# CTRW approximations for fractional equations with variable order

V. N. Kolokoltsov111Department of Statistics, University of Warwick, Coventry
CV4 7AL UK, ,222 National Research University Higher School of Economics,
Moscow 333Email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

The standard diffusion processes are known to be obtained as the limits of
appropriate random walks. These prelimiting random walks can be quite
different however. The diffusion coefficient can be made responsible for the
size of jumps or for the intensity of jumps. The ”rough” diffusion limit does
not feel the difference. The situation changes, if we model jump-type
approximations via CTRW with non-exponential waiting times. If we make the
diffusion coefficient responsible for the size of jumps and take waiting times
from the domain of attraction of an $\alpha$-stable law with a constant
intensity $\alpha$, then the standard scaling would lead in the limit of small
jumps and large intensities to the most standard fractional diffusion
equation. However, if we choose the CTRW approximations with fixed jump sizes
and use the diffusion coefficient to distinguish intensities at different
points, then we obtain in the limit the equations with variable position-
dependent fractional derivatives. In this paper we build rigorously these
approximations and prove their convergence to the corresponding fractional
equations for the cases of multidimensional diffusions and more general Feller
processes.

Key words: variable order fractional equations, continuous time random walks
(CTRW), subordinated Markov processes.

##  1 Introduction

The standard diffusion equation

$\frac{\partial u}{\partial
t}(t,x)=\frac{1}{2}a(x)\frac{\partial^{2}u}{\partial x^{2}}(t,x),$ (1)

with a positive continuous function $a(x)$, and the corresponding diffusion
process are known to be obtained as the limit of appropriate random walks.
These prelimiting random walks can be quite different however. For example,
approximating random walks in continuous time can be chosen as jump type
processes with the generators

$L_{h}f(x)=\frac{1}{2h^{2}}[f(x+h\sqrt{a(x)})+f(x-h\sqrt{a(x)})-2f(x)],$

or as jump type processes with the generators

$\tilde{L}_{h}f(x)=\frac{a(x)}{2h^{2}}[f(x+h)+f(x-h)-2f(x)],$

since both $L_{h}$ and $\tilde{L}_{h}$ tend to $(1/2)a(x)(d^{2}/dx^{2})$, as
$h\to 0$. In the first approximation, the diffusion coefficient $a(x)$ is
responsible for the size of jumps (with constant intensity), and in the second
approximation, it is responsible for the intensity of jumps (with constant
sizes). The ”rough” diffusion limit does not feel the difference. The
situation changes, if we model jump-type approximations via CTRW with non-
exponential waiting times. If we make the diffusion coefficient responsible
for the size of jumps and take waiting times from the domain of attraction of
an $\alpha$-stable law with a constant intensity $\alpha$, then the standard
scaling would lead (in the limit of small jumps and large intensities) to the
most standard fractional diffusion equation

$D^{\alpha}_{0+*}u(t,x)=\frac{1}{2}a(x)\frac{\partial^{2}u}{\partial
x^{2}}(t,x),$ (2)

where $D^{\alpha}_{0+*}$ is the so-called Caputo-Dzerbashyan fractional
derivative.

However, if we will use the CTRW approximations with fixed jump sizes and use
$a(x)$ to distinguish intensities at different points, then (as will be shown)
we get in the limit the equation with a variable position-dependent fractional
derivative:

$D^{\alpha a(x)}_{0+*}u(t,x)=\frac{1}{2}\frac{\partial^{2}u}{\partial
x^{2}}(t,x).$ (3)

Of course, for any decomposition $a(x)=b(x)c(x)$ with positive $b(x),c(x)$,
one can use $b(x)$ as a function controlling the intensity of jumps and $c(x)$
as a function controlling the spread of the jumps. In this scenario we get in
the limit the equation

$D^{\alpha b(x)}_{0+*}u(t,x)=\frac{1}{2}c(x)\frac{\partial^{2}u}{\partial
x^{2}}(t,x).$ (4)

From this point of view, all equations (4) are equally legitimate fractional
extensions of the diffusion equation (1).

Similarly (though technically more cumbersome), for a time dependent diffusion
equation

$\frac{\partial u}{\partial
t}(t,x)=\frac{1}{2}a(t,x)\frac{\partial^{2}u}{\partial x^{2}}(t,x),$ (5)

factorising $a(t,x)=b(t,x)c(t,x)$ we get by the CTRW approximation the
fractional equation with variable order, which is time- and position-
dependent:

$D^{\alpha b(t,x)}_{0+*}u(t,x)=\frac{1}{2}c(t,x)\frac{\partial^{2}u}{\partial
x^{2}}(t,x).$ (6)

In this paper we aim to develop these CTRW approximations in a general
multidimensional case. Namely, let us look at the CTRW approximations and
possible corresponding fractional limits for the diffusion processes generated
by the operator

$L_{dif}f(x)=\frac{1}{2}{\rm{tr}}\left(G(x)\frac{\partial^{2}f}{\partial
x^{2}}(x)\right)=\frac{1}{2}\sum_{ij}G_{ij}(x)\frac{\partial^{2}f}{\partial
x_{i}\partial x_{j}}(x),\quad x\in\mathbf{R}^{d},$ (7)

with $G(x)$ a positive $d\times d$-matrix, and stable-like processes generated
by the operator

$L_{st}^{\beta}f(x)=\int_{S^{d-1}}|(s,\nabla f(x)|^{\beta}\mu(x,s)\,ds,\quad
x\in\mathbf{R}^{d},$ (8)

with $\beta\in(0,2)$, $\mu(x,s)$ an even in $s$ positive function on
$\mathbf{R}^{d}\times S^{d-1}$ and $ds$ Lebesgue measure on the unit sphere
$S^{d-1}$. In particular, if $\mu(x,s)=1$ (that is, the spectral measure
$\mu(x,s)\,ds$ is uniform), then $L_{st}^{\beta}=\sigma|\Delta|^{\beta/2}$
(with $\sigma$ a constant depending on $d$) becomes a standard fractional
Laplacian. To avoid unnecessary repetitions, we shall speak about
$L_{st}^{\beta}$ with $\beta\in(0,2]$ assuming that for $\beta=2$ the
diffusion operator (7) is meant.

We shall look at the CTRW approximations arising from the equations

$\frac{\partial u}{\partial t}(t,x)=a(t,x)L_{st}^{\beta}u(t,x),$ (9)

with some positive function $a(x)$. We stress that the separation of a
multiplier $a(t,x)$ for a diffusion or a stable generator is not intrinsic, it
can be done in arbitrary way. Once the choice is made, we shall analyse the
jump-type approximations to the processes governed by $a(t,x)L_{st}^{\beta}$,
where $a(t,x)$ stands for the intensity of jumps and $L_{st}^{\beta}$ stands
for the distribution of jumps. We shall show that the limiting processes are
governed by the equations of type

$D^{\alpha a(t,x)}_{0+*}u(t,x)=L_{st}^{\beta}u(t,x),$

extending (6).

Our results are extendable to equations

$\frac{\partial u}{\partial t}(t,x)=a(t,x)Lu(t,x),$ (10)

with a generator of a general Feller process. We shall mostly work with (9),
where concrete assumptions on coefficients are easy to check, and then comment
on regularity assumptions required for a general $L$.

