued processes concerns Markovian lifts
of stochastic Volterra processes (see, e.g., Abi Jaber (2019); Cuchiero and
Teichmann (2019, 2020)), which became recently very popular in view of rough
volatility modeling initiated by Gatheral et al. (2018). Stochastic portfolio
theory, introduced in Fernholz (2002) (see also Fernholz and Karatzas (2009)),
is another area of applications where one deals with potentially high or
infinite dimensional markets. Hence a measure-valued analog of the finite
dimensional setup considered e.g. in Fernholz and Karatzas (2005); Cuchiero
(2019) is of interest to study asymptotic relative arbitrages. Measure-valued
processes can also be used as state processes for stochastic optimal control
problems with applications including model-independent derivatives pricing and
two player games with asymmetric information (see, e.g., Cox et al. (2021a)).

The advantage of using measure-valued processes instead of function-valued
ones is that many spatial stochastic processes do not fall into the framework
of stochastic partial differential equations (SPDEs). Indeed, it is often
easier to establish existence of a measure-valued process than of an analogous
stochastic partial differential equation, say in some Hilbert space, which
would for instance correspond to its Lebesgue density, but which does not
necessarily exist. For an introduction, important concepts and results related
to measure-valued processes we refer to Etheridge (2000); Dawson (1993) and Li
(2010). For recent results in the probability measure valued case on
generalized Wasserstein spaces see Larsson and Svaluto-Ferro (2020).

Beyond the general suitability of measure-valued processes for infinite
dimensional dynamic modeling, the current polynomial framework allows for high
tractability and the possibility to transfer many famous finite dimensional
models, such as the Black-Scholes model or the Feller diffusion, to an
infinite dimensional measure-valued setting. It offers a unified treatment of
a large class of processes that includes these well-known examples but also
goes far beyond them as we shall illustrate in Theorem 5.9.

Let us now describe the precise setup in more detail. We denote by $X$
measure-valued polynomial diffusions taking values in the space of (non-
negative) measures on a locally compact Polish space $E$. We define them as
solutions of martingale problems with continuous trajectories for certain
operators $L$ acting on classes of cylindrical functions, i.e. functions $f$
of the form

$\textstyle
f(\nu)=\phi\left(\int_{E}g_{1}(x)\nu(dx),\ldots,\int_{E}g_{m}(x)\nu(dx)\right),$

where $\phi\in C^{\infty}(\mathbb{R}^{m})$, $g_{1},\ldots,g_{m}$ are
continuous and bounded, and the argument $\nu$ is a (non-negative) measure.
The defining property of a measure-valued polynomial diffusion is that when
$L$ is applied to a cylindrical polynomial, i.e. $\phi$ is a polynomial on
$\mathbb{R}^{m}$, then $Lf$ is again a polynomial with the same degree as $f$.
Precise definitions are given in Section 4. For related concepts in general
Banach spaces we refer to the inspiring paper Benth et al. (2020).

As a consequence of the defining property of polynomial diffusion we can then
prove the following results:

  * •

We show that the conditional marginal moments of measure-valued polynomial
diffusions can be represented via a system of _finite dimensional linear PDEs_
, whose (maximal) spatial dimension corresponds to the degree of the moment,
see Theorem 5.5. This means that the so-called _moment formula_ holds and that
the tractability of finite dimensional polynomial processes can be preserved
in this setting. Indeed, for polynomial terminal conditions the infinite
dimensional Kolmogorov backward equation can be reduced to finite dimensional
system of PDEs.

  * •

We give necessary and sufficient conditions for the existence of measure-
valued polynomial diffusions by analyzing the corresponding martingale
problem. When the domain of the respective operators is large enough, we
obtain a representation of the drift and diffusion part analogous to
polynomial diffusions on $\mathbb{R}^{m}_{+}$, analyzed in Filipović and
Larsson (2016), see Theorem 5.11. Otherwise the infinite dimensional setting
allows for much more generality by going strictly beyond this representation,
see Theorem 5.9.

  * •

One key ingredient in these proofs is a characterization of the positive
maximum principle of the respective operators. To this end we also obtain new
optimality conditions which are applicable for all martingale problems on the
space of (non-negative) measures, see Section 3.

  * •

By specifying affine type operators in Section 6 we recover Dawson-Watanabe-
type superprocesses as the affine subclass of polynomial diffusions. In this
case, additionally to the moment formula, the Laplace transform of the
process’ marginals is exponentially affine in the initial state. As well-known
the corresponding characteristic exponent can then be computed by solving the
associated Riccati partial differential equations (see, e.g., Li (2010)). Let
us here also mention that infinite dimensional affine processes, however with
values in Hilbert spaces, have recently been studied in Schmidt et al. (2020);
Cox et al. (2021b).

  * •

We also show under which conditions exponential moment exist, so that
uniqueness in law holds and the corresponding martingale problems are well-
posed, see Theorem 5.15. This condition is satisfied by the subclass of
measure-valued affine diffusions, whose marginal laws are then characterized
by the solution of the associated Riccati partial differential equations.

  * •

We provide several concrete specifications in Section 7. Indeed, we consider
as underlying space $E$ a finite set of points and recover thus the results of
Filipović and Larsson (2016). Moreover, we analyze in detail the case when
$E\subseteq\mathbb{R}$, exploiting the characterization of strongly continuous
positive groups provided in Arendt et al. (1986), which allows us to specify a
concrete and easily applicable form of the infinitesimal generator. We also
connect our findings with measure-valued analogs of Pearson diffusions (see,
e.g., Forman and Sørensen (2008)) and the Black-Scholes model.

This shows in particular how to construct measure-valued versions of the most
famous finite dimensional models in finance and hence why measure-valued
affine and polynomial diffusions qualify as tractable models for all high
dimensional financial markets potentially involving an infinite number of
assets. As already mentioned above, this concerns especially term structures
arising from fixed income markets with possibly multiple yield curves (see,
e.g., Cuchiero et al. (2016a)), equity derivatives (see Kallsen and Krühner
(2015) and the references therein) or forward contracts in energy markets (see
Benth et al. (2008); Benth and Krühner (2014)). Time to maturity (or the
strike dimension in case of equity derivatives) then takes the role of the
spatial structure.

Our results can most notably be used for energy market modeling, in particular
for electricity and gas markets, whose essential products are futures and
forward contracts as well as options written on these. We concentrate on
futures (also called swaps) with delivery over a time interval
$[\tau_{1},\tau_{2}]$. Their price at time $t\geq 0$ is denoted by
$F(t,\tau_{1},\tau_{2})$ at $t\leq\tau_{1}$. Following Benth et al. (2008),
$F(t,\tau_{1},\tau_{2})$ can be written as a weighted integral of
instantaneous forward prices $f(t,u)$ with delivery at time $\tau_{1}\leq
u\leq\tau_{2}$, i.e.

$F(t,\tau_{1},\tau_{2})=\int_{\tau_{1}}^{\tau_{2}}w(u,\tau_{1},\tau_{2})f(t,u)du,$

where $w(u,\tau_{1},\tau_{2})$ denotes some weight function. The crucial
reason for using measure-valued processes in electricity and gas modeling now
comes from the fact that there is no trading with the instantaneous forwards
$f(t,u)$ for obvious reasons. Thus, rather than using $f(t,u)du$ in the
expression of the future prices, we can also use a measure, which cannot
necessarily be evaluated pointwise. In the companion paper Cuchiero et al.
(2021) we establish a _Heath-Jarrow-Morton_ (HJM) approach (see Heath et al.
(1992)) with measure-valued processes instead of processes taking values in
some Hilbert space of functions. We then obtain a HJM-drift condition that
restricts the choice of the process since the drift part is completely
determined, but we are free to specify its martingale part as long as we do
not leave the state space of (non-negative) measures. Here, the full
specification of affine and polynomial diffusions provided in Section 5 comes
into play, since it allows us to obtain a rich and flexible class of models
while preserving tractability. Indeed, the martingale part given by $Q_{1}$
and $Q_{2}$ in Theorem 5.9 depends on continuous functions satisfying certain
admissibility conditions. For calibration and pricing purposes these functions
can then be parametrized by neural networks. This in turn leads to fast
calibration procedures where on the one hand the analytic tractability coming
from the affine and polynomial nature and on the other hand (stochastic)
gradient descent methods for neural networks can be exploited.

The remainder of the paper is organized as follows. In Section 1.1 we
introduce frequently used notation and basic definitions. In Section 2 we
define polynomials and cylindrical functions of measure arguments, while in
Section 3 we prove optimality conditions for these function, which we need to
apply the positive maximum principle later on. In Section 4 we introduce the
notion of polynomial operators in the current setting and characterize them as
second order differential operator with affine drift and quadratic diffusion
part. Section 5 contains the main results, i.e. the moment formula, sufficient
conditions for existence and uniqueness and a full characterization when the
domain of the infinitesimal generator is rich enough. Section 6 is dedicated
to the analysis of the affine subclass and Section 7 concludes with exemplary
specifications. In Appendix A we prove necessary conditions for the positive
maximum principle, that in turn implies existence of solutions to the
martingale problem, see Appendix B. In Appendix C we recall a characterization
of generators of strongly continuous positive groups proved in Arendt et al.
(1986).

###  1.1 Notation and basic definitions

Throughout this article, $E$ is a locally compact Polish space endowed with
its Borel $\sigma$-algebra.

  * •

$M_{+}(E)$ denotes the set of finite non-negative measures on $E$, and
$M(E)=M_{+}(E)-M_{+}(E)$ the space of signed measures of the form
$\nu_{+}-\nu_{-}$ with $\nu_{+},\nu_{-}\in M_{+}(E)$. We usually leave “non-
negative” away and just say finite measures for $M_{+}(E)$. Both $M(E)$ and
$M_{+}(E)$ are equipped with the topology of weak convergence, which turns
$M_{+}(E)$ into a Polish space. For $\mu,\nu\in M(E)$ we write $\mu\leq\nu$ if
$\nu-\mu\in M_{+}(E)$ and $|\nu|$ for $\nu_{+}+\nu_{-}$.

  * •

$C(E)$, $C_{b}(E)$, $C_{0}(E)$, $C_{c}(E)$ denote, respectively, the spaces of
continuous, bounded, vanishing at infinity, and compactly supported real
functions on $E$. We equip the latter three with the topology of uniform
convergence, and denote by $\|\,\cdot\,\|$ the supremum norm.

  * •

If $E$ is non-compact, then $E^{\Delta}=E\cup\\{\Delta\\}$ denotes the one-
point compactification, i.e. a compact Polish space. If $E$ is compact we
write $E^{\Delta}=E$. We also define

$\displaystyle C_{\Delta}(E^{k}):=\big{\\{}f|_{E^{k}}\ :\ f\in
C((E^{\Delta})^{k})\big{\\}},$ (1.1)

which is a closed subspace of $C_{b}(E^{k})$. The spaces $C_{\Delta}(E)$ and
$C(E^{\Delta})$ can be identified, and we sometimes regard elements of the
former as elements of the latter, and vice versa. When $E$ is compact,
$C(E)=C_{b}(E)=C_{0}(E)=C_{c}(E)=C_{\Delta}(E)$ holds true and we thus simply
write $C(E)$. Note that the constant function $1$ lies in $C_{\Delta}(E)$, but
of course not in $C_{0}(E)$. This is one reason why we introduce the function
space $C_{\Delta}(E^{k})$.

  * •

One important underlying space, taking in the above definition the role of
$E$, will be $M_{+}(E)$ if $E$ is compact, and $M_{+}(E^{\Delta})$ otherwise.
Recall that $M_{+}(E)$ is locally compact when $E$ is compact (see, e.g.,
(Luther, 1970, Remark 1.2.3)), which however does not hold true when $E$ is
non-compact. Therefore we shall consider $M_{+}(E^{\Delta})$ in the non-
compact case. Identifying $E^{\Delta}$ with $E$, when $E$ is compact, we just
write $M_{+}(E^{\Delta})$ for both cases. In order to distinguish between the
different one-point compactifications, we denote the one-point
compactification of $M_{+}(E^{\Delta})$ by
$M^{\mathfrak{\Delta}}_{+}(E^{\Delta})$ and identify it with measures of
infinite mass. The function space $C_{\mathfrak{\Delta}}(M_{+}(E^{\Delta}))$
is defined analogously to (1.1) and can be identified with
$C(M^{\mathfrak{\Delta}}_{+}(E^{\Delta}))$.

  * •

$\widehat{C}_{\Delta}(E^{k})$ is the closed subspace of $C_{\Delta}(E^{k})$
consisting of symmetric functions $f$, i.e.,
$f(x_{1},\ldots,x_{k})=f(x_{\sigma(1)},\ldots,x_{\sigma(k)})$ for all
$\sigma\in\Sigma_{k}$, the permutation group of $k$ elements.
$\widehat{C}_{0}(E^{k})$ and $\widehat{C}(E^{k})$ are defined similarly. For
any $g\in\widehat{C}_{\Delta}(E^{k}),h\in\widehat{C}_{\Delta}(E^{\ell})$ we
denote by $g\otimes h\in\widehat{C}_{\Delta}(E^{k+\ell})$ the symmetric tensor
product, given by

$\displaystyle(g\otimes h)$ $\displaystyle(x_{1},\ldots,x_{k+\ell})$ (1.2)
$\displaystyle=\frac{1}{(k+\ell)!}\sum_{\sigma\in\Sigma_{k+\ell}}g\big{(}x_{\sigma(1)},\ldots,x_{\sigma(k)}\big{)}h\big{(}x_{\sigma(k+1)},\ldots,x_{\sigma(k+\ell)}\big{)}.$

We emphasize that only symmetric tensor products are used in this paper.

  * •

Throughout the paper we let

$D\subseteq C_{\Delta}(E)$

be a dense linear subspace containing the constant function 1. We set
$D^{\otimes 2}=D\otimes D:=\text{span}\\{g\otimes g\ :\ g\in D\\}$. This
generalizes of course to $D^{\otimes k}$ for $k>2$.

Two key notions that we shall often use are the _positive maximum principle,_
and the _positive minimum principle_ for certain linear operators.

######  Definition 1.1.

Fix a Polish space ${\mathcal{X}}$ and a subset
${\mathcal{S}}\subseteq{\mathcal{X}}$. Moreover, let $D({\mathcal{A}})$ be a
linear subspace of $C({\mathcal{X}})$.

  * •

An operator ${\mathcal{A}}$ with domain $D({\mathcal{A}})$ is said to satisfy
the positive maximum principle on ${\mathcal{S}}$ if

$f\in D({\mathcal{A}})$, $x\in{\mathcal{S}}$, $\sup_{\mathcal{S}}f=f(x)\geq
0\quad$ implies $\quad{\mathcal{A}}f(x)\leq 0$.

  * •

An operator ${\mathcal{A}}$ with domain $D({\mathcal{A}})$ is said to satisfy
the positive minimum principle on ${\mathcal{S}}$ if

$0\leq g\in
D({\mathcal{A}}),\,x\in{\mathcal{S}},\,\inf_{{\mathcal{S}}}g=g(x)=0\quad\text{implies}\quad\mathcal{A}g(x)\geq
0.$

######  Remark 1.2.

Note that the positive maximum principle implies the positive minimum
principle. Indeed, let $g\geq 0$ with $\inf_{\mathcal{S}}g=g(x)=0$ and set
$f=-g$ (which is possible since $D$ is a linear space). Then the positive
maximum principle yields $\mathcal{A}f(x)=-\mathcal{A}g(x)\leq 0$.

Moreover, let ${\mathcal{A}}$ be an operator with domain $D({\mathcal{A}})$
satisfying the positive minimum principle on ${\mathcal{S}}$ and suppose that
$1\in D({\mathcal{A}})$. Then ${\mathcal{A}}g={\mathcal{B}}g+mg$ for some map
$m$ and some operator ${\mathcal{B}}$ satisfying ${\mathcal{B}}1=0$ and the
positive maximum principle on ${\mathcal{S}}$. ${\mathcal{B}}$ and $m$ can be
explicitly constructed by setting ${\mathcal{B}}g(x)={\mathcal{A}}(g-g(x))(x)$
and $m(x):={\mathcal{A}}1(x)$.

These notions shall play an important role throughout the paper. Indeed, the
positive maximum principle (combined with conservativity) is essentially
equivalent to the existence of an ${\mathcal{S}}$-valued solution to the
martingale problem for ${\mathcal{A}}$, see (Ethier and Kurtz, 2009, Theorem
4.5.4). Here it is crucial that ${\mathcal{S}}$ is locally compact. In our
setting the most important case is ${\mathcal{S}}=M_{+}(E^{\Delta})$ which is
locally compact.

The positive minimum principle will be used to characterize certain operators
appearing in the generator of measure-valued polynomial diffusions. Note here
that for generators of strongly continuous semigroups on $C(\mathcal{X})$ the
positive minimum principle is equivalent to generating a _positive_ semigroup,
if $\mathcal{X}$ is compact, see (Arendt et al., 1986, Theorem B.II.1.6).

Finally, we indicate bounded pointwise limits by $\text{bp-}\lim$ in the sense
of (Ethier and Kurtz, 2009, Appendix 3).

##  2 Polynomials and cylindrical functions of measure arguments

In this section we recall the notion of polynomials of measure arguments
similarly as in Cuchiero et al. (2019).

###  2.1 Monomials and polynomials

A _monomial_ on $M(E)$ is defined via

$\langle
g,\nu^{k}\rangle=\int_{E^{k}}g(x_{1},\ldots,x_{k})\nu(dx_{1})\cdots\nu(dx_{k})$

for some $k\in\mathbb{N}_{0}$, where $g\in\widehat{C}_{\Delta}(E^{k})$ is
referred to as the _coefficient_ of the monomial; see, e.g., (Dawson, 1993,
Chapter 2). We identify $\widehat{C}_{\Delta}(E^{0})$ with $\mathbb{R}$, so
that for $k=0$ we have $\langle g,\nu^{0}\rangle=g\in\mathbb{R}$. The map
$\nu\mapsto\langle g,\nu^{k}\rangle$ is clearly homogeneous of degree $k$, and
$g\mapsto\langle g,\nu^{k}\rangle$ is linear. Furthermore, the identity
$\langle g,\nu^{k}\rangle\langle h,\nu^{\ell}\rangle=\langle g\otimes
h,\nu^{k+\ell}\rangle$, holds true, where the symmetric tensor product
$g\otimes h$ is defined in (1.2).

A polynomial on $M(E)$ is now defined as a (finite) linear combination of
monomials,

$p(\nu)=\sum_{k=0}^{m}\langle g_{k},\nu^{k}\rangle,$ (2.1)

with coefficients $g_{k}\in\widehat{C}_{\Delta}(E^{k})$, and we shall denote
the coefficients vector $(g_{0},\ldots,g_{m})$ of a polynomial by
$\vec{g}=(g_{0},\ldots,g_{m})\in\bigoplus_{k=0}^{m}\widehat{C}_{\Delta}(E^{k})$.

The degree of a polynomial $p$, denoted by $\deg(p)$, is the largest $k$ such
that $g_{k}$ is not the zero function, and $-\infty$ if $p$ is the zero
polynomial. The representation (2.1) is unique; see Corollary 2.4 in Cuchiero
et al. (2019).

The following example shows that the notion of polynomials coincides with the
usual one on $\mathbb{R}^{m}$ when $E$ is a finite set with $m$ elements.

######  Example 2.1.

Let $E=\\{1,\ldots,m\\}$ be a finite set. Then every element $\nu\in M(E)$ is
of the form

$\nu=c_{1}\delta_{1}+\cdots+c_{m}\delta_{m},\qquad(c_{1},\ldots,c_{m})\in\mathbb{R}^{m},$

where $\delta_{i}$ is the Dirac mass concentrated at $\\{i\\}$. Monomials take
the form

$\langle
g,\nu^{k}\rangle=\sum_{i_{1},\ldots,i_{k}}g(i_{1},\ldots,i_{k})\,c_{i_{1}}\cdots
c_{i_{k}},$

where the summation ranges over $E^{k}=\\{1,\ldots,m\\}^{k}$. Therefore, as
$g$ can be any symmetric function on $E^{k}$, we recover all homogeneous
polynomials of total degree $k$ in the $m$ variables $c_{1},\ldots,c_{m}$.

In the following we define the function space of polynomials on $M(E)$.

######  Definition 2.2.

Let

$P:=\left\\{\nu\mapsto p(\nu)\colon\text{$p$ is a polynomial on
$M(E)$}\right\\}$

denote the algebra of all polynomials on $M(E)$ regarded as real-valued maps,
equipped with pointwise addition and multiplication.

The subsequent lemma asserts that polynomials on $M(E)$ can be uniquely
extended to polynomials on $M(E^{\Delta})$, which we shall apply regularly.
For its proof we refer to Lemma 2.3.1 in Cuchiero et al. (2019)

######  Lemma 2.3.

Each $p\in P$ is continuous on $M_{+}(E)$, sequentially continuous on $M(E)$,
and can be uniquely extended to a polynomial on $M(E^{\Delta})$.111It can be
shown that sequential continuity cannot be strengthened to continuity.

###  2.2 Directional derivatives

The notion of derivatives of a function on $M(E)$ that we apply throughout the
paper is a directional one. More precisely, a function $f\colon
M(E)\to\mathbb{R}$ is called differentiable at $\nu$ in direction $\delta_{x}$
for $x\in E$ if

$\partial_{x}f(\nu):=\lim_{\varepsilon\to
0}\frac{f(\nu+\varepsilon\delta_{x})-f(\nu)}{\varepsilon}$

exists. We write $\partial f(\nu)$ for the map $x\mapsto\partial_{x}f(\nu)$,
and we use the notation

$\partial^{k}_{x_{1}x_{2}\cdots
x_{k}}f(\nu):=\partial_{x_{1}}\partial_{x_{2}}\cdots\partial_{x_{k}}f(\nu)$

for iterated derivatives. We write $\partial^{k}f(\nu)$ for the corresponding
map from $E^{k}$ to $\mathbb{R}$. Observe that for a linear map $p\in P$ of
the form $p(\nu)=\langle g,\nu\rangle$ we get

$\partial_{x}p(\nu)=\lim_{{\varepsilon}\to 0}\frac{1}{{\varepsilon}}\int
g(y){\varepsilon}\delta_{x}(dy)=g(x)$

for each $x\in E$.

###  2.3 Cylindrical polynomials with regular coefficients

In order to be able to consider certain differential operators later on, let
us introduce subspaces of polynomials with more regular coefficients. Let
$\textup{Pol}(\mathbb{R}^{m})$ denote the set of polynomials on
$\mathbb{R}^{m}$ and recall that $D\subseteq C_{\Delta}(E)$ is a dense linear
subspace containing the constant function 1. We then call _cylindrical
polynomial_ an element of the set

$P^{D}:=\Big{\\{}\nu\mapsto p(\nu)=\phi(\langle
g_{1},\nu\rangle,\ldots,\langle
g_{m},\nu\rangle)\colon\phi\in\textup{Pol}(\mathbb{R}^{m}),\,g_{k}\in
D,\,m\in\mathbb{N}_{0}\Big{\\}}.$

Note that $P^{D}$ is the subalgebra of $P$ consisting of all (finite) linear
combinations of the constant polynomial and “rank-one” monomials $\langle
g\otimes\cdots\otimes g,\nu^{k}\rangle=\langle g,\nu\rangle^{k}$ with $g\in
D$.

Since for $g_{k_{1}}\in\widehat{C}_{\Delta}(E^{k_{1}})$ and
$g_{k_{2}}\in\widehat{C}_{\Delta}(E^{k_{2}})$, it holds $\langle
g_{k_{1}},\nu^{k_{1}}\rangle\langle g_{k_{2}},\nu^{k_{2}}\rangle=\langle
g_{k_{1}}\otimes g_{k_{2}},\nu^{k_{1}+k_{2}}\rangle$, we can equivalently
write

$P^{D}=\left\\{\nu\mapsto\sum_{k=0}^{m}\langle g_{k},\nu^{k}\rangle\colon
m\in\mathbb{N}_{0},\ g_{k}\in D^{\otimes k}\right\\},$

where $D^{\otimes k}$ was introduced right after (1.2). For the sake of
completeness, let us restate parts of Lemma 2.7 of Cuchiero et al. (2019),
showing that the derivative of $x\mapsto\partial_{x}p(\nu)$ has the same
regularity as the coefficients of $p$, the latter being our main motivation to
introduce $P^{D}$.

######  Lemma 2.4.

For any $p\in P^{D}$ and $\nu\in M(E)$, we have $\partial^{k}p(\nu)\in
D^{\otimes k}$.

###  2.4 Cylindrical functions

Similarly to cylindrical polynomials we also consider cylindrical functions.
Define

$F^{D}=\Big{\\{}\nu\mapsto f(\nu)=\phi(\langle g_{1},\nu\rangle,\ldots,\langle
g_{m},\nu\rangle)\colon,\phi\in C^{\infty}_{0}(\mathbb{R}^{m}),g_{k}\in
D,\,m\in\mathbb{N}_{0}\Big{\\}}.$

Since concatenations of continuous functions are continuous, cylindrical
functions satisfy the following lemma.

######  Lemma 2.5.

Each $f\in F^{D}$ is continuous on $M_{+}(E)$, sequentially continuous on
$M(E)$, and can be uniquely extended to $M(E^{\Delta})$.

For further use we now define the restriction of $F^{D}$ to
$M_{+}(E^{\Delta})$, i.e.

$F^{D}(M_{+}(E^{\Delta})):=\\{f|_{M_{+}(E^{\Delta})}\colon f\in F^{D}\\}.$

Observe that functions in $F^{D}$ are first extended to $M(E^{\Delta})$ using
Lemma 2.5 and then restricted to $M_{+}(E^{\Delta})$. Since elements in
$F^{D}(M_{+}(E^{\Delta}))$ do not need to be compactly supported we also
define

$F^{D}_{c}:=F^{D}(M_{+}(E^{\Delta}))\cap C_{c}(M_{+}(E^{\Delta})).$

######  Lemma 2.6.

For any $f\in F^{D}$ and $\nu\in M(E)$, we have $\partial^{k}f(\nu)\in
D^{\otimes k}$. Moreover $F^{D}_{c}$ is dense in $C_{0}(M_{+}(E^{\Delta}))$.

###### Proof.

For $f(\nu):=\langle g,\nu\rangle$ we have $\partial^{k}f(\nu)=g^{\otimes
k}\in D^{\otimes k}$. The first part follows by the chain rule. Since
$F^{D}_{c}$ is a point separating algebra that vanishes nowhere, the locally
compact version of the Stone–Weierstrass theorem yields the second part. ∎

Finally, we introduce a set that will be used as domain for linear operators:

$\displaystyle\mathcal{D}:=\operatorname{span}\\{P^{D},F^{D}\\}.$ (2.2)

Observe that we here do not specify the Banach space containing $\mathcal{D}$.

##  3 Optimality conditions

We now develop optimality conditions for cylindrical functions and polynomials
of measure arguments, which we need in order to apply the positive maximum
principle on $M_{+}(E^{\Delta})$. Our first result, Theorem 3.1, extends the
classical first and second order Karush–Kuhn–Tucker conditions for functions
on $\mathbb{R}_{+}^{m}$ (see, e.g., Bertsekas (1997)).

Throughout the section we let ${\mathcal{D}}$ be the set defined in (2.2).
Note that we use Lemma 2.4 and Lemma 2.5 to extend the polynomials and
cylindrical functions from $M_{+}(E)$ to $M_{+}(E^{\Delta})$.

######  Theorem 3.1.

Let $f\in{\mathcal{D}}$ and $\nu^{*}$ satisfy $f(\nu^{*})=\max_{\nu\in
M_{+}(E^{\Delta})}f(\nu)$. Then the following first and second order
optimality conditions are satisfied.

  1. (i)

For all $\mu,\nu\in M(E^{\Delta})$ with
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$, it holds that

$\langle\partial f(\nu^{*}),\mu\rangle=0\qquad\text{and}\qquad\langle\partial
f(\nu^{*}),\nu\rangle\leq 0.$

In particular, $\partial_{x}f(\nu^{*})\leq 0$ for all $x\in E^{\Delta}$ and
vanishes for all $x\in\operatorname{supp}(\nu^{*})$.

  2. (ii)

For all $\mu\in M(E^{\Delta})$ with
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$, it holds that

$\langle\partial^{2}f(\nu^{*}),\mu^{2}\rangle\leq 0.$

In particular, $\partial^{2}_{xx}f(\nu^{*})\leq 0$ for all
$x\in\operatorname{supp}(\nu^{*})$.

###### Proof.

(i): Let $x\in\operatorname{supp}(\nu^{*})$. We shall first consider the
following perturbation $\nu^{*}-\varepsilon_{n}\nu_{n}$ with
$(\nu_{n})_{n\in\mathbb{N}}$ a sequence of non-negative measures converging to
$\delta_{x}$ and $\varepsilon_{n}>0$ converging to $0$. To this end we need to
construct $(\nu_{n})_{n\in\mathbb{N}}$ and
$(\varepsilon_{n})_{n\in\mathbb{N}}$ such that
$\nu^{*}-\varepsilon_{n}\nu_{n}$ is a non-negative measure for all
$n\in\mathbb{N}$.

For each $n\in\mathbb{N}$, let $A_{n}$ be the ball of radius $\frac{1}{n}$
centered at $x$, intersected with $\operatorname{supp}(\nu^{*})$. Since
$A_{n}\subset\operatorname{supp}(\nu^{*})$, we have $\nu^{*}(A_{n})>0$ and the
measures $\nu_{n}:=\nu^{*}(\,\cdot\,\cap A_{n})/\nu^{*}(A_{n})$ converge
weakly to $\delta_{x}$ as $n\rightarrow\infty$. Let
$\varepsilon_{n}\in(0,\nu^{*}(A_{n}))$ and note that for all
$B\in\mathcal{B}(E^{\Delta})$

$\nu^{*}(B)-\varepsilon_{n}\frac{\nu^{*}(B\cap
A_{n})}{\nu^{*}(A_{n})}\geq\frac{\nu^{*}(B\cap
A_{n})}{\nu^{*}(A_{n})}(\nu^{*}(A_{n})-\varepsilon_{n})\geq 0,$

where the last inequality follows from the fact that
$\varepsilon_{n}\in(0,\nu^{*}(A_{n}))$. Hence $\nu^{*}-\varepsilon_{n}\nu_{n}$
is a non-negative measure. Moreover, using that $\nu_{n}$ is a probability
measure, we can compute

$\|(\langle g_{1},\varepsilon_{n}\nu_{n}\rangle,\ldots,\langle
g_{k},\varepsilon_{n}\nu_{n}\rangle)\|^{2}\leq\varepsilon_{n}^{2}\sum_{i=1}^{k}\|g_{i}\|^{2}.$
(3.1)

By the maximality of $\nu^{*}$ and the Taylor expansion of $\phi$ we then have

$\displaystyle 0$ $\displaystyle\geq
f(\nu^{*}-\varepsilon_{n}\nu_{n})-f(\nu^{*})$ $\displaystyle=\phi(\langle
g_{1},\nu^{*}-\varepsilon_{n}\nu_{n}\rangle,\ldots,\langle
g_{k},\nu^{*}-\varepsilon_{n}\nu_{n}\rangle)-\phi(\langle
g_{1},\nu^{*}\rangle,\ldots,\langle g_{k},\nu^{*}\rangle)$
$\displaystyle=-\varepsilon_{n}\sum_{i=1}^{k}\langle
g_{i},\nu_{n}\rangle\partial_{i}\phi(\langle
g_{1},\nu^{*}\rangle,\ldots,\langle g_{k},\nu^{*}\rangle)+o(\varepsilon_{n})$
$\displaystyle=-\varepsilon_{n}\langle\partial
f(\nu^{*}),\nu_{n}\rangle+o(\varepsilon_{n}).$

Dividing by $\varepsilon_{n}$, sending $n$ to infinity, and using the fact
that $\nu_{n}$ converges weakly to $\delta_{x}$ we obtain
$\partial_{x}f(\nu^{*})\geq 0$ for $x\in\operatorname{supp}(\nu^{*})$. For
general $x\in E^{\Delta}$, it holds that $\nu^{*}+\varepsilon\delta_{x}$ is a
non-negative measure. By the maximimality of $\nu^{*}$ we thus have

$\displaystyle 0$ $\displaystyle\geq
f(\nu^{*}+\varepsilon\delta_{x})-f(\nu^{*})=$ $\displaystyle=\phi(\langle
g_{1},\nu^{*}+\varepsilon\delta_{x}\rangle,\ldots,\langle
g_{k},\nu^{*}+\varepsilon\delta_{x}\rangle)-\phi(\langle
g_{1},\nu^{*}\rangle,\ldots,\langle g_{k},\nu^{*}\rangle)$
$\displaystyle=\varepsilon\sum_{i=1}^{k}g_{i}(x)\partial_{i}\phi(\langle
g_{1},\nu^{*}\rangle,\cdots,\langle g_{k},\nu^{*}\rangle)+o(\varepsilon)$
$\displaystyle=\varepsilon\partial_{x}f(\nu^{*})+o(\varepsilon).$

Dividing again by $\varepsilon$ and sending it to $0$, we deduce that
$\partial_{x}f\leq 0$ for all $x\in E^{\Delta}$.

Hence $\partial_{x}f=0$ for all $x\in\operatorname{supp}(\nu^{*})$ and
$\partial_{x}f\leq 0$ for all $x\in
E^{\Delta}\setminus\operatorname{supp}(\nu^{*})$. This proves assertion (i) as
for all signed measures $\mu\in M(E^{\Delta})$ with
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$

$\langle\partial
f(\nu^{*}),\mu\rangle=\int_{\operatorname{supp}(\mu)}\partial_{x}f(\nu^{*})\mu(dx)=0.$

(ii): Let $\mu\in M_{+}(E^{\Delta})$ such that
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$. Since
$\nu^{*}$ is a maximum for $f$, we obtain from the second order Taylor
expansion of $\phi$, the estimate (3.1), and by using (i)

$\begin{split}0&\geq f(\nu^{*}+\varepsilon\mu)-f(\nu^{*})=\\\
&=\varepsilon\sum_{i=1}^{k}\partial_{i}\phi(\langle
g_{1},\nu^{*}\rangle,\ldots,\langle g_{k},\nu^{*}\rangle)\langle
g,\mu\rangle\\\
&\quad+\frac{1}{2}\varepsilon^{2}\sum_{i,j=1}^{n}\partial_{ij}\phi(\langle
g_{1},\nu^{*}\rangle,\ldots,\langle g_{k},\nu^{*}\rangle)\langle g_{i}\otimes
g_{j},\mu^{2}\rangle+o(\varepsilon^{2})\\\
&=\frac{1}{2}\varepsilon^{2}\langle\partial^{2}f(\nu^{*}),\mu^{2}\rangle+o(\varepsilon^{2}).\end{split}$
(3.2)

Therefore, $\langle\partial^{2}f(\nu^{*}),\mu^{2}\rangle\leq 0$ for $\mu\in
M_{+}(E^{\Delta})$ with
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$. In
particular, for $x\in\operatorname{supp}(\nu^{*})$ and choosing
$\mu=\delta_{x}$ we have

$\partial^{2}_{xx}f(\nu^{*})=\sum_{i,j=1}^{n}g_{i}(x)g_{j}(x)\partial_{ij}\phi(\langle
g_{1},\nu^{*}\rangle,\ldots,\langle g_{k},\nu^{*}\rangle)\leq 0.$

For the general signed measure case, define

$\mu_{n}:=\sum_{i=1}^{m}\lambda_{i}\nu_{n,i},$

for $m\in\mathbb{N}$, $\lambda_{1},\ldots,\lambda_{m}\in\mathbb{R}$ and
$\nu_{n,i}$ constructed as $\nu_{n}$ above with $x$ replaced by
$x_{i}\in\operatorname{supp}(\nu^{*})$. Letting $\varepsilon_{n}$ decrease to
$0$ sufficiently fast (note that only the negative $\lambda_{i}$ have to be
taken into account) yields $\nu^{*}+\varepsilon_{n}\mu_{n}\in
M_{+}(E^{\Delta})$. The same reasoning as in (3.2) replacing $\varepsilon\mu$
by $\varepsilon_{n}\mu_{n}$ and passing to the limit implies that

$\langle\partial^{2}f(\nu^{*}),\Big{(}\sum_{i=1}^{m}\lambda_{i}\delta_{x_{i}}\Big{)}^{2}\rangle\leq
0.$

Passing to the weak closure yields
$\langle\partial^{2}f(\nu^{*}),\mu^{2}\rangle\leq 0$ for all $\mu\in
M(E^{\Delta})$ with
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$. This implies
the assertions of (ii). ∎

######  Remark 3.2.

To compare the results of Theorem 3.1 with the classical conditions for
functions on $\mathbb{R}^{m}_{+}$, let $E=\\{1,\ldots,m\\}$. Then $M_{+}(E)$
can be identified with $\mathbb{R}^{m}_{+}$. Condition (i) translates to

$\partial_{i}f(\nu^{*})=0,\quad i\in E,$

if the $i$-th component of $\nu^{*}$ is strictly positive and
$\partial_{i}f(\nu^{*})\leq 0$ otherwise. Condition (ii) corresponds to

$y^{\top}\partial^{2}f(\nu^{*})y\leq 0$

for all $y\in\mathbb{R}^{m}$ such that $y_{j}=0$ if $\nu^{*}_{j}=0$. In
particular

$\partial^{2}_{ii}f(\nu^{*})\leq 0,\quad i\in E,$

in the case where the $i$-th component of $\nu^{*}$ is strictly positive.
These come from the classical Karush-Kuhn-Tucker conditions when the
inequality constraints describe $\mathbb{R}_{m}^{+}$.

Our next optimality condition is an slight adaptation of Theorem 3.4 in
Cuchiero et al. (2019). In comparison with that result we do not need to work
with a group of positive isometries, but just with a positive group. Let us
recall here the definition of a positive group (see, e.g., Arendt et al.
(1986)).

######  Definition 3.3.

A group $\\{T_{t}\\}_{t\in\mathbb{R}}$ on $C_{\Delta}(E)$ is positive if for
any $0\leq g\in C_{\Delta}(E)$, we have $T_{t}g\geq 0.$

We shall use the tensor notation $A\otimes A$ to denote the linear operator
from $D\otimes D$ to $\widehat{C}_{\Delta}(E^{2})$ determined by

$(A\otimes A)(g\otimes g):=(Ag)\otimes(Ag)$

for a given linear operator $A\colon D\to C_{\Delta}(E)$.

######  Theorem 3.4.

Fix $f\in{\mathcal{D}}$ and $\nu_{*}\in M_{+}(E^{\Delta})$ with
$f(\nu_{*})=\max_{\nu\in M_{+}(E^{\Delta})}f(\nu)$. Let $A$ be the generator
of a strongly continuous positive group on $C_{\Delta}(E)$, and assume the
domain of $A$ contains both $D$ and $A(D)$. Then

$\langle A^{2}(\partial f(\nu_{*})),\mu\rangle+\langle(A\otimes
A)(\partial^{2}f(\nu_{*})),\mu^{2}\rangle\leq 0$

for every $\mu\in M_{+}(E^{\Delta})$ with $\mu\leq\nu_{*}$ and
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu_{*})$.