The CTRW models summing independent terms were analysed mathematically in many
papers, see e. g. [13], following their development in physics literature, see
e.g. [18]. The CTRW approximation for general models of Markov processes
subordinated by their monotone coordinate turning in the scaling limit to a
generalised fractional equation (of variable order) were seemingly first
discussed mathematically in [7]. There is now a heavy body of literature on
the variable order fractional equations. We are not trying to review it here,
but rather refer to extensive reviews [2] and [17] dealing with both
mathematical and applied questions. Let us mention specifically paper [16],
where CTRW approximations to equation (3) were discussed in detail, and paper
[15], with careful derivation of well-posedness for rather general variable
order fractional equations. As basic references to general rules of fractional
calculus and its applications we mention books [5], [14], [1].

The rest of the paper is organised as follows. In Section 2 we introduce
carefully various CTRW approximations for diffusion or stable-like processes,
when their coefficients can be used either to distinguish the sizes of jumps
or their intensities. In Section 3 our main results are formulated devoted to
the convergence of these approximations to processes solving fractional
equations of variable order, and to the explicit representations of the
solutions of these equations. The main result is actually Theorem 3.2 on the
CTRW approximations and specific representation for their limit. The proof is
based on the ideas from [7], where they were presented however rather sketchy.
Theorem 3.3 is essentially known. Its detailed proof is given in [15]. We
present here a different proof adding two details. Namely, our equation is a
bit more general (variable fractional order depends on both space and time),
and moreover we prove well-posedness of classical solutions (unlike only the
mild solutions of [15]). In Section 4 we derive a handy general representation
for the distributions of Markov processes subordinated by its monotone
coordinate. It supplies an important ingredient to the proof of our main
results, but may be of independent interest. Section 5 presents a well known
result on the convergence of the standard CTRWs, but enhanced by the exact
rates of (weak) convergence. Sections 6 \- 9 contain proofs of our main
results.

Notice finally that similar approach to interacting particle systems leading
to the new variable order fractional kinetic equations was developed in [11]
and [12]. Quite different variable order kinetic equations are analysed in
[3].

We shall use standard notations for the spaces of smooth function:
$C(\mathbf{R}^{d})$ the Banach space of continuous functions equipped with
sup-norm, $C^{k}(\mathbf{R}^{d})$, $k\in\mathbf{N}$, the Banach space of
$k$-times continuously differentiable functions with bounded derivative of
order up to $k$, $C_{\infty}(\mathbf{R}^{d})$ the closed subspace of
$C(\mathbf{R}^{d})$ consisting of functions vanishing at infinity,
$C^{k}_{\infty}(\mathbf{R}^{d})$ the closed subspace of
$C^{k}(\mathbf{R}^{d})$ consisting of functions such that all its derivatives
up to order $k$ belong to $C(\mathbf{R}^{d})$.

##  2 CTRW approximation

We are looking for the CTRW approximations to the equations (9).

For the operator $L_{st}^{\beta}$, we choose standard random walk
approximations

$L_{st,\tau}^{\beta}f(x)=\frac{1}{\tau}\int(f(x+\tau^{1/\beta}z)-f(x))p(x,dz),$
(11)

with $p(x,dz)$ a symmetric probability kernel in $\mathbf{R}^{d}$ (i.e. $\int
f(z)p(x,dz)=\int f(-z)p(x,dz)$), so that

$\sup_{x}|(L_{st}^{\beta}-L_{st,\tau}^{\beta})f(x)|\to
0,\quad\text{as}\quad\tau\to 0,$ (12)

uniformly for $f$ from any bounded subset of $C^{2}(\mathbf{R}^{d})$.

The operator $L_{st,\tau}^{\beta}$ generates a jump-type Markov process with
the intensity $\tau^{-1}$ and the distribution of jumps given by $p(x,dz)$.

######  Remark 1.

Of course, such approximations are not unique. The existence is however more
or less obvious. For the diffusion operator (7) the measures $p(x,dz)$ can be
(and usually are) taken as discrete, for instance, of the type

$\sum_{i}(\delta_{a_{i}e_{i}}+\delta_{-a_{i}e_{i}})+\sum_{i>j}(\delta_{b_{ij}(e_{i}+e_{j})}+\delta_{-b_{ij}(e_{i}+e_{j})}),$

with some numbers $a_{i},b_{ij}$, where $\\{e_{i}\\}$ is the standard basis in
$\mathbf{R}^{d}$. For the stable generator one can take as $p(x,dz)$ a
probability measure with an appropriate power tail, see e.g. [13] or Section
8.3 from [8].

For a generator $L$ of an arbitrary Feller process with a core
$C^{k}_{\infty}(\mathbf{R}^{d})$, where $k$ equals one or two, on can use an
arbitrary approximation $L_{\tau}$ of the type

$L_{\tau}f(x)=\frac{1}{\tau}\int(f(x+z)-f(x))p_{\tau}(x,dz),$ (13)

with $p_{\tau}(x,dz)$ a family of probability kernels in $\mathbf{R}^{d}$ such
that

$\sup_{x}|(L-L_{\tau})f(x)|\to 0,\quad\text{as}\quad\tau\to 0,$ (14)

uniformly for $f$ from any bounded subset of $C^{k}_{\infty}(\mathbf{R}^{d})$.

We shall now construct a process with waiting times having power tails. For
exponential waiting times, the intensity $a(s,x)$ governs the tail
distribution $\mathbf{P}(T>t)=e^{-ta(s,x)\tau}$ with a scaling parameter
$\tau$. Hence, for the power tail the analogous dependence will be the power
law

$\mathbf{P}(T>t)\sim\frac{1}{a(s,x)\alpha}t^{-a(s,x)\alpha},\quad t\to\infty.$
(15)

To simplify presentation we shall make (15) more precise. Namely, we assume
that these distributions have continuous densities $Q_{s,x}(r)$ such that

$Q_{s,x}(r)=r^{-1-a(s,x)\alpha}\,\,\text{for}\,\,r\geq
B,\,\,\text{and}\,\,Q_{s,x}(r)\leq 1\,\,\text{for all}\,\,r,$ (16)

with some $B>0$ uniformly for all $s,x$.

######  Remark 2.

(i) The coefficients at the power of $t$ in (15) are chosen for convenience.
If not used here, then it would appear in (16) and in the limiting equation.
(ii) Simplification arising from (16) is not essential and can be dispensed
with.