###### Proof.

In what follows we adapt the proof of Theorem 3.4 in Cuchiero et al. (2019) to
the case $M_{+}(E^{\Delta})$. Let $\\{T_{t}\\}_{t\in\mathbb{R}}$ be the group
generated by $A$. For any $\mu\in M_{+}(E^{\Delta})$, the group induces a flow
of measures $\mu_{t}\in M(E^{\Delta})$ via the formula $\langle
g,\mu_{t}\rangle=\langle T_{t}g,\mu\rangle$ for $g\in C_{\Delta}(E)$. The
positivity implies that $\mu_{t}$ is non-negative. Indeed, $\langle
T_{t}g,\mu\rangle\geq 0$ for all $g\geq 0$, whence $\mu_{t}$ is a non-negative
measure.

Therefore, assuming henceforth that $\mu\leq\nu_{*}$, it follows that
$\nu_{*}+\mu_{t}-\mu$ is a non-negative measure. Since $\|T_{t}g-g\|=O(t)$ for
every $g\in D$, we have $\langle g,(\mu_{t}-\mu)^{k}\rangle=O(t^{k})$ for
every $g\in D^{\otimes k}$. Maximality of $\nu_{*}$ and Taylor’s formula then
yield

$\displaystyle 0$ $\displaystyle\geq
f(\nu_{*}+\mu_{t}-\mu)-f(\nu_{*})=\langle\partial
f(\nu_{*}),\mu_{t}-\mu\rangle+\frac{1}{2}\langle\partial^{2}f(\nu_{*}),(\mu_{t}-\mu)^{2}\rangle+o(t^{2})$
$\displaystyle=\langle(T_{t}-\text{id})\partial
f(\nu_{*}),\mu\rangle+\frac{1}{2}\langle(T_{t}\otimes
T_{t}-2\,T_{t}\otimes\text{id}+\text{id}\otimes\text{id})\partial^{2}f(\nu_{*}),\mu^{2}\rangle+o(t^{2}).$
(3.3)

We claim that both $A$ and $-A$ satisfy the positive minimum principle (see
Definition 1.1) on $E^{\Delta}$. Indeed, for $0\leq g\in D$ and $x\in
E^{\Delta}$ with $g(x)=\inf_{E^{\Delta}}g=0$, we have

$\begin{split}Ag(x)&=\lim_{t\to 0}\frac{T_{t}g(x)-g(x)}{t}=\lim_{t\to
0}\frac{T_{t}g(x)}{t}\geq 0,\\\ -Ag(x)&=\lim_{t\to
0}\frac{T_{-t}g(x)-g(x)}{t}=\lim_{t\to 0}\frac{T_{-t}g(x)}{t}\geq
0;\end{split}$ (3.4)

which can just be seen as a consequence of (Arendt et al., 1986, Theorem
B.II.16). Since $-\partial f(\nu^{*})\geq 0$ and $\partial_{x}f(\nu_{*})=0$
for all $x\in\operatorname{supp}(\nu^{*})$ due to Theorem 3.1, it follows that
$A(\partial f(\nu_{*}))(x)=0$ for all such $x$. As a result, using that
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu_{*})$ and that the
domain of $A$ contains $A(D)$, we get

$\langle(T_{t}-\text{id})\partial
f(\nu_{*}),\mu\rangle=\langle(T_{t}-\text{id}-tA)\partial
f(\nu_{*}),\mu\rangle=\frac{1}{2}t^{2}\langle A^{2}(\partial
f(\nu_{*})),\mu\rangle+o(t^{2}).$ (3.5)

Furthermore, using that

$(T_{t}\otimes
T_{t}-2\,T_{t}\otimes\text{id}+\text{id}\otimes\text{id})(g\otimes
g)=(T_{t}g-g)\otimes(T_{t}g-g)$

for all $g\in D$, we deduce that

$\langle(T_{t}\otimes
T_{t}-2\,T_{t}\otimes\text{id}+\text{id}\otimes\text{id})g,\mu^{2}\rangle=t^{2}\langle(A\otimes
A)g,\mu^{2}\rangle+o(t^{2})$ (3.6)

for all $g\in D\otimes D$. Inserting (3.5) and (3.6) into (3.3), dividing by
$t^{2}$, and sending $t$ to zero yields

$0\geq\frac{1}{2}\langle A^{2}(\partial
f(\nu_{*})),\mu\rangle+\frac{1}{2}\langle(A\otimes
A)\partial^{2}f(\nu_{*}),\mu^{2}\rangle.$

This completes the proof. ∎

The optimality condition obtained in Theorem 3.4 involves the generator of a
strongly continuous positive group and is more subtle than the previous ones.
In particular in the finite-dimensional case the generator of a strongly
continuous positive group can only be a diagonal matrix, see Lemma 3.6. For
the properties of such an operator in the general case, we refer to Remark 3.7
and Appendix C.

######  Remark 3.5.

We claim that for $A$ as in Theorem 3.4, the operator $A^{2}$ satisfies the
positive minimum principle on $E^{\Delta}$. Indeed, let $g\in D$ and $x\in
E^{\Delta}$ with $g(x)=\inf_{E^{\Delta}}g=0$. Then, as $A$ and $-A$ satisfy
the positive minimum principle $Ag(x)=0$. Hence $A^{2}g(x)=\lim_{t\downarrow
0}(T_{t}g(x)-g(x)-Af(x))/t\geq 0$, which proves the claim.

The following lemma illustrates how the conditions of Theorem 3.4 translate to
the case where $D=C(E^{\Delta})$. This includes in particular the finite
dimensional case when $E$ consists of $n\in\mathbb{N}$ points.

######  Lemma 3.6.

  1. (i)

Let $A:C(E^{\Delta})\to C(E^{\Delta})$ be the generator of a strongly
continuous positive group, then

$Ag(x)=a(x)g(x)$

for some $a\in C_{\Delta}(E)$. Moreover $(A\otimes A)g(x,y)=a(x)a(y)g(x,y)$
for $g\in\widehat{C}_{\Delta}(E^{2})$.

  2. (ii)

Let $E$ consist of $n\in\mathbb{N}$ points and let $A$ be the generator of a
strongly continuous positive group on $C_{\Delta}(E)\cong\mathbb{R}^{n}$. Then
$A$ is a diagonal matrix.

###### Proof.

As shown in equation (3.4), $A$ and $-A$ satisfy the positive minimum
principle. Moreover, as $D=C(E^{\Delta})$, Corollary B.II.1.12 in Arendt et
al. (1986) yields that $A$ is bounded and that

$-\|A\|g\leq Ag\leq\|A\|g,\quad\text{for all }g\geq 0,$ (3.7)

where $\|A\|$ denotes the operator norm. Observe that (3.7) yields

$A((g-g(x))^{+})(x)=A((g-g(x))^{-})(x)=0,$

showing that $Ag(x)$ can only depend on $g(x)$. The statement follows from the
fact that $A$ is a bounded linear operator. The form of $(A\otimes A)g(x,y)$
is then obvious.

The second part directly follows from the first one. Let us underline that a
matrix in $\mathbb{R}^{n\times n}$ satisfies the positive minimum principle if
and only if the off-diagonal elements are non-negative; see (Arendt et al.,
1986, Example B.II.1.4). Since both $A$ and $-A$ satisfy the positive minimum
principle on $E$, we conclude that $A$ is a diagonal matrix. ∎

######  Remark 3.7.

Generators of strongly continuous positive groups on $C_{\Delta}(E)$ have been
fully characterized in (Arendt et al., 1986, Theorem B.II.3.14). In Appendix C
we recall the main tools behind their characterization by introducing
definitions such as flows and cocycles.

##  4 Polynomial operators

Recall that $E$ is a locally compact Polish space and $D\subseteq
C_{\Delta}(E)$ is a dense linear subspace containing the constant function 1.
Set ${\mathcal{D}}$ as in (2.2). We now define polynomial operators, which
constitute a class of possibly unbounded linear operators acting on
cylindrical functions and polynomials. For the moment we define these
polynomial operators for polynomials on general subsets
${\mathcal{S}}\subseteq M(E)$, similarly as in Section 4 of Cuchiero et al.
(2019). To avoid unnecessary loss of generality we work with the set
${\mathcal{M}}(M(E))$ of measurable real valued maps on $M(E)$.

######  Remark 4.1.

In this section, and also more generally, we often work just with subsets of
$M(E)$ despite the fact that the results hold also for subsets of
$M(E^{\Delta})$. This is due to the fact that the second can be deduced from
the first by means of a direct argument. Whenever this is not the case, we
will explicitly state it.

######  Definition 4.2.

Fix ${\mathcal{S}}\subseteq M(E)$. A linear operator
$L:{\mathcal{D}}\to{\mathcal{M}}(M(E))$ is called ${\mathcal{S}}$-polynomial
if $L$ maps $P^{D}$ to $P$ such that for every $p\in P^{D}$ there is some
$q\in P$ with $q|_{{\mathcal{S}}}=Lp|_{{\mathcal{S}}}$ and
$\deg(q)\leq\deg(p).$

###  4.1 Characterization of polynomial operators in the diffusion setting

For a linear operator $L$, the associated _carré-du-champ operator_ (see,
e.g., Bakry et al. (2013)) is the symmetric bilinear map
$\Gamma:\mathcal{D}\times\mathcal{D}\to{\mathcal{M}}(M(E))$

$\Gamma(p,q)=L(pq)-pLq-qLp.$ (4.1)

Related to the carré-du-champ operator is the notion of a _derivation_ , which
we recall in the following definition and which can be used to characterize
the form of $L$, as a second-order differential operator, typical for
diffusion processes.

######  Definition 4.3.

Fix ${\mathcal{S}}\subseteq M(E)$. A symmetric bilinear map
$\Gamma:\mathcal{D}\times\mathcal{D}\to{\mathcal{M}}(M(E))$ is called an
${\mathcal{S}}$-derivation if for all $p,q,r\in\mathcal{D}$,
$\Gamma(pq,r)=p\Gamma(q,r)+q\Gamma(p,r)$ on ${\mathcal{S}}$.

######  Remark 4.4.

As explained in Remark 4.1 we consider only subsets of $M(E)$ and not of
$M(E^{\Delta})$. Hence for finite measure we shall only speak of
$M_{+}(E)$-derivations (even if $\mathcal{S}=M_{+}(E^{\Delta})$).

######  Lemma 4.5.

The carré-du-champs operator $\Gamma$ of a linear operator
$L:\mathcal{D}\to{\mathcal{M}}(M(E))$ is an $M_{+}(E)$-derivation if and only
if $Lf(\nu)=0$ for each $f$ such that $\partial f(\nu)=0$ and
$\partial^{2}f(\nu)=0$.

###### Proof.

Observe first that if $\Gamma$ is an $M_{+}(E)$-derivation, it holds
$\Gamma(1,1)=2\Gamma(1,1)$, hence $L(1)=-\Gamma(1,1)=0$. Since for each
$p,q,r\in{\mathcal{D}}$ and $\nu\in M_{+}(E)$ it holds

$\displaystyle\Gamma(pq,r)(\nu)$
$\displaystyle-p(\nu)\Gamma(q,r)(\nu)-q(\nu)\Gamma(p,r)(\nu)-p(\nu)q(\nu)r(\nu)L1(\nu)$
$\displaystyle=L\Big{(}(p-p(\nu))(q-q(\nu))(r-r(\nu))\Big{)}(\nu),$

we get that $\Gamma$ is a derivation if and only if $L1=0$ and

$L\Big{(}(p-p(\nu))(q-q(\nu))(r-r(\nu))\Big{)}(\nu)=0,$ (4.2)

for each $p,q,r\in{\mathcal{D}}$ and $\nu\in M_{+}(E)$. Observe now that each
$p\in{\mathcal{D}}$ admits the representation

$p(\nu)=\phi(\langle g_{1},\nu\rangle,\ldots,\langle g_{n},\nu\rangle)$

for some $\phi\in C^{\infty}(\mathbb{R}^{n})$ and some linearly independent
$g_{1},\ldots,g_{n}\in D$. Assuming that $\partial p(\nu)=0$ and
$\partial^{2}p(\nu)=0$ and applying Taylor’s theorem to $\phi$ yields

$p(\mu)=\sum_{k_{1},k_{2}=1}^{n}\langle g_{k_{1}},\mu-\nu\rangle\langle
g_{k_{2}},\mu-\nu\rangle\tilde{p}_{k_{1},k_{2}}(\mu)$

for some $\tilde{p}_{k_{1},k_{2}}\in{\mathcal{D}}$ such that
$\tilde{p}_{k_{1},k_{2}}(\nu)=0$. The claim then follows by (4.2). ∎

For a finite-dimensional diffusion it is known that its generator is
polynomial if and only if the drift and diffusion coefficients are polynomial
of first and second degree, respectively, see Cuchiero et al. (2012) and
Filipović and Larsson (2016). The following result is the generalization of
this fact to the measure-valued setting.

######  Theorem 4.6.

A linear operator $L:\mathcal{D}\to{\mathcal{M}}(M(E))$ is
$M_{+}(E)$-polynomial and its carré-du-champ operator $\Gamma$ is an
$M_{+}(E)$-derivation if and only if $L$ admits a representation

$\displaystyle Lf(\nu)=$ $\displaystyle B_{0}(\partial f(\nu))+\langle
B_{1}(\partial f(\nu)),\nu\rangle$ (4.3)
$\displaystyle+\frac{1}{2}\Big{(}Q_{0}(\partial^{2}f(\nu))+\langle
Q_{1}(\partial^{2}f(\nu)),\nu\rangle+\langle
Q_{2}(\partial^{2}f(\nu)),\nu^{2}\rangle\Big{)},\quad f\in{\mathcal{D}},\nu\in
M_{+}(E)$

for some linear operators $B_{0}:D\to\mathbb{R}$, $B_{1}:D\to C_{\Delta}(E)$,
$Q_{0}:D\otimes D\to\mathbb{R}$, $Q_{1}:D\otimes D\to C_{\Delta}(E)$,
$Q_{2}:D\otimes D\to\widehat{C}_{\Delta}(E^{2})$. These operators are uniquely
determined by $L$.

###### Proof.

Theorem A.1 in Cuchiero et al. (2019) yields the result when ${\mathcal{D}}$
is replaced by $P^{D}$. The extension of the first implication from $P^{D}$ to
${\mathcal{D}}$ is then direct. For the converse implication, observe that for
each $f\in{\mathcal{D}}$ and $\nu\in M_{+}(E)$ there is a $p\in P^{D}$ such
that $\partial(p-f)(\nu)=0$ and $\partial^{2}(p-f)(\nu)=0$. The claim then
follows by Lemma 4.5. ∎

###  4.2 Dual operators

We would now like to associate to an ${\mathcal{S}}$-polynomial operator $L$ a
family of so-called _dual_ operators $(L_{k})_{k\in\mathbb{N}}$ which are
linear operators mapping the coefficients vector of $p$ to the coefficients
vector of $Lp$. To this end we recall Definition 2.3 of Cuchiero and Svaluto-
Ferro (2021).

######  Definition 4.7.

Fix $m\in\mathbb{N}_{0}$ and let $L$ be an ${\mathcal{S}}$-polynomial
operator. An _$m$ -th dual operator_ corresponding to $L$ is a linear operator
$L_{m}:\bigoplus_{k=0}^{m}D^{\otimes
k}\to\bigoplus_{k=0}^{m}\widehat{C}_{\Delta}(E^{k})$ such that
$L_{m}\vec{g}=:(L_{m}^{0}\vec{g},\ldots,L_{m}^{m}\vec{g})$ satisfies

$Lp(\nu)=\sum_{k=0}^{m}\langle L^{k}_{m}\vec{g},\nu^{k}\rangle\qquad\text{for
all }\nu\in{\mathcal{S}},$

where $p(\nu)=\sum_{k=0}^{m}\langle g_{k},\nu^{k}\rangle$. Whenever $L_{m}$ is
a closable operator222We refer to (Ethier and Kurtz, 2009, Chapter 1) for the
precise definition., we still denote its closure by
$L_{m}:\mathcal{D}(L_{m})\to\bigoplus_{k=0}^{m}\widehat{C}_{\Delta}(E^{k})$
and its domain by
$\mathcal{D}(L_{m})\subseteq\bigoplus_{k=0}^{m}\widehat{C}_{\Delta}(E^{k})$.

If $\mathcal{S}=M_{+}(E)$, then the dual operator is unique. Indeed, in this
case the representation (2.1) is unique, which is a consequence of Corollary
2.4 in Cuchiero et al. (2019), and we can thus identify each polynomial with
its coefficients vector. We therefore call it _the_ dual operator. It is
interesting to note that if $L$ satisfies (4.3), then

$\displaystyle L_{m}^{m}(g^{\otimes m})$ $\displaystyle=mB_{1}(g)\otimes
g^{\otimes(m-1)}+\frac{m(m-1)}{2}Q_{2}(g\otimes g)\otimes g^{\otimes(m-2)},$
$\displaystyle L_{m}^{m-1}(g^{\otimes m})$
$\displaystyle=mB_{0}(g)g^{\otimes(m-1)}+\frac{m(m-1)}{2}Q_{1}(g\otimes
g)\otimes g^{\otimes(m-2)},$

$L_{m}^{m-2}(g^{\otimes m})=\frac{m(m-1)}{2}Q_{0}(g\otimes
g)g^{\otimes(m-2)}$, and $L_{m}^{k}(g^{\otimes m})=0$ for each $k<m-2$.

##  5 Moment formula and existence of polynomial diffusions on $M_{+}(E)$

Let $L$ be a linear operator acting on $\mathcal{D}$ for ${\mathcal{D}}$
specified in (2.2). In this section we study existence and uniqueness of non-
negative measure valued polynomial diffusions which we introduce via the
martingale problem. In the following let $\mathcal{S}$ stand either for
$M_{+}^{\mathfrak{\Delta}}(E^{\Delta})$, $M_{+}(E^{\Delta})$ or $M_{+}(E)$.

An $\mathcal{S}$-valued process $X$ with càdlàg paths defined on some filtered
probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\geq
0},{\mathbb{P}})$ is called an $\mathcal{S}$-valued solution to the martingale
problem for $L$ with initial condition $\nu\in\mathcal{S}$ if $X_{0}=\nu$
${\mathbb{P}}$-a.s. and

$N^{f}_{t}=f(X_{t})-f(X_{0})-\int_{0}^{t}Lf(X_{s})ds$ (5.1)

defines a local martingale for every $f$ in the domain of $L$. Uniqueness of
solutions to the martingale problem is always understood in the sense of law.
The martingale problem for $L$ is well–posed if for every initial condition
$\nu\in\mathcal{S}$ there exists a unique $\mathcal{S}$-valued solution to the
martingale problem for $L$ with initial condition $\nu$.

We are mainly interested in $M_{+}(E)$-valued solutions with continuous paths
(with respect to the topology of weak convergence) corresponding to polynomial
operators.

######  Definition 5.1.

Let $L$ be $M_{+}(E)$-polynomial. Any continuous $M_{+}(E)$-valued solution to
the martingale problem for $L$ is called a _measure-valued polynomial
diffusion_.

The following lemma relates path continuity of $M_{+}(E)$-and
$M_{+}(E^{\Delta})$-valued solutions to the martingale problem to the carré-
du-champ operator being a derivation. This explains why we consider
derivations in Theorem 4.6.

######  Lemma 5.2.

If the carré-du-champ operator $\Gamma$ associated to $L$ is an
$M_{+}(E)$-derivation, then any $M_{+}(E)$\- or $M_{+}(E^{\Delta})$-valued
solution to the martingale problem for $L$ has continuous paths and the
corresponding quadratic covariation structure is given by

$d[N^{f},N^{g}]_{t}=2\Gamma(f,g)(X_{t})dt,$

for every $f,g\in{\mathcal{D}}$.

Conversely, if for every initial condition $\nu\in M_{+}(E)$ (or
$M_{+}(E^{\Delta})$ respectively) there is an $M_{+}(E)$\- (or
$M_{+}(E^{\Delta})$ respectively) valued solution to the martingale problem
for $L$ with continuous paths, then the carré-du-champ operator $\Gamma$
associated to $L$ is an $M_{+}(E)$-derivation.

###### Proof.

We here state the proof only for $M_{+}(E)$ since the $M_{+}(E^{\Delta})$-case
follows by the same arguments extending $f\in\mathcal{D}$ to $M(E^{\Delta})$.

Let $X$ be an $M_{+}(E)$-valued solution to the martingale problem for $L$. By
Proposition 2 in Bakry and Émery (1985), the real-valued process $f(X)$ is
continuous for every $f\in\mathcal{D}$, in particular for every linear
monomial $f(\nu)=\langle g,\nu\rangle$ with $g\in D$. Since $D$ is dense in
$C_{\Delta}(E)$, we can conclude that $X$ is continuous with respect to the
topology of weak convergence on $M_{+}(E)$.

Conversely, if $X$ is a $M_{+}(E)$-valued solution to the martingale problem
for $L$ with continuous paths, then, by Lemma 2.3 and Lemma 2.5, the map
$t\mapsto f(X_{t})$ is continuous for all $f\in\mathcal{D}$. The result now
follows by Proposition 1 in Bakry and Émery (1985). ∎

###  5.1 Moment formula

Polynomial diffusions are of particular interest in many applications because
they satisfy a moment formula, which allows to compute the process’ moments in
a tractable way. If $E$ is a finite set, the moment formula always holds, but
technical conditions, in particular on the dual operators, are needed in the
general case. We apply here Theorem 3.4 and Remark 3.20 of Cuchiero and
Svaluto-Ferro (2021) (see also Example 3.21 and Lemma 3.2 in Cuchiero and
Svaluto-Ferro (2021)) to get Theorem 5.5.

Let $L$ be an $M_{+}(E)$-polynomial operator, fix $m\in\mathbb{N}_{0}$, and
let $L_{m}$ be the closable $m$-th dual operator corresponding to $L$ with
domain $\mathcal{D}(L_{m})$. Before stating the theorem, we extend the domain
of $L$ to polynomials with coefficients in $\mathcal{D}(L_{m})$ by setting

$p_{\vec{g}}=\sum_{k=0}^{m}\langle g_{k},\nu^{k}\rangle\quad\text{and}\quad
Lp_{\vec{g}}=\sum_{k=0}^{m}\langle L^{k}_{m}\vec{g},\nu^{k}\rangle$ (5.2)

for all $\vec{g}\in\mathcal{D}(L_{m})$ and $m\in\mathbb{N}_{0}$. As in the
finite dimensional case the moment formula corresponds to a solution of a
system of linear ODEs. Note that we shall usually call these differential
equations ODEs, even though they correspond in many cases to PDEs or PIDEs
depending on the underlying space and the specification of the operators. In
the current infinite dimensional setting we also need to make the solution
concept precise. The following is an adaptation of Definition 3.3 in Cuchiero
and Svaluto-Ferro (2021) to the current measure-valued setting.

######  Definition 5.3.

We call a function $t\mapsto\vec{g}_{t}$ with values in $\mathcal{D}(L_{m})$ a
solution of the $m+1$ dimensional system of ODEs

$\partial_{t}\vec{g}_{t}=L_{m}\vec{g}_{t},\quad\vec{g}_{0}=g,$

if for every $t>0$ it holds

$\displaystyle\sum_{k=0}^{m}\langle
g_{t,k},\nu^{k}\rangle=\sum_{k=0}^{m}\langle
g_{0,k},\nu^{k}\rangle+\int_{0}^{t}\sum_{k=0}^{m}\langle
L^{k}_{m}\vec{g}_{s},\nu^{k}\rangle ds$ (5.3)

for all $\nu\in M_{+}(E^{\Delta})$.

######  Remark 5.4.

Note that the above solution concept reduces to a more classical one if we
take $\nu=\delta_{x_{1}}+\cdots+\delta_{x_{k}}$ with $x_{i}\in E$,
$i=1,\ldots,k$ and $k=1,\ldots,m$. Indeed, by polarization (5.3) can be
transformed into

$\displaystyle g_{t,0}$
$\displaystyle=g_{0,0}+\int_{0}^{t}L_{m}^{0}\vec{g}_{s}ds$ $\displaystyle
g_{t,1}(x_{1})$
$\displaystyle=g_{0,1}(x_{1})+\int_{0}^{t}L_{m}^{1}\vec{g}_{s}(x_{1})ds$
$\displaystyle\vdots$ $\displaystyle g_{t,m}(x_{1},\ldots,x_{m})$
$\displaystyle=g_{0,m}(x_{1},\ldots,x_{m})+\int_{0}^{t}L^{m}_{m}\vec{g}_{s}(x_{1},\ldots,x_{m})ds$

and thus reduces to a classical (except of the integral form) solution of a
multivariate P(I)DE.

######  Theorem 5.5 (Dual moment formula).

Let $L$ be an $M_{+}(E)$-polynomial operator, fix $m\in\mathbb{N}_{0}$, let
$L_{m}$ be the $m$-th dual operator corresponding to $L$, and assume that
$L_{m}$ is closable with domain $\mathcal{D}(L_{m})$. Suppose that an
$M_{+}(E^{\Delta})$-valued solution $(X_{t})_{t\geq 0}$ to the martingale
problem for $L$ exists. Fix a coefficients vector
$\vec{g}=(g_{0},\ldots,g_{m})\in\mathcal{D}(L_{m})$ and suppose that the
following condition holds true.

  * •

There is a solution in the sense of Definition 5.3 of the $m+1$ dimensional
system of linear ODEs on $[0,T]$ given by

$\partial_{t}\vec{g}_{t}=L_{m}\vec{g}_{t},\qquad\vec{g}_{0}=\vec{g}.$ (5.4)

Then, for all $0\leq t\leq T$ the representation

$\mathbb{E}\left[\sum_{k=0}^{m}\langle
g_{k},X_{T}^{k}\rangle\,\big{|}\,\mathcal{F}_{t}\right]=\sum_{k=0}^{m}\langle
g_{T-t,k},X_{t}^{k}\rangle$

holds almost surely.

###### Proof.

This is simply a consequence of Theorem 3.4 in Cuchiero and Svaluto-Ferro
(2021) and Lemma 5.7 below. ∎

######  Example 5.6.

Let $L$ be an $M_{+}(E)$-polynomial operator, fix $m\in\mathbb{N}_{0}$, and
let $L_{m}$ be the closable $m$-th dual operator corresponding to $L$. Assume
that $L_{m}^{0}=0,\ldots,L_{m}^{m-1}=0$ and that $L_{m}^{m}|_{D^{\otimes m}}$
satisfies the positive minimum principle. If the closure
$\overline{L}_{m}^{m}$ of $L_{m}^{m}$ generates a strongly continuous positive
semigroup $(Y^{k}_{t})_{t\geq 0}$, then $\vec{g}_{t}:=(Y_{t}^{k}g)\vec{e}_{k}$
satisfies (5.4) for every $g$ in the domain of
$\overline{L}_{m}^{m}|_{D^{\otimes m}}$. We provide now a simple example to
illustrate this mechanism.

Consider the process given by $X_{t}=S_{t}\mu$ where $dS_{t}=\sigma
S_{t}dW_{t}$, $S_{0}=1$, and $\mu\in M_{+}(E)$. Since $\langle g,X\rangle$ is
a local martingale and

$\langle g,X_{t}\rangle^{2}=(S_{t}\langle
g,\mu\rangle)^{2}=\frac{1}{2}\int_{0}^{t}\langle\sigma^{2}2g\otimes
g,X_{s}^{2}\rangle ds+\text{(local martingale)},$

the generator of $X$ is given by
$Lp(\nu)=\frac{1}{2}\langle\sigma^{2}\partial^{2}p(\nu),\nu^{2}\rangle.$
Observe that

$L(\langle g,\,\cdot\,\rangle^{n})(\nu)=\langle L_{n}^{n}g^{\otimes
n},\nu^{n}\rangle,$

where $L_{n}^{n}g^{\otimes n}=\sigma^{2}\frac{n(n-1)}{2}g^{\otimes n}$ and
note that $L_{n}^{n}$ is the generator of the semigroup given by

$Y_{t}g^{\otimes n}(z):=\mathbb{E}[g^{\otimes
n}(Z^{(n)}_{t})\exp(\int_{0}^{t}m(Z^{(n)}_{s})ds)|Z^{(n)}_{0}=z],$

where $Z^{(n)}$ is the constant process (hence the process generated by 0) and
$m(z)=\sigma^{2}\frac{n(n-1)}{2}$. This in particular yields

$\langle Y_{t}g^{\otimes n},X_{0}^{n}\rangle=\langle
g,\mu\rangle^{n}\exp({\sigma^{2}t}\frac{n(n-1)}{2})=\mathbb{E}[\langle
g,\mu\rangle^{n}S_{t}^{n}]=\mathbb{E}[\langle g,X_{t}\rangle^{n}].$

######  Lemma 5.7.

Let $L$ be an $M_{+}(E)$-polynomial operator and suppose that an
$M_{+}(E^{\Delta})$-valued solution $(X_{t})_{t\geq 0}$ to the martingale
problem for $L$ exists.

  1. (i)

Then, for every $p\in P^{D}$,

$\mathbb{E}[p(X_{t})^{2}]<Ce^{Ct},$

for some $C>0$. Moreover, $N^{p}$ as defined in (5.1) is a true martingale.

  2. (ii)

Moreover, if there is a solution to (5.4) in the sense of Definition 5.3, then
Condition (ii) and (iii) of Theorem 3.4 in Cuchiero and Svaluto-Ferro (2021)
also hold true.

###### Proof.

Let $\|\,\cdot\,\|_{M(E^{\Delta})}$ denote the total variation norm on
${M}(E^{\Delta})$. Then for all $\nu\in M_{+}(E^{\Delta})$, it holds
$\|\nu\|_{M(E^{\Delta})}=\nu(E^{\Delta})$ and hence

$1+\|\nu\|_{M(E^{\Delta})}^{2m}=1+\nu(E^{\Delta})^{2m},\quad\nu\in
M_{+}(E^{\Delta}).$

Thus, for any $p=\sum_{k=0}^{m}\langle g_{k},\nu^{k}\rangle\in P$, we can
estimate

$p^{2}\leq C(1+\nu(E^{\Delta})^{2m})$

and

$|L(1+\nu(E^{\Delta})^{2m})|\leq C(1+\nu(E^{\Delta})^{2m})$

for some constant $C>0$. According to Definition 3.18 in Cuchiero and Svaluto-
Ferro (2021), this means that every $p$ of degree $m$ is
$(C,1+\nu(E^{\Delta})^{2m})$ bounded. Lemma 3.19 in Cuchiero and Svaluto-Ferro
(2021) thus yields that for every $p\in P$,

$\mathbb{E}[p(X_{t})^{2}]<Ce^{Ct}$

and the local martingale $N^{p}$ is actually a true (even square integrable)
martingale.

Concerning part (ii), Lemma 3.19 in Cuchiero and Svaluto-Ferro (2021) also
yields Condition (ii) and (iii) of Theorem 3.4 as long as we have solution to
(5.4). ∎

###  5.2 Existence

Our first main result of this section gives sufficient conditions for the
existence of solutions to the martingale problem. Applications of this result
are discussed in Section 7. Recall that $E$ is throughout a locally compact
Polish space. We start by a definition which will be used to describe the form
of $Q_{2}$ in the representation of $L$ given by (4.3).

######  Definition 5.8.

We say that a linear operator $C$ admits a $(\beta,\pi)$-representation if

$\displaystyle
C(g)(x,y)=\frac{1}{2}(\pi(x,y)g(x,x)+\pi(y,x)g(y,y)+2\beta(x,y)g(x,y)),\quad
g\in D\otimes D$ (5.5)

where

  * •

$\beta:(E^{\Delta})^{2}\to\mathbb{R}$ is a symmetric function such that
$\beta(x,x)\geq 0$ for all $x\in E^{\Delta}$;

  * •

$\pi:(E^{\Delta})^{2}\to\mathbb{R}_{+}$ is a non-negative function such that
$\pi(x,x)=0$ for all $x\in E^{\Delta}$;

  * •

for all $n\in\mathbb{N}$, $x_{1},\ldots,x_{n}\in E^{\Delta}$,
$c_{1},\ldots,c_{n}\in\mathbb{R}_{++}$, the $n\times n$ matrix

$\displaystyle
A^{(n)}:=\beta_{n}+\begin{pmatrix}\sum_{j=1}^{n}\frac{c_{j}}{c_{1}}\pi(x_{1},x_{j})&&\\\
&\ddots&\\\
&&\sum_{j=1}^{n}\frac{c_{j}}{c_{n}}\pi(x_{n},x_{j})\end{pmatrix}\in\mathbb{S}^{n}_{+},$
(5.6)

where $\beta_{n}\in\mathbb{S}^{n}$ with entries
$\beta_{n,ij}=\beta(x_{i},x_{j})$;

  * •

the map
$(x,y)\mapsto\frac{1}{2}(\pi(x,y)g(x,x)+\pi(y,x)g(y,y)+2\beta(x,y)g(x,y))$
lies in $\widehat{C}_{\Delta}(E^{2})$ for all $g\in D\otimes D$.

Accordingly, our sufficient conditions for the existence of the martingale
problem now read as follows.

######  Theorem 5.9.

Suppose that for $i=1,\ldots n$, each $A_{i}$ is the generator of a strongly
continuous positive group on $C_{\Delta}(E)$ such that its domain contains
both $D$ and $A_{i}(D)$. Let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear
operator of form (4.3), where

  1. (i)

$B_{0}:D\to\mathbb{R}$ is given by $B_{0}(g)=\langle g,b\rangle$ with $b\in
M_{+}(E^{\Delta})$;

  2. (ii)

$B_{1}-\frac{1}{2}\sum_{i=1}^{n}A_{i}^{2}:D\to C_{\Delta}(E)$ satisfies the
positive minimum principle on $E^{\Delta}$;

  3. (iii)

$Q_{0}\equiv 0$;

  4. (iv)

$Q_{1}$ is of the form

$Q_{1}(g)=\alpha\operatorname{diag}(g),\quad\text{ that is,}\quad
Q_{1}(g)(x)=\alpha(x)g(x,x),\quad g\in D\otimes D,$

where $\alpha\in C_{\Delta}(E)$ with values in $\mathbb{R}_{+}$;

  5. (v)

$Q_{2}$ is of the form

$Q_{2}(g)=C(g)+\sum_{i=1}^{n}(A_{i}\otimes A_{i})(g),\quad g\in D\otimes D,$

where $C$ admits a ($\beta,\pi$)-representation.

Then $L$ is $M_{+}(E)$-polynomial and its martingale problem has an
$M_{+}(E^{\Delta})$-valued solution with continuous paths for every initial
condition $\nu\in M_{+}(E^{\Delta})$.

If additionally, the measure $b$ and the initial condition $\nu$ both lie in
$M_{+}(E)$ and if there exists some function $m\in C_{\Delta}(E)$ such that

$\text{bp-}\lim_{n\to\infty}(B_{1}(1-g_{n})-m(1-g_{n}))\leq 0,$

for some sequence $(g_{n})_{n\in\mathbb{N}}\in D\cap C_{0}(E)$ satisfying
$\text{bp-}\lim_{n\to\infty}g_{n}=1_{E}$, then any solution to the martingale
problem takes values in $M_{+}(E)$.

######  Remark 5.10.

  1. (i)

When the state space are probability measures $M_{1}(E)$, Theorem 5.6 in
Cuchiero et al. (2019) provides sufficient conditions guaranteeing existence
of solutions to the corresponding martingale problem. These conditions are
stronger and imply the current ones. Indeed, we identify $B_{1}$ and $Q_{2}$
here with $B$ and $Q$ there. More specifically, $C(g)$ is identified with
$\alpha\Psi(g)$ and $A_{i}$ with $A_{i}$. All other quantities are set to $0$.
The conditions on $B$ and $A_{i}$ are clearly stronger. In particular, the
positive maximum principle implies the positive minimum principle on
$E^{\Delta}$. Moreover, $\alpha\Psi$ admits a $(\beta,\pi)$-representation for
$\pi(x,y)=\pi(y,x)=-\beta(x,y)=\alpha(x,y)1_{\\{x\neq y\\}}$. Observe indeed
that in this case the matrix $A^{(n)}$ appearing in (5.6) is given by

$A^{(n)}_{ij}:=-\alpha(x_{i},x_{j})1_{\\{x_{i}\neq
x_{j}\\}}+\sum_{k=1}^{n}\frac{c_{k}}{c_{i}}\alpha(x_{i},x_{k})1_{\\{x_{i}\neq
x_{k}\\}}1_{\\{i=j\\}},$

which is positive semidefinite since
$\sum_{ij=1}^{n}A^{(n)}_{ij}v_{i}v_{j}=\sum_{i=1}^{n}\alpha(x_{i},x_{j})\frac{c_{j}}{c_{j}}\big{(}v_{i}-v_{j}\frac{c_{j}}{c_{i}}\big{)}^{2}\geq
0$ for each $v_{i}\in\mathbb{R}$.

  2. (ii)

Note that (4.3) imposes the implicit condition that $C(g)$ has to lie in
$\widehat{C}_{\Delta}(E^{2})$ for every $g\in D\otimes D$. If
$D=C_{\Delta}(E)$, then this necessarily yields boundedness conditions on
$\pi$ and $\beta$, as is seen from Theorem 5.11 below. However, this does not
hold for general $D\subseteq C_{\Delta}(E)$, as one can see by considering
$E=\mathbb{R}$, $D\subseteq C^{1}_{\Delta}(\mathbb{R})$, and
$\pi(x,y)=\pi(y,x)=-\beta(x,y)=|x-y|^{-1}1_{\\{x\neq y\\}}$.

###### Proof of Theorem 5.9.

Theorem 4.6 implies that $L$ is $M_{+}(E)$-polynomial. Lemma B.2(i) yields the
existence of an $M_{+}(E^{\Delta})$-valued continuous solution to the
martingale problem for any initial condition provided that $L$ satisfies the
positive maximum principle on $M_{+}(E^{\Delta})$, which we shall check below.

Let us now verify that the positive maximum principle on $M_{+}(E^{\Delta})$
holds true. Let therefore $\nu^{*}\in M_{+}(E^{\Delta})$ be a maximizer of
$f\in\mathcal{D}$ over $M_{+}(E^{\Delta})$. Then the optimality conditions in
Theorem 3.1 yield

$\begin{split}\partial_{x}f(\nu^{*})\leq 0,\quad\langle\partial
f(\nu^{*}),\nu^{*}\rangle=0,\quad\partial^{2}_{xx}f(\nu^{*})\leq
0\quad\text{and}\quad\langle\partial^{2}f(\nu^{*}),\mu^{2}\rangle\leq
0,\end{split}$ (5.7)

for each $x\in E$ and $\mu\in M(E^{\Delta})$ with
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\nu^{*})$. We now
analyze $Lf(\nu^{*})$, which reads due the conditions (i) \- (v) as follows:

$\displaystyle Lf(\nu^{*})$ $\displaystyle=\langle\partial
f(\nu^{*}),b\rangle+\langle B_{1}(\partial
f(\nu^{*})),\nu^{*}\rangle+\frac{1}{2}\langle\alpha\operatorname{diag}(\partial^{2}f(\nu^{*})),\nu^{*}\rangle$
$\displaystyle\quad+\frac{1}{2}\langle
C(\partial^{2}f(\nu^{*})),(\nu^{*})^{2}\rangle+\frac{1}{2}\langle\sum_{i=1}^{n}(A_{i}\otimes
A_{i})(\partial^{2}f(\nu^{*})),(\nu^{*})^{2}\rangle.$

By (5.7) and the positivity of $b$ and $\alpha$, we have

$\langle\partial f(\nu^{*}),b\rangle\leq 0\qquad\text{ and
}\qquad\langle\alpha\operatorname{diag}(\partial^{2}f(\nu^{*})),\nu^{*}\rangle\leq
0.$ (5.8)

Next, observe that choosing the signed measure in (5.7) to be
$\sum_{i=1}^{n}\lambda_{i}\delta_{x_{i}}$ for some $\lambda_{i}\in\mathbb{R}$
yields
$\sum_{i,j=1}^{n}\lambda_{i}\lambda_{j}\partial^{2}_{x_{i}x_{j}}f(\nu^{*})\leq
0$, which implies that the matrix $(H_{n,ij})_{ij=1}^{n}$ given by
$H_{n,ij}:=\partial^{2}_{x_{i}x_{j}}f(\nu^{*})$ is negative semidefinite.
Taking $\mu\in M_{+}(E^{\Delta})$ such that $\operatorname{supp}(\mu)$
consists of $n$ points $x_{1},\ldots,x_{n}\in\operatorname{supp}(\nu^{*})$, we
thus have $\mu=\sum_{i=1}^{n}c_{i}\delta_{x_{i}}$ for some
$c_{1},\ldots,c_{n}>0$.