As usual for CTRW modelling, we start by building an auxiliary scaled enhanced
Markov chain with spatial jumps distributed according to $p(x,dz)$ and with an
additional coordinate $r$ specifying the total waiting times. In the usual
CTRW scaling (see e.g. [13] and [8]) one scales discrete times by $\tau$ and
the waiting times by the multiplier $\tau^{1/(\alpha a(s,x))}$ (see also
(41)). Therefore, we define the Markov chain
$(X^{\tau}_{x,s},S^{\tau}_{x,s})(k\tau)$ evolving in discrete times $\tau k$,
$k\in\mathbf{N}$, such that $(X^{\tau}_{x,s},S^{\tau}_{x,s})(0)=(x,s)$ and, if
a state of the chain in some discrete time $k\tau$ is $(q,v)$, then in the
next moment $(k+1)\tau$ it will turn to the state

$(q+\tau^{1/\beta}y,v+\tau^{1/(\alpha a(v,q))}r),$

with $y$ distributed according to $p(x,dy)$ and $r$ distributed according to
$Q_{v,q}$. We obtain the Markov chain with transition operators in time $\tau$
given by the formula

$U^{\tau}F(x,s)=\int_{\mathbf{R}_{+}}\int_{\mathbf{R}^{d}}F(x+\tau^{1/\beta}y,s+\tau^{1/(\alpha
a(s,x))}r)Q_{s,x}(r)\,dr\,p(x,dy).$ (17)

When working with an arbitrary generator $L$ (rather than more concrete
$L^{\beta}_{st}$) the corresponding chain is defined by the equation

$U^{\tau}F(x,s)=\int_{\mathbf{R}_{+}}\int_{\mathbf{R}^{d}}F(x+y,s+\tau^{1/(\alpha
a(s,x))}r)Q_{s,x}(r)\,dr\,p_{\tau}(x,dy).$ (18)

We are interested in the value of the first coordinate $X^{\tau}_{x,s}$
evaluated at the random time $k\tau$ such that the total waiting time
$S^{\tau}_{x,s}(k\tau)$ reaches $t$, that is, at the time

$k\tau=T_{x,s}^{\tau}(t)=\inf\\{m\tau:S^{\tau}_{x,s}(m\tau)\geq t\\},$

so that $T_{x,s}^{\tau}$ is the inverse process to $S^{\tau}_{x,s}$. We define
the scaled CTRW approximation for the process with the intensity of jumps
governed by $a(x)$ and the distribution of jumps governed by $p(x,dy)$ as the
(non-Markovian) process

$\tilde{X}^{\tau}_{x,s}(t)=X^{\tau}_{x,s}(T_{x,s}^{\tau}(t)).$ (19)

We are going to identify the weak limit of the process
$\tilde{X}^{\tau}_{x,s}(t)$, as $\tau\to 0$, and to show that the limiting
processes are governed by the equations extending (6). As an auxiliary step,
we shall identify the weak limit of the Markov chain
$(X^{\tau}_{x,s},S^{\tau}_{x,s})$, that is, the weak limit of the chains with
transitions $[U^{\tau}]^{[t/\tau]}$, where $[t/\tau]$ denotes the integer part
of the number $t/\tau$, as $\tau\to 0$. It is known (see e.g. Theorem 19.28 of
[4] or Theorem 8.1.1 of [8]) that if such chain converges to a Feller process,
then the generator of this limiting process can be obtained as the limit

$\Lambda F=\lim_{\tau\to 0}\frac{1}{\tau}(U^{\tau}F-F).$ (20)

##  3 Main results

Let us assume the strict non-degeneracy condition:

$g_{1}\leq G(x)\leq g_{2},\,\,\text{or}\,\,m_{1}\leq m(x,s)\leq m_{2}$ (21)

for the spatial part and

$a_{1}\leq a(t,x)\leq a_{2}$ (22)

for the temporal part, with some positive constants $g_{i},m_{i},a_{i}$ such
that $a_{2}\alpha<1$, where the first inequality (21) is understood in the
sense of matrix, and the smoothness condition:

$a(.),G(.),m(.,s)\in C^{4}(\mathbf{R}^{d}),$ (23)

with $m(.,s)$ having a bounded norm uniformly in $s$. Let us assume (12) for
the approximating jumps and (16) for the distributions of waiting times.

######  Theorem 3.1.

(i) For $F\in C^{2}_{\infty}(R^{d+1})$ the limit in (20) exists and

$\Lambda F(x,s)=\lim_{\tau\to 0}\frac{1}{\tau}(U^{\tau}F-F)(x,s)$
$=\int_{0}^{\infty}\frac{F(x,s+r)-F(x,s)}{r^{1+\alpha
a(s,x)}}dr+L_{st}^{\beta}F(x,s).$ (24)

(ii) The operator $\Lambda$ defined by the r.h.s. of (24) generates a
(conservative time-homogeneous) Feller process $(X_{x,s},S_{x,s})(t)$ in
$\mathbf{R}^{d+1}$ (with the initial condition $(X_{x,s},S_{x,s})(0)=(x,s)$)
with continuous (in fact, even smooth) transition probability density
$G(t;x,s;y,v)$, $t>0$, and the corresponding Feller semigroup in
$C_{\infty}(\mathbf{R}^{d+1})$ such that the space
$C^{2}_{\infty}(\mathbf{R}^{d+1})$ is its invariant core. The process
$(X_{x,s},S_{x,s})(t)$ belongs to the class of processes usually referred to
as stable-like processes.

(iii) The chains with transitions $[U^{\tau}]^{[t/\tau]}$, with $U^{\tau}$
given by (17), converge weakly to the process $(X_{x,s},S_{x,s})(t)$, and the
operators $[U^{\tau}]^{[t/\tau]}$ converge to the transition operators of the
process $(X_{x,s},S_{x,s})(t)$ strongly and uniformly for compact intervals of
time.

######  Theorem 3.2.

(i) The marginal distributions of the scaled CTRW (19) converge to the
marginal distributions of the process

$\tilde{X}_{x,s}(t)=X_{x,s}(T_{x,s}(t)),$ (25)

where

$T_{x,s}(t)=\inf\\{r:S_{x,s}(r)\geq t\\},$

that is, for a bounded continuous function $F(x)$, it holds that

$\lim_{\tau\to
0}\mathbf{E}F(\tilde{X}^{\tau}_{x,s}(t))=\mathbf{E}F(\tilde{X}_{x,s}(t)).$
(26)

(ii) For any $F\in C_{\infty}(\mathbf{R}^{d})$ and arbitrary $K>0$,

$\mathbf{E}[F(\tilde{X}_{x,s}(t))\mathbf{1}(T_{x,s}(t)\in[1/K,K])]$ $=\int
dy\int_{1/K}^{K}du\int_{s}^{t}dv\frac{(t-v)^{-a(v,y)\alpha}}{a(v,y)\alpha}\,G(u;x,s;y,v)F(y),$
(27)

where $\theta_{\geq t}$ is the indicator function of the interval
$[t,\infty)$.

This formula extends to infinite $K$, that is

$\mathbf{E}[F(\tilde{X}_{x,s}(t))]=\int
dy\int_{0}^{\infty}du\int_{s}^{t}dv\frac{(t-v)^{-a(v,y)\alpha}}{a(v,y)\alpha}\,G(u;x,s;y,v)F(y).$
(28)

######  Theorem 3.3.

For any $F\in C^{2}(\mathbf{R}^{d})$, the evolution of averages
$F(x,s)=\mathbf{E}F(\tilde{X}_{x,s}(t))$ represents the unique solution to the
mixed fractional differential equation

$D_{t-*}^{\alpha a(s,x)}F(x,s)=L_{st}^{\beta}F(x,s),\quad s\in[0,t],$ (29)

with the terminal condition $F(x,t)=F(x)$, where the right fractional
derivative (of Caputo-Dzerbashyan type) acting on the variable $s\leq t$ of
$F(x,s)$ is defined as

$D_{t-*}^{\alpha a(s,x)}g(s)=-\int_{0}^{t-s}\frac{g(s+r)-g(s)}{r^{1+\alpha
a(s,x)}}dy-(g(t)-g(s))\frac{(t-s)^{-\alpha a(s,x)}}{a(s,x)\alpha}.$ (30)

######  Remark 3.

The limiting equation (30) is written in terms of the right fractional
derivative, because we have chosen the forward motion for the auxiliary
coordinate $S_{s,x}$ of the Markov process $(X_{x,s},S_{x,s})(t)$. The
analogous equations with left derivatives, like in (4), can be obtained by
changing the direction of $S_{s,x}$.

######  Remark 4.

For the case of a general operator $L$ the results above remain true (with $L$
instead of $L_{st}^{\beta}$) if we assume that the process generated by
$\Lambda$ from (24) has a continuous transition probability density, which is
sufficiently smooth so that smooth functions form a core for its Feller
semigroup, and which has integrability condition for small and large times
ensuring convergence of the integral in (27).

##  4 A general representation for a subordination

In this section we derive a representation for Markov processes subordinated
by its monotone coordinate. It supplies the proof for the part (ii) of Theorem
3.1, but may be of independent interest.

######  Lemma 4.1.

Let $Y_{s}$ be an adapted process on a stochastic basis
$(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ and $\Sigma$ a stopping time. Let the
pairs $(Y_{s},\Sigma)$ have joint densities $g_{s}(y,\sigma)$ for $s>0$ such
that, for $s$ from any bounded interval separated from zero, $g_{s}(y,\sigma)$
is bounded and right continuous in $s$. Then, for any $K>0$ and a continuous
bounded function $F(y)$,

$\mathbf{E}[F(Y_{\Sigma})\mathbf{1}(\Sigma\in[1/K,K])]=\int
dy\int_{1/K}^{K}d\sigma\,g_{\sigma}(y,\sigma)F(y).$ (31)

###### Proof.

Let us choose the discrete approximations $\Sigma_{\tau}=\tau[\Sigma/\tau+1]$
(square brackets denoting the integer part), so that $\Sigma_{\tau}=k\tau$
when $(k-1)\tau\leq\Sigma<k\tau$. Thus $\Sigma_{\tau}$ are right continuous
stopping times such that $\Sigma_{\tau}\geq\Sigma$ for all $\tau$, they depend
monotonically on $\tau$ and $\Sigma_{\tau}\to\Sigma$, as $\tau\to 0$. Then we
can write

$\mathbf{E}[F(Y_{\Sigma})\mathbf{1}(\Sigma\in[1/K,K])]=\lim_{\tau\to
0}\mathbf{E}[F(Y_{\Sigma_{\tau}})\mathbf{1}(\Sigma\in[1/K,K])]$
$=\lim_{\tau\to
0}\sum_{k=1}^{\infty}\mathbf{E}[F(Y_{\Sigma_{\tau}})\mathbf{1}(\Sigma\in[1/K,K])\mathbf{1}(\Sigma_{\tau}=k\tau)]$
$=\lim_{\tau\to
0}\sum_{k=1}^{\infty}\mathbf{E}\left[F(Y_{k\tau})\mathbf{1}(\Sigma\in[1/K,K])\mathbf{1}(\Sigma\in[(k-1)\tau,k\tau))\right]$
$=\lim_{\tau\to 0}\int
dy\int_{1/K}^{K}d\sigma\,g_{\sigma_{\tau}}(y,\sigma)F(y).$

And therefore

$\mathbf{E}[F(Y_{\Sigma})\mathbf{1}(\Sigma\in[1/K,K])]=\int
dy\int_{1/K}^{K}d\sigma\,g_{\sigma}(y,\sigma)F(y)$ $+\lim_{\tau\to 0}\int
dy\int_{1/K}^{K}d\sigma\,(g_{\sigma_{\tau}}-g_{\sigma})(y,\sigma)F(y).$

Since the last term tends to zero (by the right continuity of $g$ and
dominated convergence), we obtain (31).

∎

######  Remark 5.

(i) To get the same without the restrictions $[1/K,K]$ one needs some
integrability conditions for $g_{s}$ for small and large $s$. (ii) The
existence of the density $g_{s}(y,\sigma)$ can be weaken to the condition

$\mathbf{P}(Y_{s}\in dy,\Sigma\in d\sigma)=g_{s}(\sigma,dy)d\sigma$

such that $g_{s}(\sigma,dy)$ weakly right continuous in $s$ uniformly in
$\sigma$ (on bounded intervals).

######  Lemma 4.2.

Let $(Y_{s},V_{s})$, $Y\in\mathbf{R}^{d}$, $V\in\mathbf{R}$ be a Markov
process with the generator $L+A$, where $L$ acts on the first variable and
does not depend on the second one (so that $Y_{s}$ is itself a Markov
process), and $A$ acts on the second variable, but may have coefficients
depending on the first variable. Let the process $(Y_{s},V_{s})$ have a
continuous (for $s>0$) transition density $G(s;y_{0},v_{0};y,v)$ and let
$G_{Y}(s,y_{0},y)$ be the transition density for the process $Y$ so that

$G_{Y}(u,y_{0},y)=\int G(u;y_{0},v_{0};y,v)dv$

for any $v_{0}$.

Then the random vector $(Y_{s},V_{u})$ has a density
$\phi_{y_{0},v_{0}}(s,u;y,v)$ such that, for $s>u>0$,

$\phi_{y_{0},v_{0}}(s,u;y,v)=\int G_{Y}(s-u,p,y)G(u;y_{0},v_{0};p,v)\,dp.$
(32)

Moreover,

$\frac{\partial}{\partial u}\phi_{y_{0},v_{0}}(s,u;y,v)=\int
G_{Y}(s-u,p,y)A^{*}_{(v;p)}G(u;y_{0},v_{0};p,v)\,dp,$ (33)

where $A^{*}$ denotes the adjoint operator to $A$ and the lower index $(v;p)$
indicates that this operator acts on the variable $v$, while the value of the
spacial variable is $p$.

######  Remark 6.

Generally speaking, formula (33) may hold only in the sense of generalized
functions. To ensure that it holds as an equation for functions, some
additional mild regularity for transitions must be assumed.

###### Proof.

For a bounded continuous function $F$, we have

$\mathbf{E}F(Y_{s},V_{u})=\int\mathbf{E}(F(Y_{s},V_{u})|Y_{u}=p,V_{u}=v)G(u;y_{0},v_{0};p,v)\,dpdv$
$=\int\mathbf{E}(F(Y_{s},v)|Y_{u}=p)G(u;y_{0},v_{0};p,v)\,dpdv=\int
F(y,v)G_{Y}(s-u,p,y)G(u;y_{0},v_{0};p,v)\,dydpdv,$

implying (32).