Since $C(g)$ admits a ($\beta,\pi$)-representation, using the notation of
Definition 5.8 we can thus write

$\langle
C(\partial^{2}f(\nu^{*})),\mu^{2}\rangle=\sum_{i,j=1}^{n}(A^{(n)}\circ
H_{n})_{ij}c_{i}c_{j},$

where $\circ$ denotes the Hadamard product, i.e. the componentwise
multiplication. As $A^{(n)}$ is by assumption positive semidefinite and as the
Hadmard product between a positive semidefinite and negative semidefinite
matrix is negative definite, we can conclude that

$\displaystyle\langle C(\partial^{2}f(\nu^{*})),\mu^{2}\rangle\leq 0.$ (5.9)

Passing to the weak closure yields (5.9).

Finally, observe that from (5.8) and (5.9) we get that

$Lf(\nu^{*})\leq\langle B_{1}(\partial
f(\nu^{*})),\nu^{*}\rangle+\frac{1}{2}\langle\sum_{i=1}^{n}(A_{i}\otimes
A_{i})(\partial^{2}f(\nu^{*})),(\nu^{*})^{2}\rangle.$

As $B_{1}-\frac{1}{2}\sum_{i=1}^{n}A_{i}^{2}$ satisfies the positive minimum
principle, we have by (5.7)

$\langle B_{1}(\partial
f(\nu^{*}))-\frac{1}{2}\sum_{i=1}^{n}A_{i}^{2}(\partial
f(\nu^{*})),\nu^{*}\rangle\leq 0.$

Therefore,

$Lf(\nu^{*})\leq\frac{1}{2}\langle\sum_{i=1}^{n}A_{i}^{2}(\partial
f(\nu^{*})),\nu^{*}\rangle+\frac{1}{2}\langle\sum_{i=1}^{n}(A_{i}\otimes
A_{i})(\partial^{2}f(\nu^{*})),(\nu^{*})^{2}\rangle\leq 0,$

where the last inequality follows from Theorem 3.4. This proves the positive
maximum principle on $M_{+}(E^{\Delta})$ and thus the existence of an
$M_{+}(E^{\Delta})$-valued solution. The remaining statement on
$M_{+}(E)$-valued solutions to the martingale problem follows by Lemma
B.2(ii). ∎

Even though Theorem 5.9 only gives sufficient conditions for the existence of
solutions to the martingale problem for general coefficient domains $D$, the
next result shows that the conditions are sharp. Indeed, we consider the case
$D=C_{\Delta}(E)$ and get the following characterization.

######  Theorem 5.11.

Let $D=C_{\Delta}(E)$ and let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear
operator. Then $L$ is $M_{+}(E)$-polynomial, there exists an
$M_{+}(E^{\Delta})$-valued solution to the martingale problem for all initial
conditions in $M_{+}(E^{\Delta})$ and all solutions have continuous paths, if
and only if $L$ satisfies (4.3) with

  1. (i)

$B_{0}:C_{\Delta}(E)\to\mathbb{R}$ is given by $B_{0}(g)=\langle g,b\rangle$
with a Radon measure $b\in M_{+}(E^{\Delta})$;

  2. (ii)

For every $g\in C_{\Delta}(E)$

$B_{1}g(x)=\int(g(\xi)-g(x))\nu_{B}(x,d\xi)+m(x)g(x)$ (5.10)

for some non-negative finite kernel $\nu_{B}$ from $E^{\Delta}$ to
$E^{\Delta}$ and some $m\in C_{\Delta}(E)$;

  3. (iii)

$Q_{0}\equiv 0$;

  4. (iv)

$Q_{1}$ is of the form

$Q_{1}(g)(x)=\alpha(x)g(x,x),\quad g\in C_{\Delta}(E)\otimes C_{\Delta}(E)$
(5.11)

where $\alpha\in C_{\Delta}(E)$ with values in $\mathbb{R}_{+}$;

  5. (v)

$Q_{2}$ admits a ($\beta,\pi$)-representation. Moreover, the parameters $\pi$
and $\beta$ are bounded and continuous on $(E^{\Delta})^{2}\setminus\\{x=y\\}$
and $\pi+\overline{\pi}+2\beta\in\widehat{C}_{\Delta}(E^{2})$, where
$\overline{\pi}(x,y)=\pi(y,x)$.

For the state space $M_{+}(E)$, the following equivalence holds.

A linear operator $L:\mathcal{D}\to C(M_{+}(E))$ is $M_{+}(E)$-polynomial,
there exists an $M_{+}(E)$-valued solution to the martingale problem for all
initial conditions in $M_{+}(E)$ and all solutions have continuous paths, if
and only if $L$ satisfies (i)-(v), $b\in M_{+}(E)$, and $\nu_{B}$ is a kernel
from $E$ to $E$.

######  Remark 5.12.

  1. (i)

Condition (ii) for the bounded operator $B_{1}$ is equivalent to the
requirement that $B_{1}g+\|B_{1}\|g\geq 0$ for all $g\geq 0$. This in turn is
also equivalent to $(\exp(tB_{1}))_{t\geq 0}$ being a positive semigroup on
$C_{\Delta}(E)$ and to $B_{1}$ satisfying the positive minimum principle; see
Theorem B.II.1.3 in Arendt et al. (1986). When $E$ is finite this exactly
means that the matrix $B_{1}$ has non-negative off-diagonal elements. The
decomposition as of Condition (ii) then means that such a matrix is decomposed
into a transition rate matrix where the diagonal elements are defined such
that the rows sum up to $0$ and a diagonal matrix (corresponding to $m$) where
this procedure is compensated.

  2. (ii)

Note that in the case of $D=C_{\Delta}(E)$, adding to $Q_{2}$ operators of the
form

$(A\otimes A)g,$

where $A$ is a generator of a strongly continuous positive group on
$C_{\Delta}(E)$ does not yield more generality, since $A$ is necessarily of
the form $Ag(x)=a(x)g(x)$ for some $a\in C_{\Delta}(E)$ as proved in Lemma
3.6. Therefore, $(A\otimes A)(g)(x,y)=a(x)a(y)g(x,y)$. This however can be
absorbed in the function $\beta$.

###### Proof of Theorem 5.11.

Assume that $L$ satisfies (4.3) with conditions (i) to (v). By Lemma A.3
condition (ii) implies that $B_{1}$ satisfying the positive minimum principle
on $E^{\Delta}$. Hence, all conditions of Theorem 5.9 are satisfied and we get
that $L$ is $M_{+}(E)$-polynomial and that its martingale problem has an
$M_{+}(E^{\Delta})$ valued solution with continuous paths for every initial
condition $\nu\in M_{+}(E^{\Delta})$. By Theorem 5.9 we also get that any
solution with initial value $\nu\in M_{+}(E)$ takes values in $M_{+}(E)$
provided that $b\in M_{+}(E)$, $\nu_{B}$ is a kernel from $E$ to $E$, and
$m\in C_{0}(E)$. Indeed, it remains to verify that
$\text{bp-}\lim_{n\to\infty}B_{1}g_{n}\geq B_{1}1$ where $g_{n}\in C_{0}(E)$
is a sequence such that $\text{bp-}\lim_{n\to\infty}g_{n}=1_{E}$. The claim
then follows by the dominated convergence theorem.

We now prove the opposite implication. Assume that $L$ is
$M_{+}(E)$-polynomial, then there exists an $M_{+}(E^{\Delta})$-valued
solution to its martingale problem for each initial condition in
$M_{+}(E^{\Delta})$ and all solutions have continuous paths. Theorem 4.6 and
Lemma 5.2 imply that $L$ satisfies (4.3), and due to Lemma B.1 also the
positive maximum principle on $M_{+}(E^{\Delta})$. An application of Lemma A.2
then yields the result.

Next, assume that $L$ satisfies (i)-(v), $b\in M_{+}(E)$, and $\nu_{B}$ is a
kernel from $E$ to $E$. By the first part of the theorem it holds that $L$ is
$M_{+}(E)$-polynomial, there exists an $M_{+}(E^{\Delta})$-valued solution to
the martingale problem for all initial conditions in $M_{+}(E)$, and all
solutions have continuous paths. By considering any sequence of functions
$g_{n}\in C_{0}(E)$ with $0\leq g_{n}(x)\uparrow 1$ for all $x\in E$ the claim
follows by Lemma B.2(ii).

Finally, assume that $L$ is $M_{+}(E)$-polynomial, there exists an
$M_{+}(E)$-valued solution to the martingale problem for all initial
conditions in $M_{+}(E)$ and all solutions have continuous paths. By the same
argument as above $L$ satisfies (4.3) and due to Lemma B.1 the positive
maximum principle on $M_{+}(E)$. Conditions (i)-(v) are then satisfied by
Lemma A.2.

It thus remains to prove that $b\in M_{+}(E)$, $\nu_{B}$ is a kernel from $E$
to $E$. As a first step, observe that

$\mathbb{E}[\langle 1,X_{t}\rangle]\leq\langle
1,X_{0}\rangle+\int_{0}^{t}\langle 1,b\rangle+\|m\|\mathbb{E}[\langle
1,X_{s}\rangle]ds$

and Lemma 5.7(i) we can apply the Gronwall inequality to get that by

$\mathbb{E}[\langle
1,X_{t}\rangle]\leq(X_{0}(E^{\Delta})+b(E^{\Delta})t)\exp(\|m\|t)=:m_{1}(X_{0},t).$

Next, since $\widetilde{B}_{1}g:=B_{1}g-\|m\|g$ is a bounded dissipative
operator on $C_{\Delta}(E)$ we know that the range of
$(\ell-\widetilde{B}_{1})$ is given by $C_{\Delta}(E)$ for each $\ell>0$. This
implies that the range of $(K-B_{1})$ is given by $C_{\Delta}(E)$ for each
$K>\|m\|$. Fix now $K>\|m\|$, $(h_{n})_{n\in\mathbb{N}}\subseteq
C_{\Delta}(E)$ converging to $1_{\Delta}$ in the bounded pointwise sense, and
choose $(g_{n})_{n\in\mathbb{N}}\subseteq C_{\Delta}(E)$ such that
$(K-B_{1})g_{n}=h_{n}$. We then have that

$\displaystyle\langle g_{n},X_{0}\rangle e^{0}$
$\displaystyle=-\mathbb{E}[\langle
g_{n},X_{t}\rangle]e^{-Kt}-\int_{0}^{t}\mathbb{E}[-K\langle g_{n},X_{s}\rangle
e^{-Ks}+(\langle g_{n},b\rangle+\langle B_{1}g_{n},X_{s}\rangle)e^{-Ks}]ds$
$\displaystyle=-\mathbb{E}[\langle
g_{n},X_{t}\rangle]e^{-Kt}+\frac{1}{K}\langle
g_{n},b\rangle(e^{-Kt}-1)-\int_{0}^{t}\mathbb{E}[-\langle h_{n},X_{s}\rangle
e^{-Ks}]ds.$

Since $|\mathbb{E}[\langle g_{n},X_{t}\rangle]|\leq\|g_{n}\|\mathbb{E}[\langle
1,X_{t}\rangle]\leq\|g_{n}\|m_{1}(X_{0},t)$ sending $t$ to infinity we get

$\langle g_{n},X_{0}\rangle=-\frac{1}{K}\langle
g_{n},b\rangle+\int_{0}^{\infty}\mathbb{E}[\langle h_{n},X_{s}\rangle
e^{-Ks}]ds,$

and hence

$\lim_{n\to\infty}\langle
g_{n},X_{0}\rangle=-\frac{1}{K}b(\Delta)+\int_{0}^{\infty}\mathbb{E}[X_{s}(\Delta)]e^{-Ks}ds=-\frac{1}{K}b(\Delta),$

showing that $\lim_{n\to\infty}g_{n}(x)=b(\Delta)=0$ for each $x\in E$. Using
that

$|\langle
g_{n},X_{0}\rangle|\leq\sup_{n\in\mathbb{N}}\|h_{n}\|\int_{0}^{\infty}\mathbb{E}[m_{1}(X_{0},s)e^{-Ks}]ds<\infty$

we can conclude that $(g_{n})_{n\in\mathbb{N}}$ is a bounded sequence.

Next, observe that
$\lim_{n\to\infty}(B_{1}g_{n}(x)-Kg_{n}(x))=\lim_{n\to\infty}-h_{n}(x)=-1_{\\{x=\Delta\\}}\leq
0$ for each $x\in E^{\Delta}$. Inserting the from of $B_{1}$ yields

$(\lim_{n\to\infty}g_{n}(x))\Big{(}\nu_{B}(x,\Delta)-1_{\\{x=\Delta\\}}(\nu_{B}(\Delta,E^{\Delta})+K-m(\Delta))\Big{)}=-1_{\\{x=\Delta\\}}.$

Since inserting $x=\Delta$ we get that $(\lim_{n\to\infty}g_{n}(x))\neq 0$, we
can conclude that $\nu_{B}(x,\Delta)=0$ for each $x\in E$ proving the claim. ∎

######  Example 5.13.

In accordance with Example 5.6, suppose that $L$ satisfies the conditions of
Theorem 5.11 for $B_{0}=0$ and $Q_{1}=0$. Then the dual operator $L_{m}$
satisfies $L_{m}^{0}=0,\ldots,L_{m}^{m-1}=0$ and

$L_{m}^{m}(g^{\otimes m})=mB_{1}(g)\otimes
g^{\otimes(m-1)}+\frac{m(m-1)}{2}Q_{2}(g\otimes g)\otimes g^{\otimes(m-2)}.$

Observe that $L_{m}^{m}$ satisfies positive minimum principle. Since $B_{1}$
and $Q_{2}$ are bounded the operator $L_{m}^{m}$ generates a strongly
continuous positive semigroup and the moment formula can thus be applied.

A quick look at the generator $L$ of the process $(S_{t}\mu)_{t\geq 0}$
studied in Example 5.6 shows that $L$ satisfies the conditions of Theorem 5.11
for $B_{0}=0$,$B_{1}=0$, $Q_{1}=0$, and $Q_{2}(g)=\sigma^{2}g$ and is thus of
the given form for $C=0$, $n=1$, and $A_{1}g=\sigma g$.

A similar reasoning can be applied to compute the generator $L$ of
$(Y_{t}\mu)_{t\geq 0}$ for $dY_{t}=\sigma\sqrt{Y}_{t}dW_{t}$. The obtained
operator $L$ does not satisfy the conditions of Example 5.6 but satisfies the
conditions of Theorem 5.11 for the parameters $B_{0}=0$, $B_{1}=0$,
$Q_{1}(g)(x)=\sigma^{2}g(x,x)$, and $Q_{2}=0$.

###  5.3 Uniqueness in law

Having established existence to the martingale problem, we are now concerned
with uniqueness. As in the finite dimensional case, uniqueness in law follows
if the (marginal) moments determine the finite-dimensional marginal
distributions. This property is known as _determinacy_ of the moment problem.
Whenever this holds true, we can then conclude uniqueness in law by relying on
the moment formula. The results of this section generalize Theorem 4.2 in
Filipović and Larsson (2016) and provide conditions ensuring the existence of
exponential moments. These conditions are satisfied by measure-valued affine
diffusions treated in Section 6 below.

######  Lemma 5.14.

Let $L$ be an $M_{+}(E)$-polynomial operator and $(X_{t})_{t\geq 0}$ an
$M_{+}(E^{\Delta})$-valued solution to the corresponding martingale problem
with initial value $X_{0}=\nu$. Suppose that the conditions of Theorem 5.5 are
satisfied for each $g\in\bigoplus_{k=0}^{m}D^{\otimes k}$ and each
$m\in\mathbb{N}$. If for each $t\geq 0$ there exists $\varepsilon>0$ with

$\mathbb{E}[e^{\varepsilon\langle 1,X_{t}\rangle}]<\infty,$ (5.12)

then the law of $X$ is uniquely determined by $L$ and $\nu$.

###### Proof.

Observe that condition (5.12) yields $\mathbb{E}[e^{\varepsilon|\langle
g,X_{t}\rangle|}]<\infty$ for each $g\in D$ with $|g|\leq 1$. Following proof
of Lemma 4.1 in Filipović and Larsson (2016) we thus get that the law of

$(\langle g_{1},X_{t}\rangle,\ldots,\langle g_{n},X_{t}\rangle)_{t\geq 0}$

is uniquely determined by $L$ and $\nu$ for each $g_{1},\ldots,g_{n}\in D$.
Using that $D$ is a dense subspace of $C_{\Delta}(E)$ the claim follows. ∎

If $L$ admits a representation as described in Theorem 5.9 explicit sufficient
conditions for (5.12) can be provided. Indeed, the following theorem assumes a
linear growth condition, meaning that $Q_{2}\equiv 0$, to ensure existence of
exponential moments.

######  Theorem 5.15.

Let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear operator satisfying the
assumptions of Theorem 5.9 and $X$ be an $M_{+}(E^{\Delta})$-valued solution
of the corresponding martingale problem. Assume that $Q_{2}\equiv 0$ and that
$B_{1}:D\to C_{\Delta}(E)$ is the generator of a strongly continuous semigroup
$(P_{t})_{t\geq 0}$. Then for each $t\geq 0$ there exists $\varepsilon>0$ such
that condition (5.12) holds true and thus the martingale problem for $X$ is
well-posed.

The proof of Theorem 5.15 relies on the following generalization of Theorem
1.3 in Hajek (1985).

######  Lemma 5.16.

Let $(Y_{t})_{t\geq 0}$ be a stochastic process satisfying
$dY_{t}=b^{Y}_{t}dt+dM_{t}$ for some continuous local martingale $M$ whose
quadratic variation is given by $d[M]_{t}=(\sigma^{Y}_{t})^{2}dt$ and some
continuous stochastic processes $b^{Y}$ and $\sigma^{Y}$. Suppose that
$\sup_{t\geq 0}|b^{Y}_{t}|\leq m$ and $\sup_{t\geq 0}|\sigma^{Y}_{t}|\leq\rho$
almost surely for some constants $m,\rho$. Then for any non-decreasing convex
function $\Phi$ on $\mathbb{R}$ it holds

$\mathbb{E}[\Phi(Y_{t})]\leq\mathbb{E}[\Phi(V)],$

where $V$ is a Gaussian random variable with mean $Y_{0}+mt$ and variance
$\rho^{2}t$.