Next,

$\frac{\partial}{\partial
u}\phi_{y_{0},v_{0}}(s,u;y,v)=-\int(L_{(p)}G_{Y}(s-u,p,y))G(u;y_{0},v_{0};p,v)\,dp$
$+\int G_{Y}(s-u,p,y))(L^{*}_{(p)}+A^{*}_{(v;p)})G(u;y_{0},v_{0};p,v)\,dp.$

The key observation is that the terms with $L_{(p)}$ cancel implying (33). ∎

######  Lemma 4.3.

Under the assumption of Lemma 4.2, let

$A\phi(v)=\int_{0}^{\infty}(\phi(v+w)-\phi(v))\nu(v,w;p)\,dw,$ (34)

where $p$ is the value of the spatial variable, where $\nu$ is a strictly
positive function for $w>0$ such that

$\sup_{v,p}\int\min(1,w)\nu(v,w;p)\,dw<\infty.$

Then the coordinate $V_{t}$ is a strictly increasing process a.s., so that its
generalized inverse (or hitting times) process

$Z_{t}=\sup\\{u\geq 0:V(u)\leq t\\}=\inf\\{u\geq 0:V(u)>t\\},\quad t>v_{0},$

is continuous; and for the density $g_{y_{0},v_{0}}(s,t;y,z)$ of the pair
$(Y_{s},Z_{t})$, $s>0$, $t>v_{0}$, the following formula holds:

$g_{y_{0},v_{0}}(s,t;y,z)=\frac{\partial}{\partial
z}\int_{t}^{\infty}\phi_{y_{0},v_{0}}(s,z;y,v_{0})\,dv$
$=\int_{t}^{\infty}dv\int_{\mathbf{R}^{d}}dp\,G_{Y}(s-z,p,y)A^{*}_{(v;p)}G(z;y_{0},v_{0};p,v).$
(35)

In particular,

$\lim_{z\to
s}g_{y_{0},v_{0}}(s,t;y,z)=\int_{t}^{\infty}A^{*}_{(v;y)}G(s;y_{0},v_{0};y,v)\,dv.$
(36)

###### Proof.

The fact that $V_{t}$ is strictly increasing follows from (34), which in turn
implies the continuity of $Z_{t}$.

Consequently, since $(Z_{t}\leq z)=(V_{z}\geq t)=(V_{z}>t)$ a.s., it follows
that

$g_{y_{0},v_{0}}(s,t;y,z)=\frac{\partial^{2}}{\partial y\partial
z}\mathbf{P}(Y_{s}\leq y,V_{z}\geq t)$ $=\frac{\partial^{2}}{\partial
z\partial
y}\int_{-\infty}^{y}\int_{t}^{\infty}\phi_{y_{0},v_{0}}(s,z;\tilde{y},v)\,dvdy=\frac{\partial}{\partial
z}\int_{t}^{\infty}\phi_{y_{0},v_{0}}(s,z;y,v)\,dv.$

Substituting (33), we get (35). ∎

######  Theorem 4.1.

Under the assumptions of Lemma 4.2

$\mathbf{E}[F(Y_{Z_{t}})\mathbf{1}(\mathbf{Z}_{t}\in[1/K,K])]=\int
dy\int_{1/K}^{K}ds\int_{t}^{\infty}dv\,A^{*}_{(v;y)}G(s;y_{0},v_{0};y,v)F(y),$
(37)

and also

$\mathbf{E}[F(Y_{Z_{t}})\mathbf{1}(\mathbf{Z}_{t}\in[1/K,K])]=\int
dy\int_{1/K}^{K}ds\int_{0}^{t}dv\,(A_{(v;y)}\theta_{\geq
t})(v)G(s;y_{0},v_{0};y,v)F(y)$ $=\int
dy\int_{1/K}^{K}ds\int_{v_{0}}^{t}dv\int_{t-v}^{\infty}\nu(v,w;y)\,dw\,G(s;y_{0},v_{0};y,v)F(y),$
(38)

where $\theta_{\geq t}$ is the indicator function of the interval
$[t,\infty)$.

######  Remark 7.

(i) Formula (37) in this general form was seemingly first derived in [7]
(though the proof was a bit sketchy there) and was reproduced in monograph
[8]. The advantage of its modification (38) is that for its validity only the
continuity of the density $G(s;y_{0},v_{0};y,v)$ is required (and even this
condition can be weaken), and not any smoothness that may be needed to make
sense out of $A^{*}_{(v;y)}G(s;y_{0},v_{0};y,v)$.

(ii) It follows that the density of the random variable $Y_{Z_{t}}$ equals

$\int_{0}^{\infty}ds\int_{v_{0}}^{t}dv\,(A_{(v;y)}\theta_{\geq
t})(v)\,G(s;y_{0},v_{0};y,v),$ (39)

whenever this integral is well defined.

###### Proof.

Formula (37) follows from (36) and (31). Next, by duality (effectively by the
shift of the variable of integration),

$\int_{t}^{\infty}A^{*}_{(v;y)}G(s;y_{0},v_{0};y,v)\,dv=\int_{-\infty}^{\infty}(\theta_{\geq
t})(v)A^{*}_{(v;y)}G(s;y_{0},v_{0};y,v)\,dv$
$=\int_{-\infty}^{\infty}(A_{(v;y)}\theta_{\geq
t})(v)G(s;y_{0},v_{0};y,v)\,dv=\int_{v_{0}}^{t}dv\int_{t-v}^{\infty}\nu(v,w;y)\,dw\,G(s;y_{0},v_{0};y,v),$

implying (38). ∎

##  5 On the rate of convergence for the standard CTRW

It is well known (see e.g. [13] and [8]) that the Markov chains with jumps
distributed like (15) converge after appropriate scaling to stable
subordinators. In the following result we give exact rates of weak convergence
for the corresponding generators under (15).

######  Proposition 5.1.

Let $p(y)$ be a probability density on $\mathbf{R}_{+}$ such that
$p(y)=y^{-1-\alpha}$ for $y\geq B$ with some $\alpha\in(0,1)$ and $B>0$. Then,
for any continuous $f$ on $\mathbf{R}_{+}$ such that $f(0)=f(\infty)=0$ and
$f$ is Lipschitz at zero so that $|f(y)|\leq Ly$ for $y\in[0,Bh]$ and some
constant $L$, it follows that

$\left|h^{-\alpha}\int_{0}^{\infty}f(hy)p(y)dy-\int_{0}^{\infty}\frac{f(y)dy}{y^{1+\alpha}}\right|\leq
C_{B}Lh^{1-\alpha},$ (40)

with

$C_{B}=\frac{B^{1-\alpha}}{1-\alpha}+\int_{0}^{B}yp(y)dy.$

######  Remark 8.

If, instead of assuming Lipschitz at zero, one assumes Hölder continuity at
zero, that is $|f(y)|\leq Ly^{\beta}$ for $y\in[0,Bh]$ and some $L>0$,
$\beta>\alpha$, then inequality (40) holds with the r.h.s.