###### Proof.

Without loss of generality we can assume the existence of an auxiliary
Brownian motion $W$ independent of $Y$. Set then

$Z^{\pm}_{t}:=Y_{0}+mt+M_{t}\pm\int_{0}^{t}(\rho^{2}-(\sigma^{Y}_{t})^{2})^{1/2}dW_{t}$

and observe that
$\frac{1}{2}(Z^{+}_{t}+Z^{-}_{t})-Y_{t}=mt-\int_{0}^{t}b^{Y}_{s}ds\geq 0.$
Moreover, since

$[Z^{\pm}]_{t}=[M]_{t}+\int_{0}^{t}(\rho^{2}-(\sigma^{Y}_{s})^{2})ds=\rho^{2}t,$

by Lévy characterization theorem we get that
$(\frac{1}{\rho}(Z^{\pm}_{t}-Y_{0}-mt))_{t\geq 0}$ is a standard Brownian
motion. This in particular implies that $Z_{t}^{+}$ and $Z^{-}_{t}$ are both
Gaussian random variables with mean $Y_{0}+mt$ and variance $\rho^{2}t$. We
can thus use the properties of $\Phi$ to conclude

$\mathbb{E}[\Phi(Y_{t})]\leq\mathbb{E}[\Phi(\frac{1}{2}(Z^{+}_{t}+Z^{-}_{t}))]\leq\frac{1}{2}\mathbb{E}[\Phi(Z^{+}_{t})+\Phi(Z^{-}_{t})]=\mathbb{E}[\Phi(V)].$

∎

We are now ready to prove Theorem 5.15. The proof is an adaptation of the
proof of Lemma C.1 in Filipović and Larsson (2016) to the current setting.

###### Proof of Theorem 5.15.

Fix $T\geq 0$ and note that setting $f_{s}:=\langle
P_{T-s}1,\,\cdot\,\rangle$, by (5.1) we have that

$N^{f_{s}}_{t}={f_{s}}(X_{t})-{f_{s}}(\nu)-\int_{0}^{t}L{f_{s}}(X_{r})dr$

is a local martingale for each $s\in[0,T]$. By Lemma 5.2 we know that

$[N^{f_{s_{1}}},N^{f_{s_{2}}}]_{t}=2\int_{0}^{t}\Gamma(f_{s_{1}},f_{s_{2}})(X_{r})dr=\int_{0}^{t}\langle\alpha(f_{s_{1}})\alpha(f_{s_{2}}),X_{t}\rangle
dt.$

Set then $Y_{t}:=\langle P_{T-t}1,X_{t}\rangle-A_{t}$ for $A_{t}:=\langle
P_{T}1,X_{0}\rangle+\int_{0}^{t}\langle P_{T-s}1,b\rangle ds.$ Since

$d\langle P_{T-t}1,X_{t}\rangle=-\langle B_{1}P_{T-t}1,X_{t}\rangle
dt+Lf_{t}(X_{t})dt+(\text{local martingale})$

we know that $Y$ is a local martingale whose quadratic variation is given by

$d[Y]_{t}=\langle\alpha(P_{T-t}1)^{2},X_{t}\rangle dt=:\sigma_{t}^{2}dt.$

Using that $\sigma_{t}^{2}\leq(\sup_{t\in[0,T]}\|\alpha\|\|P_{T-t}1\|)\langle
P_{T-t}1,X_{t}\rangle$ and the fact that $A$ is deterministic and thus bounded
on $[0,T]$ we get

$\sigma_{t}^{2}\leq C(1+|Y_{t}|).$ (5.13)

Following now the proof of Lemma C.1 in Filipović and Larsson (2016) we can
apply Lemma 5.16 to conclude that

$\mathbb{E}[e^{\varepsilon|Y_{T}|}]<\infty$

for some ${\varepsilon}>0$. The claim follows by noting that $\langle
1,X_{T}\rangle=Y_{T}$. ∎

##  6 Affine diffusions on $M_{+}(E)$

We now define operators of _affine type_ and consider solutions to the
associated martingale problems. As we will show these solutions constitute a
subclass of polynomial processes, which correspond to classical measure-valued
branching Markov diffusions analyzed e.g. in Fitzsimmons (1988); Etheridge
(2000); Li (2010) and thus range in the class of measure-valued affine
processes. They can be considered as generalizations of the Dawson-Watanabe
superprocess (in the terminology of Etheridge (2000)333Note that in Li (2010)
“Dawson-Watanabe superprocess” is used for a class of measure-valued branching
processes which can also exhibit jumps. Essentially the class of processes
that we obtain here is the subset of processes with continuous trajectories
therein.), also called super-Brownian motion, where the constant diffusion
coefficient is replaced by a function and where the spatial motion is governed
by an operator satisfying the positive minimum principle. As it is well-known
for affine processes, additionally to the moment formula, the Laplace
transform of the process’ marginals is exponentially affine in the initial
state and the characteristic exponent can be computed by solving a Riccati
partial differential equation, which we shall introduce in Section 6.2.

###  6.1 Operators of affine type and their characterization in the diffusion
setting

Let us here start by defining operators of affine type, inspired by the
classical form of the infinitesimal generator of affine processes on finite
dimensional state spaces, see, e.g., Kawazu and Watanabe (1971); Duffie et al.
(2003); Cuchiero et al. (2011, 2016b).

######  Definition 6.1.

We say that a linear operator $L:{\mathcal{D}}\to{\mathcal{M}}(M(E))$ is of
_affine type_ on $M_{+}(E)$ if there exist maps $F:D\to\mathbb{R}$ and $R:D\to
C_{\Delta}(E)$ such that

$L\exp(\langle g,\,\cdot\,\rangle)(\nu)=(F(g)+\langle
R(g),\nu\rangle)\exp(\langle g,\nu\rangle)$

for all $g\in D_{-}$ and $\nu\in M_{+}(E)$, where $D_{-}$ denotes all function
in $D$ with values in $\mathbb{R}_{-}$.

For a finite-dimensional diffusions it is well-known that its generator is of
affine type if and only if the drift and diffusion coefficients are affine
functions (see, e.g., Kawazu and Watanabe (1971); Duffie et al. (2003)). The
following assertion states the same result for the measure-valued setting. We
report and prove it here for the reader convenience. It implies in particular
that (in the current diffusion setting) operators of affine type constitute a
subclass of polynomial operators.

######  Theorem 6.2.

A linear operator $L:\mathcal{D}\to{\mathcal{M}}(M(E))$ is of affine type on
$M_{+}(E)$ and its carré-du-champs operator $\Gamma$ is an
$M_{+}(E)$-derivation if and only if $L$ admits a representation

$\begin{split}Lf(\nu)=&B_{0}(\partial f(\nu))+\langle B_{1}(\partial
f(\nu)),\nu\rangle\\\ &+\frac{1}{2}\Big{(}Q_{0}(\partial^{2}f(\nu))+\langle
Q_{1}(\partial^{2}f(\nu)),\nu\rangle\Big{)},\quad f\in{\mathcal{D}},\nu\in
M_{+}(E)\end{split}$ (6.1)

for some linear operators $B_{0}:D\to\mathbb{R}$, $B_{1}:D\to C_{\Delta}(E)$,
$Q_{0}:D\otimes D\to\mathbb{R}$, $Q_{1}:D\otimes D\to C_{\Delta}(E)$. The
restriction of these operators on $D_{-}$ and $D_{-}\otimes D_{-}$,
respectively, are uniquely determined by $L$ and the following relations hold
for the functions $F$ and $R$ of Definition 6.1

$\displaystyle F(g)=B_{0}(g)+\frac{1}{2}Q_{0}(g\otimes g),$ (6.2)
$\displaystyle R(g)=B_{1}(g)+\frac{1}{2}Q_{1}(g\otimes g).$ (6.3)

###### Proof.

Assume first that $L$ satisfies (6.1). Then for $f(\nu)=\exp(\langle
g,\nu\rangle)$ with $g\in D_{-}$ and $\nu\in M_{+}(E)$ we have

$L\exp(\langle g,\,\cdot\,\rangle)(\nu)=\exp(\langle
g,\nu\rangle)(B_{0}(g)+\langle B_{1}(g),\nu\rangle+\frac{1}{2}(Q_{0}(g\otimes
g)+\langle Q_{1}(g\otimes g),\nu\rangle)).$

Hence $L$ is an affine operator with $F(g)=B_{0}(g)+\frac{1}{2}Q_{0}(g\otimes
g)$ and $R(g)=B_{1}(g)+\frac{1}{2}Q_{1}(g\otimes g)$. Hence $L$ is of affine
type. Moreover, a direct calculation yields for all $p,q\in\mathcal{D}$

$\Gamma(p,q)(\nu)=Q_{0}\left(\partial p(\nu)\otimes\partial
q(\nu)\right)+\left\langle Q_{1}(\partial p(\nu)\otimes\partial
q(\nu)),\nu\right\rangle\qquad\text{for all $\nu\in M_{+}(E)$,}$

which is an $M_{+}(E)$-derivation.

Conversely, assume that $L$ is of affine type and that its carré-du-champs
operator $\Gamma$ is an $M_{+}(E)$-derivation. We now explain how the affine
property and the fact that $\Gamma$ is a derivation imply the form (6.1) on
$f(\nu)=\exp(\langle g,\nu\rangle)$ for $g\in D_{-}$. The extension to all of
$\mathcal{D}$ follows then similarly as in the proof of Theorem 4.6. Observe
that since $L$ satisfies the affine property it holds that

$L\exp(\langle g,\nu\rangle)=(F(g)+\langle R(g),\nu\rangle)\exp(\langle
g,\nu\rangle),$

for each $g\in D_{-}$. Define then

$\displaystyle B_{0}(g)$ $\displaystyle:=2F(g)-\frac{1}{2}F(2g),$
$\displaystyle B_{1}g$ $\displaystyle:=R(g)-\frac{1}{2}R(2g),$ $\displaystyle
Q_{0}(g\otimes g)$ $\displaystyle:=F(2g)-2F(g),$ $\displaystyle Q_{1}(g\otimes
g)$ $\displaystyle:=R(2g)-2R(g),$

and observe that the claim follows by showing that these operators are linear.
Set $p_{g}(\nu):=\exp(\langle g,\nu\rangle)$ and note that

$\displaystyle B_{0}(g)+\langle B_{1}(g),\nu\rangle$
$\displaystyle=L(2p_{g}p_{g}(\nu)^{-1}-\frac{1}{2}p_{g}^{2}p_{g}(\nu)^{-2})(\nu),$
$\displaystyle Q_{0}(g\otimes g)+\langle Q_{1}(g\otimes g),\nu\rangle$
$\displaystyle=L(p_{g}^{2}p_{g}(\nu)^{-2}-2p_{g}p_{g}(\nu)^{-1})(\nu),$

for each $g\in D_{-}$. Fix now $g_{i},f_{i}\in D_{-}$ and
$\alpha_{i},\beta_{i}\in\mathbb{R}$ such that
$\sum\alpha_{i}g_{i}=\sum\beta_{i}f_{i}$ and note that

$\displaystyle\sum\alpha_{i}B_{0}(g_{i})-\sum\beta_{i}B_{0}(f_{i})+\langle\sum\alpha_{i}B_{1}(g_{i})-\sum\beta_{i}B_{1}(f_{i}),\nu\rangle=Lp(\nu)$

for
$p=\alpha_{i}(2p_{g_{i}}p_{g_{i}}(\nu)^{-1}-\frac{1}{2}p_{g_{i}}^{2}p_{g_{i}}(\nu)^{-2})-\beta_{i}(2p_{f_{i}}p_{f_{i}}(\nu)^{-1}-\frac{1}{2}p_{f_{i}}^{2}p_{f_{i}}(\nu)^{-2})$.
Since $\partial p(\nu)=0$ and $\partial p^{2}(\nu)=0$ the linearity of $B_{0}$
and $B_{1}$ follows by Lemma 4.5. Similarly, fix $g_{i},f_{i}\in D_{-}$ and
$\alpha_{i},\beta_{i}\in\mathbb{R}$ such that $\sum\alpha_{i}g_{i}\otimes
g_{i}=\sum\beta_{i}f_{i}\otimes f_{i}$ and note that

$\displaystyle\sum\alpha_{i}Q_{0}(g_{i}\otimes
g_{i})-\sum\beta_{i}Q_{0}(f_{i}\otimes
f_{i})+\langle\sum\alpha_{i}Q_{1}(g_{i}\otimes
g_{i})-\sum\beta_{i}Q_{1}(f_{i}\otimes f_{i}),\nu\rangle=Lp(\nu)$

for
$p=\alpha_{i}(p_{g_{i}}^{2}p_{g_{i}}(\nu)^{-2}-2p_{g_{i}}p_{g_{i}}(\nu)^{-1})-\beta_{i}(p_{f_{i}}^{2}p_{f_{i}}(\nu)^{-2}-2p_{f_{i}}p_{f_{i}}(\nu)^{-1})$.
Since $\partial p(\nu)=0$ and $\partial p^{2}(\nu)=0$ the linearity of $Q_{0}$
and $Q_{1}$ follows by Lemma 4.5. ∎

We now introduce the subclass of _measure-valued affine diffusions_ via affine
type operators.

######  Definition 6.3.

Let $L$ be of affine type on $M_{+}(E)$. Then any continuous $M_{+}(E)$-valued
solution to the martingale problem for $L$ is called a _measure-valued affine
diffusion_.

###  6.2 Laplace transform and Riccati (partial) differential equations

Due to the additional properties of affine diffusions, not only the moment
formula holds true but also the Laplace transform can be computed explicitly
via non-linear partial differential equations of Riccati type. We start here
by introducing the corresponding solution concept. Recall that we refer to
differential equations as ODEs even though they often correspond to PDEs or
PIDEs depending on the state space and the involved operators.

######  Definition 6.4.

Let $R$ be given by (6.3). Then we call a function $t\mapsto\psi_{t}$ with
values in $D$ a solution to the Riccati ODE

$\partial_{t}\psi_{t}=R(\psi_{t}),\quad\psi_{0}=g\in D,$

if for every $t>0$ it holds

$\displaystyle\langle\psi_{t},\nu\rangle=\langle
g,\nu\rangle+\int_{0}^{t}\langle R(\psi_{s}),\nu\rangle ds$ (6.4)

for all $\nu\in M_{+}(E^{\Delta})$.

Note that the reason why (6.4) is called Riccati differential equation is
because $g\mapsto R(g)$ is a quadratic function as seen from (6.3).

######  Remark 6.5.

Similarly as for the moment formula note that the above solution concept
reduces to a more classical solution if we take $\nu=\delta_{x}$ with $x\in
E$, Indeed, (6.4) can then be transformed into

$\displaystyle\psi_{t}(x)=g(x)+\int_{0}^{t}R(\psi_{s})(x)ds$

and thus reduces to a classical (except of the integral form) solution of a
Riccati PDE.

Note that in contrast to strong solutions, which were in the current context
for instance considered in (Iscoe, 1986, Appendix) or (Li, 2010, Section 7.1)
and where $\psi$ is required to be a continuously (Fréchet) differentiable
curve $\psi_{t}:\mathbb{R}_{+}\to D$ , Definition 6.4 corresponds to an
(analytically) weak solution concept.

In the following we prove the exponential affine property of the Laplace
transform.

######  Theorem 6.6.

Let $L$ be an operator of affine type with functions $F$ and $R$. Suppose that
an $M_{+}(E^{\Delta})$-valued solution $(X_{t})_{t\geq 0}$ to the martingale
problem for $L$ exists. Assume furthermore that there is a solution in the
sense of Definition 6.4 of the Riccati ODE on $[0,T]$ given by

$\partial_{t}\psi_{t}=R(\psi_{t}),\qquad\psi_{0}=g\in D_{-},$ (6.5)

which takes values in $D_{-}$ where $D_{-}$ denotes all functions in $D$ with
values in $\mathbb{R}_{-}$. Define furthermore

$\phi_{t}=\int_{0}^{t}F(\psi_{s})ds$ (6.6)

and suppose it takes values in $\mathbb{R}_{-}$. Then, for all $0\leq t\leq T$
the representation

$\mathbb{E}\left[\exp(\langle
g,X_{T}\rangle)\,\big{|}\,\mathcal{F}_{t}\right]=\exp(\phi_{T-t}+\langle\psi_{T-t},X_{t}\rangle)$
(6.7)

holds almost surely.

###### Proof.

Due to the fact that $X$ solves the martingale problem for $L$, we can deduce
analogously as in the proof of (Li, 2010, Theorem 7.13) that
$\exp(\phi_{T-t}+\langle\psi_{T-t},X_{t}\rangle)$ is a local martingale. By
the assumptions that $\psi$ takes values in $D_{-}$ and
$\phi\in\mathbb{R}_{-}$, this is actually a true martingale as it is bounded
by $1$ and we thus have

$\exp(\phi_{T-t}+\langle\psi_{T-t},X_{t}\rangle)=\mathbb{E}[\exp(\phi_{0}+\langle\psi_{0},X_{T}\rangle)|\mathcal{F}_{t}]=\mathbb{E}[\exp(\langle
g,X_{T}\rangle)|\mathcal{F}_{t}],$

proving the claim. ∎

###  6.3 Existence and uniqueness in law

We can now combine Theorem 6.2, Theorem 5.9 and Theorem 6.6 to get the
following existence and uniqueness in law result for measure-valued affine
diffusions. Recall that $D\subseteq C_{\Delta}(E)$ be a dense linear subspace
containing the constant function $1$.

######  Corollary 6.7.

Let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear operator of form (6.1), where

  1. (i)

$B_{0}:D\to\mathbb{R}$ is given by $B_{0}(g)=\langle g,b\rangle$ with $b\in
M_{+}(E^{\Delta})$;

  2. (ii)

$B_{1}:D\to C_{\Delta}(E)$ satisfies the positive minimum principle on
$E^{\Delta}$;

  3. (iii)

$Q_{0}\equiv 0$;

  4. (iv)

$Q_{1}$ is of the form

$Q_{1}(g)=\alpha\operatorname{diag}(g),\quad\text{ that is,}\quad
Q_{1}(g)(x)=\alpha(x)g(x,x),\quad g\in D\otimes D$

where $\alpha\in C_{\Delta}(E)$ with values in $\mathbb{R}_{+}$.

Then $L$ is of affine type on $M_{+}(E)$ and its martingale problem has an
$M_{+}(E^{\Delta})$-valued solution with continuous paths for every initial
condition $\nu\in M_{+}(E^{\Delta})$.

If additionally, the measure $b$ and the initial condition $\nu$ lie in
$M_{+}(E)$ and if there exists some function $m\in C_{\Delta}(E)$ such that

$\text{bp-}\lim_{n\to\infty}(B_{1}(1-g_{n})-m(1-g_{n}))\leq 0,$

for some sequence $g_{n}\in D\cap C_{0}(E)$ satisfying
$\text{bp-}\lim_{n\to\infty}g_{n}=1_{E}$, then any solution to the martingale
problem takes values in $M_{+}(E)$.

If, in either of the $M_{+}(E)$ or $M_{+}(E^{\Delta})$ cases,

$\displaystyle\partial_{t}\psi_{t}=B_{1}\psi_{t}+\frac{1}{2}\alpha\psi_{t}^{2},\quad\psi_{0}=g\in
D_{-}$ (6.8)

has a solution in the sense of Definition 6.4 with values in $D_{-}$ for all
$t\geq 0$, then the corresponding martingale problem is well-posed.

###### Proof.

The existence result is just a consequence of Theorem 6.2 and Theorem 5.9.
Concerning the well-posedness of the martingale problem, observe that the
function $F$ associated to the operator $L$ of affine type is of the form

$F(g)=B_{0}(g)=\langle g,b\rangle$

for $b\in M_{+}(E^{\Delta})$. Hence, $F$ maps $\psi_{t}\in D_{-}$ to
$\mathbb{R}_{-}$ and the assumptions of Theorem 6.6 are satisfied. Hence the
Laplace transform of $X_{t}$ is given by

$\mathbb{E}[\exp(\langle
g,X_{t}\rangle)]=\exp(\phi_{t}+\langle\psi_{t},X_{0}\rangle).$

Since $g\in D_{-}$ was arbitrary and $D$ is dense in $C_{\Delta}(E)$, the law
of $X_{t}$ is uniquely determined for all $t\geq 0$. From (Ethier and Kurtz,
2009, Theorem 4.4.2), we infer that $X$ is a Markov process thus the unique
solution to the martingale problem associated to $L$. ∎

######  Remark 6.8.

Let us here explain how the above result relates to the literature on
$(\xi,\phi)$\- (Dawson-Watanabe) - superprocesses as defined in (Li, 2010,
p.42). Indeed, $\xi$ is the spatial motion which is described by its
infinitesimal generator $\mathcal{A}$ that can be obtained from the current
$B_{1}$ via

$\mathcal{A}g=B_{1}g-mg,$

as explained in Remark 1.2. The branching mechanism $\phi$ (not to be confused
with the function defined via (6.6)) then corresponds to

$\phi(x,g)=-m(x)g(x)+\frac{1}{2}\alpha(x)g(x)^{2}.$

Hence, in the notation of (Li, 2010, Eq.(2.27)), $b=-m$ and
$c=\frac{1}{2}\alpha$. The $B_{0}$-part can be added as an immigration part as
considered in (Li, 2010, Section 9.3).

This identification allows us to apply the results of Li (2010) to our setup.
In view of the Riccati equations this means in particular that (6.8)
corresponds (up to a change of sign) to (Li, 2010, Eq.(7.4)), which is of the
form

$\displaystyle\partial_{t}V_{t}f$
$\displaystyle=\mathcal{A}V_{t}f-\phi(\,\cdot\,,V_{t}f)=\mathcal{A}V_{t}f+mV_{t}f-\frac{1}{2}\alpha(V_{t}f)^{2}=B_{1}V_{t}f-\frac{1}{2}\alpha(V_{t}f)^{2},$
$\displaystyle V_{0}f$ $\displaystyle=f(=-g\text{ in our notation}).$

Supposing that $\mathcal{A}$ generates a Feller semigroup $(P_{t})_{t\geq 0}$
and that $m=-b\in C_{\Delta}(E)$, then by Theorem (Li, 2010, Theorem 7.11)
this equation is in turn equivalent to the integral equation (Li, 2010,
Eq.(5.32))

$V_{t}f=P_{t}f-\int_{0}^{t}P_{t-s}\phi(\,\cdot\,,V_{s}f)ds,$

which corresponds to a mild solution. As $\psi_{t}=-V_{t}f$ with initial value
$\psi_{0}=g=-f$ and denoting the positive semigroup generated by $B_{1}$ with
$Q$ we thus get a similar equivalence between our solution concept for (6.8)
and the following integral equation in weak form (corresponding to a weakly
mild solution)

$\displaystyle\langle\psi_{t},\nu\rangle$ $\displaystyle=\langle
Q_{t}g,\nu\rangle+\frac{1}{2}\int_{0}^{t}\langle
Q_{t-s}\alpha\psi_{s}^{2},\nu\rangle ds,$ $\displaystyle=\langle
P_{t}g,\nu\rangle+\int_{0}^{t}\langle
P_{t-s}(m\psi_{s}+\frac{1}{2}\alpha\psi_{s}^{2}),\nu\rangle ds.$

By (Li, 2010, Corollary 5.17) which is a consequence of (Li, 2010, Proposition
2.20 and (Eq. 2.33)) it admits a unique negative solution in the domain of
$\mathcal{A}$ (in the strong sense as generator of the Feller semigroup
$(P_{t})_{t\geq 0}$). The latter property is a consequence of (Li, 2010,
Theorem 7.11). From this discussion we can now deduce the following corollary.

######  Corollary 6.9.

Let the Conditions (i)-(iv) of Corollary 6.7 be satisfied and suppose
additionally that $B_{1}$ is the generator of a strongly continuous positive
semigroup. Then (6.8) has a unique solution in the sense of Definition 6.4
with values in $D_{-}$ for all $t\geq 0$ and the martingale problem is well-
posed.

###### Proof.

Since $B_{1}$ is the generator of a strongly continuous positive semigroup, it
can be decomposed into $B_{1}g=\mathcal{A}g+mg$ where $m\in C_{\Delta}(E)$ and
$\mathcal{A}$ is the (strong) generator of a Feller semigroup. Hence the
assertion follows from Theorem 7.11 and Corollary 5.17 in Li (2010) as
explained in Remark 6.8. ∎

######  Remark 6.10.

Note that the conditions of Corollary 6.9 are in line with the ones of Theorem
5.15 from which uniqueness in law can be deduced as well.

Similarly as in the polynomial case we get a characterization of affine
diffusions when $D=C_{\Delta}(E)$.

######  Corollary 6.11.

Let $D=C_{\Delta}(E)$ and let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear
operator. Then $L$ is of affine type on $M_{+}(E)$, there exists an
$M_{+}(E^{\Delta})$-valued solution to the martingale problem for all initial
conditions in $M_{+}(E^{\Delta})$ and all solutions have continuous paths, if
and only if $L$ satisfies (4.3) with

  1. (i)

$B_{0}:C_{\Delta}(E)\to\mathbb{R}$ is given by $B_{0}(g)=\langle g,b\rangle$
with a Radon measure $b\in M_{+}(E^{\Delta})$;

  2. (ii)

For every $g\in C_{\Delta}(E)$

$B_{1}g(x)=\int g(\xi)-g(x)\nu_{B}(x,d\xi)+m(x)g(x)$ (6.9)

for some non-negative finite kernel $\nu_{B}$ from $E^{\Delta}$ to
$E^{\Delta}$ and some $m\in C_{\Delta}(E)$;

  3. (iii)

$Q_{0}\equiv 0$;

  4. (iv)

$Q_{1}$ is of the form

$Q_{1}(g)(x)=\alpha(x)g(x,x),\quad g\in C_{\Delta}(E)\otimes C_{\Delta}(E)$
(6.10)

where $\alpha\in C_{\Delta}(E)$ with values in $\mathbb{R}_{+}$.

For the state space $M_{+}(E)$, the following equivalence holds.

A linear operator $L:\mathcal{D}\to C(M_{+}(E))$ is of affine type on
$M_{+}(E)$, there exists an $M_{+}(E)$-valued solution to the martingale
problem for all initial conditions in $M_{+}(E)$ and all solutions have
continuous paths, if and only if $L$ satisfies (i)-(iv), $b\in M_{+}(E)$, and
$\nu_{B}$ is a kernel from $E$ to $E$.

Moreover, in either case

$\partial_{t}\psi_{t}=B_{1}\psi_{t}+\frac{1}{2}\alpha\psi_{t}^{2},\quad\psi_{0}=g\in
C_{\Delta}(E)$

has a unique solution in the sense of Definition 6.4 with values in $D_{-}$
for all $t\geq 0$, which implies (6.7) and well-posedness of the martingale
problem.

###### Proof.

All assertions except the very last one follow from combining Theorem 6.2 and
Theorem 5.11. The last one is a consequence of Corollary 6.9 since $B_{1}$ is
decomposed into $B_{1}g=\mathcal{A}g+mg$ where $\mathcal{A}g(x)=\int
g(\xi)-g(x)\nu_{B}(x,d\xi)$ generates a Feller semigroup corresponding to a
pure jump process and $m\in C_{\Delta}(E)$. ∎

##  7 Examples and applications

Having established the general setting for measure-valued affine and
polynomial diffusions, we now consider several important examples.

###  7.1 Finite underlying space

We start with a finite underlying space $E$ consisting of $m\geq 1$ points,
i.e $E=\\{1,...,m\\}$. Then $C_{\Delta}(E)=C(E)$ is finite-dimensional, hence
any dense linear subspace is equal to the whole space. In this setting, any
$M_{+}(E)$-valued process is of the form
$X_{t}=\sum_{i=1}^{m}Z_{t}^{i}\delta_{i}$ for some $\mathbb{R}^{m}_{+}$-valued
process $Z=(Z^{1},...,Z^{m})$. Hence, Theorem 5.11 corresponds to the finite
dimensional characterization of polynomial operators with state space
$\mathbb{R}^{m}_{+}$, as proved in (Filipović and Larsson, 2016, Proposition
6.4). Indeed, comparing with this proposition we recognize the following
identifications:

  1. (i)

the Radon measure $b\in M_{+}(E)$ corresponds to the vector
$\beta_{J}\in\mathbb{R}^{m}_{+}$;

  2. (ii)

the operator $B_{1}$ corresponds to the submatrix $B_{JJ}^{\top}$ and (5.10)
translates to positive off-diagonal elements;

  3. (iii)

the function $\alpha\in C_{\Delta}(E)$ with values in $\mathbb{R}_{+}$
corresponds to the vector $\phi\in\mathbb{R}^{m}_{+}$;

  4. (iv)

the functions $\pi$ and $\beta$ correspond to the matrix $\Pi$ and
$\alpha\in\mathbb{S}^{m}$ while Condition (5.6) translates to
$\alpha+\operatorname{Diag}(\Pi^{T}x)\operatorname{Diag}(x)^{-1}\in\mathbb{S}^{m}_{+}$
for all $x\in\mathbb{R}^{m}_{++}$.

###  7.2 Underlying space $E\subseteq\mathbb{R}$

We consider here the case $E^{\Delta}=[a,b]\subseteq\mathbb{R}^{\Delta}$, for
some $-\infty\leq a<b\leq\infty$ and set

$D:=\\{f|_{E}\,:\,f\in C_{c}^{\infty}(\mathbb{R})+\mathbb{R}\\}.$

Recall that we identify $E^{\Delta}$ with $E$, when $E$ is compact
corresponding to the case $a>-\infty$ and $b<\infty$. Our goal is to analyze
Theorem 5.9 in this setting. We therefore start by considering generators of
strongly continuous positive groups on $C_{\Delta}(E)$ such that their domain
contains both $D$ and $A(D)$. For results on generators of strongly continuous
positive groups we refer to Appendix C, in particular to Definition C.5 and
Definition C.7 for the notions of _admissibility_ and _lattice isomorphisms_.
In the following lemma we denote by $C_{\Delta}^{k}(E)$ the restriction of
$C^{k}(\mathbb{R})$-functions to $E$, i.e.
$C_{\Delta}^{k}(E)=\\{f|_{E}\,:\,f\in C^{k}(\mathbb{R})\cap
C_{\Delta}(\mathbb{R})\\}$.

######  Lemma 7.1.

Let $V$ be a lattice isomorphism on $C_{\Delta}(E)$, which additionally leaves
$C_{\Delta}^{1}(E)$ and $C^{2}_{\Delta}(E)$-functions invariant. Moreover, let
$h\in C^{1}_{\Delta}(E)$ and $\tau\in C_{\Delta}^{1}(E)$ be an admissible
function in the sense of Definition C.5. Define $\delta_{\tau}$ via (C.4).
Then the operator $A$ given by

$A:=V^{-1}\delta_{\tau}V+h,$ (7.1)

satisfies the conditions of Theorem 3.4. That is, $A$ is the generator of a
strongly continuous positive group of $C_{\Delta}(E)$ and its domain contains
both $D$ and $A(D)$.

###### Proof.

First of all note that the domain of $A$ is given by $\\{g\in
C_{\Delta}(E)\,|\,Vg\in D(\delta_{\tau})\\}$, where $D(\delta_{\tau})$ is
defined in (C.5). As $V$ leaves $C_{\Delta}^{1}(E)$ and
$C^{2}_{\Delta}(E)$-functions invariant and since $\tau$ and $h$ are in
$C_{\Delta}^{1}(E)$, the domain of $A$ contains $D$ and $A(D)$. Moreover, by
Theorem C.8 the operator $A$ generates a strongly continuous positive group on
$C_{\Delta}(E)$, which yields the result. ∎

######  Example 7.2.

Let $E=\mathbb{R}$. Set $Vg=pg$ for some strictly positive function $p\in
C^{\infty}_{\Delta}(E)$, $\tau\equiv 1$ and $h\in C^{1}_{\Delta}(E)$. Then all
the conditions of Lemma 7.1 are satisfied and

$Ag=\frac{1}{p}(p^{\prime}g+pg^{\prime})+hg=g^{\prime}+\widetilde{h}g,\quad
g\in D,$

where $\widetilde{h}=\frac{p^{\prime}}{p}+h$. Hence, the stochastic process
associated to $A$ is a pure drift process killed at rate $\widetilde{h}$.

To consider some bounded interval, let $E=[0,1]$. Moreover, let $\tau\in
C^{1}_{\Delta}(E)$ with such that $\tau(0)=\tau(1)=0$. Then by Remark C.6 the
function $\tau$ is admissible as it has bounded derivatives. Hence,

$Ag=\tau g^{\prime},\quad g\in D$

satisfies the conditions of Lemma 7.1.

We can now reformulate Theorem 5.9 in the current setting as follows.

######  Corollary 7.3.

Let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear operator of form (4.3), where
$B_{0}$, $Q_{0}$ and $Q_{1}$ satisfy the conditions of Theorem 5.9. Moreover,
for $i=1,\ldots n$, let $A_{i}$ be of form (7.1) such that for $V,\tau$ and
$h$ the conditions of Lemma 7.1 hold. Suppose that

  1. (i)

$B_{1}-\frac{1}{2}\sum_{i=1}^{n}A_{i}^{2}:D\to C_{\Delta}(E)$ satisfies the
positive minimum principle on $E^{\Delta}$;

  2. (ii)

$Q_{2}$ is of the form

$Q_{2}(g)=C(g)+\sum_{i=1}^{n}(A_{i}\otimes A_{i})(g),\quad g\in D\otimes D,$

where $C$ admits a ($\beta,\pi$)-representation.

Then conditions (i)-(v) of Theorem 5.9 are satisfied.

###### Proof.

This follows directly from Lemma 7.1. ∎

The rest of the section is devoted to the case $E=\mathbb{R}$. In view of
Corollary 7.3 and Remark 3.5, $B_{1}$ should satisfy the positive minum
principle. Applying Remark 1.2, we thus decompose $B_{1}$ into an operator
$B_{1}g=\mathcal{A}g+mg$ where $\mathcal{A}$ satisfies the positive maximum
principle and where $m\in C_{\Delta}(\mathbb{R})$. It is well-known, that
under this condition $\mathcal{A}$ is a Lévy type operator, i.e.

$\displaystyle\mathcal{A}g=\gamma g^{\prime}+\frac{1}{2}\Gamma
g^{\prime\prime}+\int(g(\,\cdot\,+\xi)-g-\chi(\xi)g^{\prime})F(\,\cdot\,,d\xi),\quad
g\in D,$ (7.2)

for some continuous functions $\gamma,\,\Gamma$ with $\Gamma\geq 0$, a
truncation function $\chi$ , and a kernel $F(\,\cdot\,,d\xi)$ from
$\mathbb{R}$ to $\mathbb{R}$ such that $\int(|\xi|^{2}\wedge
1)F(\,\cdot\,,d\xi)<\infty$ . Every operator of this form satisfies
$\mathcal{A}1=0$ and the positive maximum principle on $\mathbb{R}$ . The
following result expresses Corollary 7.3 within the current setting.

######  Corollary 7.4.

Let $L:{\mathcal{D}}\to C(M_{+}(E))$ be a linear operator of form (4.3), where
$B_{0}$, $Q_{0}$ and $Q_{1}$ satisfy the conditions of Theorem 5.9. Let
$B_{1}$ be given by $B_{1}g=\mathcal{A}g+mg$ where $m\in
C_{\Delta}(\mathbb{R})$ and $\mathcal{A}$ is of form $\eqref{IIeqn24}$ with
$\Gamma:=\sigma^{2}+\tau^{2}$ for some continuous function $\sigma$ and
$\tau\in C^{1}_{\Delta}{(\mathbb{R})}$. Moreover, define $Q_{2}$ via

$Q_{2}(g)(x,y)=C(g)(x,y)+\tau(x)\tau(y)g^{\prime}(x)g^{\prime}(y)+h(x)h(y)g(x)g(y),$

where $C$ admits a ($\beta,\pi$)-representation and $h\in
C^{1}_{\Delta}(\mathbb{R})$. Then conditions (i)-(v) of Theorem 5.9 hold true.

###### Proof.

We apply here Corollary 7.3 with $i=1$ and the operator by $Ag=\tau
g^{\prime}+h$. Note that $\tau$ is admissible as it is bounded and we work on
$E=\mathbb{R}$. Hence, we only have to verify that $B_{1}-\frac{1}{2}A^{2}$
satisfies the positive minimum principle on $\mathbb{R}^{\Delta}$. This is the
case because of the form of $\Gamma=\sigma^{2}+\tau^{2}$ in $\mathcal{A}$ so
that we obtain

$B_{1}g-\frac{1}{2}A^{2}g=(\gamma+\tau
h)g^{\prime}+\frac{1}{2}\sigma^{2}g^{\prime\prime}+\int(g(\,\cdot\,+\xi)-g-\chi(\xi)g^{\prime})F(\,\cdot\,,d\xi)+(m+h^{2})g,\quad
g\in D,$

which satisfies the positive minimum principle on $\mathbb{R}$ and also
$\mathbb{R}^{\Delta}$ since $D\subset C_{c}(\mathbb{R})$. ∎

######  Remark 7.5.

Note that in Corollary 7.4 a non-zero $\tau$ in the specification of $Q_{2}$
is coupled with a corresponding diffusive component in the specification (7.2)
of $\mathcal{A}$ and in turn of $B_{1}$. This is analogous to (Cuchiero et
al., 2019, Corollary 6.3 and Example 6.4) (compare also (Cuchiero et al.,
2019, Corollary 6.3 and Example 6.4)).

###  7.3 Measure-valued Pearson diffusions

Certain measure-valued polynomial diffusions resulting from Theorem 5.9 can be
considered as generalizations of one-dimensional Pearson diffusions on
$\mathbb{R}_{+}$. We recall here the definition when their state space is some
bounded or unbounded interval of $\mathbb{R}$, see, e.g., Forman and Sørensen
(2008).

######  Definition 7.6.

A real-valued stochastic process $(X_{t})_{t\geq}$ is called _Pearson
diffusion_ if it is a weak solution of the following stochastic differential
equation

$dX_{t}=b(X_{t})dt+\sigma(X_{t})dW_{t},$

where $W_{t}$ is a standard Brownian motion and $x\mapsto b(x)$ and
$x\mapsto\sigma^{2}(x)$ are polynomials respectively of at most first and
second order degree, i.e., one has

$\displaystyle b(x)$ $\displaystyle=b_{0}+b_{1}x,$ $\displaystyle q(x)$
$\displaystyle=\frac{1}{2}\sigma^{2}(x)=q_{0}+q_{1}x+q_{2}x^{2},$

for some $b_{0},b_{1},q_{0},q_{1},q_{2}\in\mathbb{R}$. In particular the
(extended) generator is defined as

$Lg(x)=b(x)g^{\prime}(x)+q(x)g^{\prime\prime}(x),\;\;\;\;\;\;g\in
C^{2}(\mathbb{R}).$

######  Remark 7.7.

Note that this definition coincides with the notion of one-dimensional
polynomial processes with continuous trajectories (see Cuchiero et al. (2012);
Filipović and Larsson (2016)).

Following Ascione et al. (2020) we can recognize six different types of
Pearson diffusions. The statements concerning the invariant measure hold true
if the drift is of the form $b(x)=b_{0}+b_{1}x=b_{1}(x+\frac{b_{0}}{b_{1}})$
with $b_{1}<0$ and $b_{0}\in\mathbb{R}$.

  1. (i)

If $q\equiv q_{0}>0$, then $X_{t}$ is a Ornstein-Uhlenbeck (OU) process with
values in $\mathbb{R}$ and the stationary distribution is a Gaussian
distribution.

  2. (ii)

If $q(x)=q_{1}x$ with $q_{1}>0$, then $X_{t}$ is a Cox-Ingersoll-Ross (CIR)
process with values in $\mathbb{R}_{+}$ and the stationary is a Gamma
distribution.

  3. (iii)

If $q(x)=q_{0}+q_{1}x+q_{2}x^{2}$ with $q_{2}<0$, then $X_{t}$ is a Jacobi
process with values in a compact interval and the stationary distribution is a
(scaled) Beta distribution.