$\left(\frac{B^{\beta-\alpha}}{\beta-\alpha}+\int_{0}^{B}y^{\beta}p(y)dy\right)Lh^{\beta-\alpha}.$

###### Proof.

Let $f=\mathbf{1}_{\geq C}$ with $C\geq Bh$. Then

$h^{-\alpha}\int_{0}^{\infty}f(hy)p(y)dy=h^{-\alpha}\int_{C/h}^{\infty}p(y)dy$
$=h^{-\alpha}\alpha^{-1}(C/h)^{-\alpha}=\alpha^{-1}C^{-\alpha}=\int_{C}^{\infty}\frac{dy}{y^{1+\alpha}}=\int_{0}^{\infty}\frac{f(y)\,dy}{y^{1+\alpha}}.$

Hence by linearity and the density argument it follows that

$h^{-\alpha}\int_{0}^{\infty}f(hy)p(y)dy=\int_{0}^{\infty}\frac{f(y)dy}{y^{1+\alpha}},$

for any bounded measurable $f$ having support on $[Bh,\infty)$. Next let $f$
have support on $[0,Bh]$. Then

$\left|h^{-\alpha}\int_{0}^{\infty}f(hy)p(dy)\right|=\left|h^{-\alpha}\int_{0}^{B}f(hy)p(y)dy\right|\leq
h^{1-\alpha}L\int_{0}^{B}yp(y)dy,$

and

$\left|\int_{0}^{\infty}\frac{f(y)dy}{y^{1+\alpha}}\right|=\left|\int_{0}^{Bh}\frac{f(y)dy}{y^{1+\alpha}}\right|\leq
L\int_{0}^{Bh}\frac{dy}{y^{\alpha}}=\frac{B^{1-\alpha}}{1-\alpha}Lh^{1-\alpha},$

implying (40). ∎

In particular, setting $\tau=h^{\alpha}$, it follows that

$\left|\tau^{-1}\int_{0}^{\infty}(f(x\pm\tau^{1/\alpha}y)-f(x))p(y)dy-\int_{0}^{\infty}\frac{(f(x\pm
y)-f(x))dy}{y^{1+\alpha}}\right|\leq C_{B}L\tau^{(1/\alpha)-1},$ (41)

where $L$ is the sup of the derivative of $f$ near $x$.

##  6 Proof of Theorem 3.1

(i) We have

$\frac{1}{\tau}(U^{\tau}F-F)(x,s)$
$=\frac{1}{\tau}\int_{\mathbf{R}_{+}}\int_{\mathbf{R}^{d}}[F(x+\tau^{1/\beta}y,s+\tau^{1/(\alpha
a(s,x))}r)-F(x,s)]Q_{s,x}(r)\,dr\,p(x,dy).$
$=\frac{1}{\tau}\int_{\mathbf{R}_{+}}[F(x,s+\tau^{1/(\alpha
a(s,x))}r)-F(x,s)]Q_{s,x}(r)\,dr+\frac{1}{\tau}\int_{\mathbf{R}^{d}}[F(x+\tau^{1/\beta}y,s)-F(x,s)]p(x,dy)+R,$
(42)

where

$R=\frac{1}{\tau}\int_{\mathbf{R}_{+}}\int_{\mathbf{R}^{d}}[g_{y}(x,s+\tau^{1/(\alpha
a(s,x))}r)-g_{y}(x,s)]Q_{s,x}(r)\,dr\,p(x,dy),$

with

$g_{y}(x,s)=F(x+\tau^{1/\beta}y,s)-F(x,s).$

By (41), the first term in (42) converges, as $\tau\to 0$, to

$\int_{0}^{\infty}\frac{F(x,s+r)-F(x,s)}{r^{1+\alpha a(s,x)}}dr,$

whenever $F$ is continuously differentiable in $s$. By (12), the second term
in (42) converges, as $\tau\to 0$, to $L_{st}^{\beta}F(x,s)$. Again by (41)
and (12) it follows that $R\to 0$, as $\tau\to 0$, implying (24).

(ii) The operator $L$ is the sum of two terms, each one of them generates a
Feller semigroup on $C_{\infty}(\mathbf{R}^{d+1})$ such that the spaces
$C^{2}_{\infty}(\mathbf{R}^{d+1})$ and $C^{4}_{\infty}(\mathbf{R}^{d+1})$
represent its invariant core. The fact that $L$ generates a Feller semigroup
on $C_{\infty}(\mathbf{R}^{d+1})$ with an invariant core
$C^{2}_{\infty}(\mathbf{R}^{d+1})$ follows from Theorems 5.2.1 and 5.2.2 of
[10]. On the other hand, there is a well known technique of building
transition probabilities for stable-like processes with the generators of type
(24). Though the author is unaware of any result for generator (24) exact,
technique of [6] (see also Chapter 7 of [8]) is fully applicable yielding the
existence of 4 times differentiable transition probabilities $G(t;x,s;y,v)$,
$t>0$, for the process $(X_{x,s},S_{x,s})(t)$. This provides also an
alternative proof of the statement about the cores.

(iii) It is a direct consequence of (i),(ii) and the standard general result
on the convergence of Markov chains, see e.g. Theorem 19.28 of [4].

##  7 Subordination for discrete Markov chains

As a starting point for the proof of Theorem 3.2, we give here some
preliminary calculations for the subordinated Markov chains in discrete times.

Let $Y_{k\tau}$ be an adapted process on a stochastic basis
$(\Omega,\mathcal{F},\mathcal{F}_{t},P)$ and $\Sigma$ a stopping time with
values in $\\{k\tau\\}$, $k\in\mathbf{N}$. Let the random variables
$Y_{m\tau}\mathbf{1}(\Sigma=k\tau))$ have densities $g_{m\tau}(y,k\tau)$.
Then, for any $K>0$ and a continuous bounded function $F(y)$,

$\mathbf{E}[F(Y_{\Sigma})\mathbf{1}(\Sigma\in[1/K,K])]=\sum_{k\tau\in[1/K,K]}\int\,g_{k\tau}(y,k\tau)F(y)dy.$
(43)

Let $(Y_{m\tau},V_{m\tau})$, $Y\in\mathbf{R}^{d}$, $V\in\mathbf{R}$, be a
Markov chain with the transition operator $U^{\tau}$:

$U^{\tau}F(y_{0},v_{0})=\int F(y,v)P_{\tau}(y_{0},v_{0};y,v)\,dydv=\int
F(y,v)P_{\tau}(y_{0},v_{0};dy,dv),$

such that $Y_{m\tau}$ is a Markov chain on its own with the transition
$U^{\tau}_{Y}$.

Then the density of the random vector $(Y_{k\tau},V_{k\tau},V_{(k-1)\tau})$ is

$\int P_{\tau}(p,v;y,w)P_{(k-1)\tau}(y_{0},v_{0};p,v)\,dp,$ (44)

because

$\mathbf{E}F(Y_{k\tau},V_{k\tau},V_{(k-1)\tau})$
$=\int\mathbf{E}(F(Y_{k\tau},V_{k\tau},v)|Y_{(k-1)\tau}=p,V_{(k-1)\tau}=v)P_{(k-1)\tau}(y_{0},v_{0};p,v)\,dpdv$
$=\int F(y,w,v)P_{\tau}(p,v;y,w)P_{(k-1)\tau}(y_{0},v_{0};p,v)\,dpdvdwdy.$

Let the coordinate $V_{k\tau}$ be strictly increasing, and let

$Z_{t}^{\tau}=\sup\\{m\tau:V(m\tau)\leq t\\}=\inf\\{m\tau:V(m\tau)>t\\},\quad
t>v_{0},$

be its (generalized) inverse (or hitting times) process.