  4. (iv)

If $q(x)=q_{1}x+q_{2}x^{2}$ with $q_{1},q_{2}>0$, then $X_{t}$ is a Fisher-
Snedecor process with values in $\mathbb{R}_{+}$ and the stationary
distribution is a Fisher-Snedecor distribution.

  5. (v)

If $q(x)=q_{2}x^{2}$ with $q_{2}>0$, then $X_{t}$ is a reciprocal Gamma (RG)
process with values in $\mathbb{R}_{+}$ and the stationary distribution is a
reciprocal Gamma distribution. Note that this corresponds exactly to diffusion
characteristic of the Black & Scholes model.

  6. (vi)

If $q(x)=q_{0}+q_{1}x+q_{2}x^{2}$ with $q_{2}>0$ and the discriminant
$\Delta_{q}<0$, then $X_{t}$ is a Student process with values in $\mathbb{R}$
and the stationary distribution is a Student distribution.

Since we are here only interested in (non-negative) measure-valued analogs of
the cases (ii), (iv) and (v) and not in deriving existence of stationary
measures, we only distinguish the following three cases in terms of the
characterization given in Theorem 5.11.

  1. (ii):

As already discussed in Section 6 the case $Q_{2}\equiv 0$ can be seen as a
generalization of Cox Ingersoll Ross processes to the measure-valued case.

  2. (iv):

Taking the general form of Theorem 5.11 yields a measure-valued analog of the
Fisher-Snedecor process.

  3. (v):

A measure-valued analog of a multivariate Black-Scholes type model corresponds
to the case $Q_{1}=0$ and $\pi\equiv 0$.

##  Appendix A Necessary conditions for the positive maximum principle

We here establish the necessary conditions implied by the positive maximum
principle for both cases $M_{+}(E^{\Delta})$ and $M_{+}(E)$. Recall that
$D\subseteq C_{\Delta}(E)$ is a dense linear subspace containing the constant
function $1$ and let ${\mathcal{D}}$ be as in (2.2).

######  Lemma A.1.

Let $L:{\mathcal{D}}\to C(M_{+}(E))$ be an $M_{+}(E)$-polynomial operator of
form (4.3) satisfying the positive maximum principle on $M_{+}(E^{\Delta})$.
Then

  1. (i)

$B_{0}$ is a positive linear functional on $D$, hence there exists a Radon
measure $b\in M_{+}(E^{\Delta})$ such that $B_{0}(g)=\langle g,b\rangle$;

  2. (ii)

$B_{1}$ satisfies the positive minimum principle on $E^{\Delta}$;

  3. (iii)

$Q_{0}(g\otimes g)=0$ for all $0\leq g\in D$;

  4. (iv)

$Q_{1}(g\otimes g)$ and $Q_{2}(g\otimes g)$ are non-negative functions for all
$g\in D$;

  5. (v)

$\langle Q_{1}(g\otimes g),\nu\rangle=0$ and $\langle Q_{2}(g\otimes
g),\nu^{2}\rangle=0$ for all $\nu\in M_{+}(E^{\Delta})$ and $0\leq g\in D$
which are $0$ on the support of $\nu$.

Replacing the positive maximum principle on $M_{+}(E^{\Delta})$ by the
positive maximum principle on $M_{+}(E)$ we obtain that conditions (i),(iii),
and (iv) still hold true and we also get that

  1. (vi)

$B_{1}$ satisfies the positive minimum principle on $E$;

  2. (vii)

$\langle Q_{1}(g\otimes g),\nu\rangle=0$ and $\langle Q_{2}(g\otimes
g),\nu^{2}\rangle=0$ for all $\nu\in M_{+}(E)$ and $0\leq g\in D$ which are
$0$ on the support of $\nu$.

###### Proof.

Consider the following function $f(\nu)=-\langle g,\nu\rangle$ with $g\in D$
such that $g\geq 0$. Then $\sup_{M_{+}(E^{\Delta})}f=f(0)=0$. By the positive
maximum principle for $L$, it follows that

$Lf(0)=-B_{0}(g)-\langle B_{1}g,0\rangle\leq 0.$

Hence, $B_{0}(g)\geq 0$ and since $g$ was arbitrary we conclude that $B_{0}$
is a positive linear functional on $D$. By assumption $D$ is a linear subspace
of $C_{\Delta}(E)$ that contains an interior point of the cone of non-negative
function, namely the constant function $1$. In this case (Schaefer, 1999,
Theorem and Corollary 2 of Chapter V.5.4) (alternatively (Bourbaki, 2003,
Section 2.3.1)) yields that every positive linear map on $D$ can be extended
to a positive linear map on $C_{\Delta}(E)$.

Let now $g$ be as above with the additional property that $g(x)=0$ for some
$x\in E^{\Delta}$, i.e., $\inf_{E^{\Delta}}g=g(x)=0$ and
$\sup_{M_{+}(E^{\Delta})}f=f(n\delta_{x})=0$ for every $n\in\mathbb{N}$. By
the positive maximum principle for $L$, we have

$Lf(n\delta_{x})=-B_{0}(g)-\langle B_{1}g,n\delta_{x}\rangle\leq 0.$

Hence

$-\langle B_{1}g,\delta_{x}\rangle\leq\frac{B_{0}(g)}{n}$

and sending $n\to\infty$, we conclude that $B_{1}g(x)\geq 0$, which implies
the positive minimum principle on $E^{\Delta}$.

Consider now the function $f(\nu)=-(\langle g,\nu\rangle)^{2}$ for $g\in D$.
As it attains its maximum at $0$ with $\partial f(0)=0$ and
$\partial^{2}f(0)=-2g\otimes g$, we have again by the positive maximum
principle

$Lf(0)=-2Q_{0}(g\otimes g)-2\langle Q_{1}(g\otimes g),0\rangle-2\langle
Q_{2}(g\otimes g),0^{2}\rangle\leq 0,$

implying that $Q_{0}(g\otimes g)\geq 0$.

On the other hand, take a function $\phi\in C_{c}^{\infty}(\mathbb{R})$
restricted to $\mathbb{R}_{+}$ of the form $\phi(x)=(x-1)^{2n}$ for
$x\in[0,\frac{1}{2}]$ and $n\in\mathbb{N}$ such that it attains is maximum at
$0$. Consider $f(\nu)=\phi(\langle g,\nu\rangle)$ for $g\in D$ with $g\geq 0$.
Then $\sup_{M_{+}(E^{\Delta})}f=f(0)=1$, $\partial f(0)=-2ng$ and
$\partial^{2}f(0)=2n(2n-1)(g\otimes g)$. By the positive maximum principle for
$L$, it thus follows that

$Lf(0)=-2nB_{0}(g)+2n(2n-1)Q_{0}(g\otimes g)\leq 0.$

Hence,

$Q_{0}(g\otimes g)\leq\frac{B_{0}(g)}{2n-1}$

and sending $n\to\infty$ yields $Q_{0}(g\otimes g)\leq 0$ for $g\geq 0$. Hence
for all $g\geq 0$, $Q_{0}(g\otimes g)=0$.

Finally, fix $g\in D$ and $\nu\in M_{+}(E^{\Delta})$ and consider the function
$f(\mu)=-(\langle g,\nu\rangle-\langle ng,\mu\rangle)^{2}$ for
$n\in\mathbb{N}$. Then $f\leq 0$, $f(\frac{\nu}{n})=0$, $\partial
f(\frac{\nu}{n})=0$ and $\partial^{2}f(\frac{\nu}{n})=-2n^{2}g\otimes g$.
Hence,

$Lf\left(\frac{\nu}{n}\right)=-2n^{2}\langle Q_{1}(g\otimes
g),\frac{\nu}{n}\rangle-2n^{2}\langle Q_{2}(g\otimes
g),\frac{\nu^{2}}{n^{2}}\rangle\leq 0.$

and thus

$\langle Q_{1}(g\otimes g),\nu\rangle\geq-\frac{\langle Q_{2}(g\otimes
g),\nu^{2}\rangle}{n},$

implying that $\langle Q_{1}(g\otimes g),\nu\rangle\geq 0$ as $n$ tends to
$\infty$. Considering the function $f(\mu)=-(\langle
g,\nu\rangle-\langle\frac{g}{n},\mu\rangle)^{2}$, yields by similar arguments
that $\langle Q_{2}(g\otimes g),\nu^{2}\rangle\geq 0$.

On the other hand, consider again a function $\phi\in
C_{c}^{\infty}(\mathbb{R})$ restricted to $\mathbb{R}_{+}$ of the form
$\phi(x)=(x-1)^{2n}$ for $x\in[0,\frac{1}{2}]$ and $n\in\mathbb{N}$ such that
it attains is maximum at $0$. Fix $0\leq g\in D$ and $\nu\in
M_{+}(E^{\Delta})$ such that $g=0$ on the support of $\nu$. Consider
$f(\mu)=\phi(\langle g,\mu\rangle)$. Then
$\sup_{M_{+}(E^{\Delta})}f=f(\nu)=1$, $\partial f(\nu)=-2ng$ and
$\partial^{2}f(\nu)=2n(2n-1)(g\otimes g)$. By the positive maximum principle
for $L$, it thus follows that

$Lf(\nu)=-2nB_{0}(g)-2n\langle B_{1}(g),\nu\rangle+2n(2n-1)\langle
Q_{1}(g\otimes g),\nu\rangle+2n(2n-1)\langle Q_{2}(g\otimes
g),\nu^{2}\rangle\leq 0.$

Hence,

$\langle Q_{1}(g\otimes g),\nu\rangle+\langle Q_{2}(g\otimes
g),\nu^{2}\rangle\leq\frac{B_{0}(g)+\langle B_{1}(g),\nu\rangle}{2n-1}$

which implies that

$\langle Q_{1}(g\otimes g),\nu\rangle+\langle Q_{2}(g\otimes
g),\nu^{2}\rangle=0.$

As both summands are non-negative we obtain

$\langle Q_{1}(g\otimes g),\nu\rangle=0\quad\text{and}\quad\langle
Q_{2}(g\otimes g),\nu^{2}\rangle=0$

for all $g\geq 0$ which are $0$ on the support of $\nu$.

The second part of the lemma can be proved analogously. ∎

######  Lemma A.2.

Suppose that $D=C_{\Delta}(E)$ and suppose that $L$ satisfies the assumptions
of Lemma A.1 with the positive maximum principle on $M_{+}(E)$ instead of
$M_{+}(E^{\Delta})$. Then conditions (i)-(v) of Theorem 5.11 are satisfied.

###### Proof.

We first apply Lemma A.1 to obtain conditions (i), (iii), (iv), (vi), and
(vii) stated there. Since its domain is given by $C_{\Delta}(E)$, the operator
$B_{0}$ is automatically continuous (see (Schaefer, 1999, Theorem V.5.5)) and
the Riesz-Markov-Kakutani theorem (e.g. (Rudin, 1987, Theorem 2.14)) yields
the representation via a Radon measure $b\in M_{+}(E^{\Delta})$ given in
condition (i) of Theorem 5.11.

Condition (ii) of Theorem 5.11 follows by Lemma A.3. Next, in order to
establish the form of $Q_{0}$ observe that by linearity condition (iii) of
Lemma A.1 extends to the cone

$K_{+}:=\left\\{\sum_{i=1}^{k}\lambda_{i}(g_{i}\otimes
g_{i})\,|\,k\in\mathbb{N},\lambda_{i}\in\mathbb{R}_{+},0\leq g_{i}\in
D\right\\}.$

Since $g=g^{+}-g^{-}$ where $0\leq g^{\pm}\in C_{\Delta}(E)$, we have

$g\otimes g=\underbrace{2g^{+}\otimes g^{+}}_{\in
K_{+}}+\underbrace{2g^{-}\otimes g^{-}}_{\in
K_{+}}-\underbrace{(g^{+}+g^{-})\otimes(g^{+}+g^{-})}_{\in K_{+}}.$

This implies that $D\otimes D=K_{+}-K_{+}$ and $Q_{0}$ is therefore a
continuous linear functional and thus $0$ on the whole of $D\otimes D$. This
yields condition (iii) of Theorem 5.11.

Next, fix $\nu\in M_{+}(E)$ and consider $C^{\nu}:=\\{g\in
C_{\Delta}(E)\,|\,g\equiv 0\text{ on}\operatorname{supp}(\nu)\\}$. Then
$C^{\nu}$ is a linear subspace of $C_{\Delta}(E)$, which can be represented by
$C^{\nu}=C^{\nu}_{+}-C^{\nu}_{+}$, where

$C^{\nu}_{+}:=\\{0\leq g\in C_{\Delta}(E)\,|\,g\equiv 0\text{
on}\operatorname{supp}(\nu)\\}.$

Similarly for

$K^{\nu}:=\left\\{\sum_{i=1}^{k}\lambda_{i}(g_{i}\otimes
g_{i})\,|\,k\in\mathbb{N},\lambda_{i}\in\mathbb{R},g_{i}\in C^{\nu}\right\\}$

we have $K^{\nu}=K^{\nu}_{+}-K^{\nu}_{+}$, where

$K_{+}^{\nu}:=\left\\{\sum_{i=1}^{k}\lambda_{i}(g_{i}\otimes
g_{i})\,|\,k\in\mathbb{N},\lambda_{i}\in\mathbb{R}_{+},g_{i}\in
C^{\nu}_{+}\right\\}$

Therefore the non-negative linear functionals $g\mapsto\langle
Q_{1}(g),\nu\rangle$ and $g\mapsto\langle Q_{2}(g),\nu^{2}\rangle$ are
continuous and thus $0$ on $K^{\nu}$, proving that

$\langle Q_{1}(g\otimes g),\nu\rangle=0\qquad\text{ and }\qquad\langle
Q_{2}(g\otimes g),\nu^{2}\rangle=0,$

for all $\nu\in M_{+}(E)$ and $g\in D$ which are $0$ on the support of $\nu$.
Lemma A.4 and Lemma A.8 then yield condition (iv) and condition (v) of Theorem
5.11, respectively. ∎

######  Lemma A.3.

Let $B:C_{\Delta}(E)\to C_{\Delta}(E)$ be a linear operator. Then $B$
satisfies the positive minimum principle on $E$ if and only if there is map
$m\in C_{\Delta}(E)$ and a non-negative, finite kernel $\nu_{B}$ from $E$ to
$E^{\Delta}$ such that (5.10) holds. In this case, $B$ is bounded and
satisfies the positive minimum principle on $E^{\Delta}$. Moreover, there is
some non-negative (finite) measure $\nu_{B}(\Delta,\,\cdot\,)$ such that
(5.10) holds also for $x=\Delta$.

###### Proof.

As explained in Remark 1.2 setting $\overline{B}g(x):=B(g-g(x))(x)$ and
$m(x):=B1(x)$ we obtain that $Bg=\overline{B}g+mg$ for a map $m\in
C_{\Delta}(E)$ and an operator $\overline{B}$ satisfying $\overline{B}1=0$ and
the positive maximum principle on $E$. The claim then follows by Lemma C.2 in
Cuchiero et al. (2019). ∎

######  Lemma A.4.

Let $D\subseteq C_{\Delta}(E)$ be a dense linear subspace containing the
constant function $1$, and let $Q_{1}\colon D\otimes D\to C_{\Delta}(E)$ be a
linear operator. Then the following conditions are equivalent:

  1. (i)

$\langle Q_{1}(g\otimes g),\nu\rangle\geq 0$ for all $g\in D$ and $\nu\in
M_{+}(E)$ with equality if $g\equiv 0$ on the support of $\nu$.

  2. (ii)

$Q_{1}(g\otimes g)(x)\geq 0$ for all $g\in D$, $x\in E$ with equality if
$g(x)=0$.

Both imply that $Q_{1}$ is of form (5.11) for some function $\alpha\in
C_{\Delta}(E)$ with values in $\mathbb{R}_{+}$.

###### Proof.

Obviously assertion (ii) implies (i). Conversely take $\nu=\delta_{x}$ to
conclude $Q_{1}(g\otimes g)(x)\geq 0$ and with equality if $g(x)=0$ as the
support of $\delta_{x}$ is just $\\{x\\}$.

For $g\in D\otimes D$ we show now that $Q_{1}(g)(x)$ depends on $g$ through
its values at $(x,x)$. To this end, fix $x\in E$ and note that the map
$(g,h)\mapsto Q_{1}(g\otimes h)(x)$ is bilinear as well as positive
semidefinite as $Q_{1}(g\otimes g)(x)\geq 0$ for all $g\in D$. Hence it
satisfies the Cauchy–Schwarz inequality

$|Q_{1}(g\otimes h)(x)|\leq\sqrt{Q_{1}(g\otimes g)(x)}\,\sqrt{Q_{1}(h\otimes
h)(x)}.$

This together with (ii) implies that $Q_{1}(g\otimes h)(x)$ depends on $g$ and
$h$ only through their values at $x$.

Bilinearity then yields $Q_{1}(g\otimes h)(x)=g(x)h(x)\alpha(x)$ for some
$\alpha:E\to\mathbb{R}$. By non-negativity we also have that $0\leq
Q_{1}(g\otimes g)(x)=g(x)^{2}\alpha(x)$ and thus that $\alpha(x)\geq 0$. Since
$Q_{1}(g)\in C_{\Delta}(E)$ and as $1\in D$, $x\mapsto\alpha(x)$ lies
necessarily in $C_{\Delta}(E)$. Extending to all of $D\otimes D$ yields
(5.11).

∎

Let us now recall the definition of a positive semidefinite kernel.

######  Definition A.5.

A symmetric function $K:(E^{\Delta})^{2}\to\mathbb{R}$ is called _positive
semidefinite kernel_ on $E^{\Delta}$ if

$\sum_{i,j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})\geq 0$

holds for any $x_{1},\dots,x_{n}\in E^{\Delta}$,
$c_{1},\dots,c_{n}\in\mathbb{R}$ and $n\in\mathbb{N}$.

For our purposes we shall need the bigger set of copositive kernels, see,
e.g., Dobre et al. (2016); Kuryatnikova and Vera (2018). This generalizes the
notion of copositive matrices, i.e. symmetric matrices
$Q\in\mathbb{R}^{n\times n}$ such that $x^{\top}Qx\geq 0$ for all
$x\in\mathbb{R}_{+}^{n}$.

######  Definition A.6.

A symmetric function $K:E^{2}\to\mathbb{R}$ is called _copositive kernel_ on
$E$ if

$\sum_{i,j=1}^{n}c_{i}c_{j}K(x_{i},x_{j})\geq 0$

holds for any $x_{1},\dots,x_{n}\in E$, $c_{1},\dots,c_{n}\in\mathbb{R}_{+}$
and $n\in\mathbb{N}$.

######  Remark A.7.

In the following lemma copositive kernels naturally arise from the condition
$\langle Q_{2}(g\otimes g),\nu^{2}\rangle\geq 0$ for all $\nu\in M_{+}(E)$.
Note that even in the finite dimensional case a simple characterization of
copositive matrices is not available, see, e.g., Hiriart-Urruty and Seeger
(2010); Vershik (1987) Indeed, only up to dimension $4$, every copositive
matrix can be represented as sum of a positive semidefinite one and a matrix
with non-negative entries. For dimensions $n\geq 5$, this is no longer true
and copositive matrices are a strict superset thereof for which no simple
characterization is known.

In our case copositivity of $Q_{2}(g\otimes g)$ (and some further conditions)
translate to the requirements of the $(\beta,\pi)$-representation, where in
particular (5.6) is a rather implicit condition. It can be easily verified
when we have the decompostion of $Q_{2}(g\otimes g)$ into a positive
semidefinite kernel and a non-negative function (see Remark A.9 below).

######  Lemma A.8.

Let $D=C_{\Delta}(E)$ be a dense linear subspace containing the constant
function $1$, and let $Q_{2}\colon D\otimes D\to\widehat{C}_{\Delta}(E^{2})$
be a linear operator. Then the following conditions are equivalent:

  1. (i)

$\langle Q_{2}(g\otimes g),\nu^{2}\rangle\geq 0$ for all $g\in D$ and $\nu\in