Then

$\mathbf{E}[F(Y_{Z_{t}^{\tau}})\mathbf{1}(Z_{t}^{\tau}\in[1/K,K])]=\sum_{k\tau\in[1/K,K]}F(Y_{k\tau})\mathbf{1}(Z^{\tau}_{t}=k\tau)=\sum_{k\tau\in[1/K,K]}F(Y_{k\tau})\mathbf{1}(V_{(k-1)\tau}<t\leq
V_{k\tau})$ $=\int F(y)\sum_{k\tau\in[1/K,K]}\int\mathbf{1}(v<t\leq
w)P_{\tau}(p,v;y,w)P_{(k-1)\tau}(y_{0},v_{0};p,v)\,dydpdvdw.$ (45)

This can be rewritten as

$\mathbf{E}[F(Y_{Z_{t}^{\tau}})\mathbf{1}(Z_{t}^{\tau}\in[1/K,K])]=\int\sum_{k\tau\in[1/K,K]}(U^{\tau}F_{v})(p,v)P_{(k-1)\tau}(y_{0},v_{0};p,v)\,dpdv,$
(46)

with $F_{v}(y,w)=F(y)\mathbf{1}(v<t\leq w)$.

##  8 Proof of Theorem 3.2

Theorem 4.1 implies formula (27):

$\mathbf{E}[F(\tilde{X}_{x,s}(t))\mathbf{1}(T_{x,s}(t)\in[1/K,K])]$ $=\int
dy\int_{1/K}^{K}du\int_{s}^{t}dv\frac{(t-v)^{-a(v,y)\alpha}}{a(v,y)\alpha}\,G(u;x,s;y,v)F(y).$
(47)

To complete the proof of statement (ii) of Theorem 3.2 one has to note that,
due to the asymptotic behavior of the transition probabilities $G(u;x,s;y,v)$
of the stable-like process $(X_{x,s},S_{x,s})(t)$ (see [6] or [8]), the
function $G(u;x,s;y,v)$ is integrable for small and large $u$, so that one can
pass to the limit $K\to\infty$ in (47) to obtain (47).

Next, by (46),

$\mathbf{E}[F(\tilde{X}^{\tau}_{x,s}(t))\mathbf{1}(T^{\tau}_{x,s}(t)\in[1/K,K])]=\int
dy\int_{0}^{t}dv\,\sum_{k\tau\in[1/K,K]}(U^{\tau}F_{v})(y,v)P_{(k-1)\tau}(x,s;y,v),$
(48)

where $F_{v}(q,w)=F(q)\mathbf{1}(v<t\leq w)$ and

$U^{\tau}F(y,v)=\int_{\mathbf{R}_{+}}\int_{\mathbf{R}^{d}}F(y+\tau^{1/\beta}q,v+\tau^{1/(\alpha
a(v,y))}r)Q_{v,y}(r)\,dr\,p(y,dq),$ (49)

with $Q_{s,x}$ satisfying (16).

Therefore,

$\mathbf{E}[F(\tilde{X}^{\tau}_{x,s}(t))\mathbf{1}(T^{\tau}_{x,s}(t)\in[1/K,K])]$
$=\int
dy\int_{0}^{t}dv\,\sum_{k\tau\in[1/K,K]}P_{(k-1)\tau}(x,s;y,v)F(y+\tau^{1/\beta}q)p(y,dq)\mathbf{1}(v<t\leq
v+\tau^{1/(\alpha a(v,y))}r)Q_{v,y}(r)\,dr.$ (50)

Since the probability of small and large jumps are small both for $S_{x,s}$
and $S^{\tau}_{x,s}$, it follows that, in order to prove Statement (i) of
Theorem 3.2, it is sufficient to show that expression (50) tends to expression
(47), as $\tau\to 0$, for any finite $K$ and any $F\in
C_{\infty}(\mathbf{R}^{d})$. By the density argument this can be further
reduced to the case of $F\in C^{2}_{\infty}(\mathbf{R}^{d})$.

The key point for the argument is that, by Theorem 3.1, the sums

$\tau\int
dy\int_{0}^{t}dv\,\sum_{k\tau\in[1/K,K]}P_{(k-1)\tau}(x,s;y,v)\Omega(y,v)$

approximate uniformly the Riemannian sums for the integral

$\int dy\int_{0}^{t}dv\,\int_{[1/K,K]}du\,G(u;x,s;y,v)\Omega(y,v),$

and thus converge to this integral, at least for $\Omega\in
C_{\infty}(\mathbf{R}^{d+1})$.

Turning concretely to (50), we note firstly that the limit must be the same
for the expression

$\int
dy\int_{0}^{t}dv\,\sum_{k\tau\in[1/K,K]}P_{(k-1)\tau}(x,s;y,v)F(y)\mathbf{1}(v<t\leq
v+\tau^{1/(\alpha a(v,y))}r)Q_{v,y}(r)\,dr.$ (51)

In fact, expressions (50) and (51) differ by
$\int(F(y+\tau^{1/\beta}q)-F(y))p(y,dq)$, which tends to zero, as $\tau\to 0$,
uniformly in $y$ (for $F\in C^{2}_{\infty}(\mathbf{R}^{d})$). Hence, if
functions (51) have a finite limit (as we are going to show), then the same
limit have functions (50).

Since

$\frac{1}{\tau}\int_{0}^{\infty}\mathbf{1}(v<t\leq v+\tau^{1/(\alpha
a(v,y))}r)Q_{v,y}(r)\,dr=\frac{1}{\tau}\int_{0}^{\infty}[\theta_{\geq
t}(v+\tau^{1/a(v,y)\alpha}r)-\theta_{\geq t}(v)]Q_{v,y}(r)\,dr,$ (52)

it tends to $A\theta_{\geq t}(v)$, by (41), as claimed. Therefore, we just
need to show that this limit goes through the Riemannian sum approximation
(51). For this end we note that the limit of (51) remains the same if we
slightly decrease the domain of integration, namely if we choose the
expression

$\int
dy\int_{[0,t-B\tau^{1/a(v,y)\alpha}]}dv\,\sum_{k\tau\in[1/K,K]}P_{(k-1)\tau}(x,s;y,v)F(y)\mathbf{1}(v<t\leq
v+\tau^{1/(\alpha a(v,y))}r)Q_{v,y}(r)\,dr.$ (53)

In fact, by (52) and (16), the difference between (51) and (53) is bounded in
magnitude by the expression

$\tau\int
dy\int_{[t-B\tau^{1/a(v,y)\alpha},t]}dv\,\sum_{k\tau\in[1/K,K]}P_{(k-1)\tau}(x,s;y,v)F(y)\Omega_{\tau}(y,v),$

with $\Omega(y,v)\leq 1/\tau$. And this expression tends to zero, as $\tau\to
0$, because

$\int_{[t-B\tau^{1/a(v,p)\alpha},t]}\Omega_{\tau}(p,v)\,dv$

is bounded by $B\tau^{(1/a(v,y)\alpha)-1}$, which is uniformly small.

Next, for $v>B\tau^{1/a(v,y)\alpha}$,

$\frac{1}{\tau}\int_{0}^{\infty}\mathbf{1}(v<t\leq v+\tau^{1/(\alpha
a(v,y))}r)Q_{v,y}(r)\,dr=A\theta_{\geq t}(v).$

(see proof of Proposition 5.1 for this simple calculation).

Hence it remains to find the limit of the expression

$\tau\int
dy\int_{[0,t-A\tau^{1/a(v,y)\alpha}]}dv\,\sum_{k\tau\in[1/K,K]}P_{(k-1)\tau}(x,s;y,v)F(y)A\theta_{\geq
t}(v).$ (54)

Since $G(u;x,s;y,v)$ is continuous in $y,v$ and $A\theta_{\geq t}(v)$ has an
integrable singularity, it follows that this expression tends to (47)
completing the proof of part (i) of Theorem 3.2.

##  9 Proof of Theorem 3.3

The statement on the probabilistic representation of the solutions to (29)
follows from the standard argument (see e.g. [9]) with Dynkin’s martingale.
Namely, for a function $F$ from the domain of the generator $\Lambda$ of the
process $(X_{x,s},S_{x,s})(t)$, the process

$M^{t}_{F}=F(X_{x,s}(t),S_{x,s}(t))-F(x,s)-\int_{0}^{t}\Lambda
F(X_{x,s},S_{x,s})(r)\,dr$

is a martingale, called Dynkin’s martingale. We are now looking at the
modification of the process $(X_{x,s},S_{x,s})(t)$ obtained by stopping it at
the attempt to cross the boundary $S_{x,s}=t$. This modification specifies a
process on $\mathbf{R}^{d}\times(-\infty,t]$ with the generator

$\tilde{\Lambda}(x,s)=L_{st}^{\beta}F(x,s)$
$+\int_{0}^{t-s}\frac{F(x,s+r)-F(x,s)}{r^{1+\alpha
a(s,x)}}dr+(F(x,t)-F(x,s))\frac{(t-s)^{-a(s,x)\alpha}}{a(s,x)\alpha},$ (55)

obtained formally from (24) by reducing it to the set of functions such that
$F(x,r)=F(x,t)$ for all $r\geq t$.

Applying Doob’s optional sampling theorem to the martingale $M^{t}_{F}$ and
the stopping time

$T_{x,s}(t)=\inf\\{r:S_{x,s}(r)\geq t\\},$

we can conclude that, if $\tilde{\Lambda}F=0$, which is equivalent to equation
(29), then

$F(x,s)=\mathbf{E}F[\tilde{X}_{x,s}(t)]=\mathbf{E}F[X_{x,s}(T_{x,s}(t))].$
(56)

Thus formula (56) gives the probabilistic representation for a solution of the
fractional boundary problem (29) implying also the uniqueness of such a
solution. To complete the proof of well-posedness it remains to check that
formula (56) really solves the problem. The fact that it satisfies the
boundary condition $F(x,t)=F(x)$ is a direct consequence of (56). Let us check
that it satisfies the equation $\tilde{\Lambda}F=0$. To this end, we shall use
the representation (28):

$F(x,s)=\mathbf{E}[F(Y_{Z_{t}})]=\int
dy\int_{0}^{\infty}du\int_{0}^{t}dv\,(A_{(v;y)}\theta_{\geq
t})(v)G(u;x,s;y,v)F(y).$

We know that $F(x,t)=F(x)=\lim_{s\to t}F(x,s)$. Setting $F(x,s)=F(x)$ for
$s\geq t$, we make the function $F(x,s)$ continuous everywhere.

We want to show that

$L_{st}^{\beta}F(x,s)+A_{(s;x)}F(s,x)=0.$ (57)

It is straightforward from (28) that $F(x,s)$ is smooth for $s<t$, so that all
terms in this equation are well defined. Let

$f(x,s;u,v)=\int dy\,G(u;x,s;y,v)F(y)(A_{(v;y)}\theta_{\geq t})(v),\quad u>0,$

so that $f(x,s;u,v)=0$ for $s\geq v$ and any $u>0$. By the Kolmogorov’s
equation for the transition density,

$\frac{\partial G}{\partial u}=L_{st}^{\beta}G+A_{(s;x)}G.$

Therefore

$\frac{\partial f}{\partial u}=L_{st}^{\beta}f+A_{(s;x)}f.$

Let

$g(x,s;v)=\int_{0}^{\infty}f(x,s;u,v)\,du\,\,\text{for}\,\,s<v\,\,\text{and}\,\,g(x,s;v)=0\,\,\text{for}\,\,s>v.$

Integrating the equation for $f$ over $u$ and using that
$G(0;x,s;y,v)=\delta(x-y)\delta(s-v)$ yields the equation

$(L_{st}^{\beta}+A_{(s;x)})g(x,s;v)+F(x)(A_{(v;x)}\theta_{\geq
t})(v)\delta(v-s)=0.$

Since

$F(x,s)=\int_{s}^{t}g(x,s;v)dv,\quad s\leq
t\,\,\text{and}\,\,F(x,s)=F(x)\,\,\text{for}\,\,s\geq t,$

it follows that

$L_{st}^{\beta}F(x,s)+\int_{s}^{t}A_{(s;x)}g(x,s;v)dv+F(x)(A_{(s;x)}\theta_{\geq
t})(s)=0.$

Here

$\int_{s}^{t}A_{(s;x)}g(x,s;v)dv+F(x)(A_{(s;x)}\theta_{\geq t})(s)$
$=\int_{s}^{t}dv\int_{0}^{t-s}dr(g(x,s+r;v)-g(x,s;v))r^{-1-a(s,x)\alpha}+(A_{(s;x)}\theta_{\geq
t})(s)(F(x)-F(x,s)).$

Consequently, in order to establish (57), it remains to show that

$\int_{s}^{t}dv\int_{0}^{t-s}dr(g(x,s+r;v)-g(x,s;v))r^{-1-a(s,x)\alpha}=\int_{0}^{t-s}dr(F(x,s+r)-F(x,s))r^{-1-a(s,x)\alpha}.$
(58)

Let $\chi_{\epsilon}(s,r)$ be positive functions that coincide with
$r^{-1-a(s,x)\alpha}$ for $r\leq\epsilon$, do not exceed these functions
everywhere, and are continuous and bounded. Then to prove (58) it is
sufficient to show

$\int_{s}^{t}dv\int_{0}^{t-s}dr(g(x,s+r;v)-g(x,s;v))\chi_{\epsilon}(s,r)=\int_{0}^{t-s}dr(F(x,s+r)-F(x,s))\chi_{\epsilon}(s,r),$
(59)

for any $\epsilon$. The second terms cancel and this equation turns to

$\int_{s}^{t}dv\int_{0}^{t-s}drg(x,s+r;v)\chi_{\epsilon}(s,r)=\int_{0}^{t-s}drF(x,s+r)\chi_{\epsilon}(s,r).$
(60)

And this holds, because the l.h.s. is

$\int_{0}^{t-s}dr\int_{s+r}^{t}dv\,g(x,s+r;v)\chi_{\epsilon}(s,r)=\int_{0}^{t-s}dr\,F(x,s+r)\chi_{\epsilon}(s,r).$

Acknowledgements.

The author is grateful to the organisers of the 50th Memorial Barrett Lectures
in the University of Tennessee, May 2021, for inviting him to give a plenary
talk, which strongly stimulated his further research in fractional calculus.

The author is grateful to the Simons foundation for the support of his
residence at INI in Cambridge during the program Fractional Differential
Equations Jan-Apr 2022.

The work was supported by the Russian Science Foundation under grant no.
20-11-20119.

## References

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