# Nonlinear Filtering of Classical and Quantum Spin Systems
111 1 U. S. Air Force Research Laboratory, Wright Patterson Air Force Base,
Ohio 45433, U. S. A. 2 NRC-Senior Research Fellow, National Academies of
Science, Engineering and Medicine, U. S. Air Force Research Laboratory, Wright
Patterson Air Force Base, Ohio 45433, U. S. A. e-mail:
<EMAIL_ADDRESS>∗Corresponding author. e-mail:
<EMAIL_ADDRESS>
Sivaguru S. Sritharan1,2* and Saba Mudaliar1 Dedicated in honor of Professor
K. R. Parthasarathy

###### Abstract.

In this paper we consider classical and quantum spin systems on discrete
lattices and in Euclidean spaces, modeled by infinite dimensional stochastic
diffusions in Hilbert spaces. Existence and uniqueness of various notions of
solutions, existence and uniqueness of invariant measures as well as
exponential convergence to equilibrium are known for these models. We
formulate nonlinear filtering problem for these classes of models, derive
nonlinear filtering equations of Fujisaki-Kallianpur-Kunita and Zakai tye, and
prove existence and uniqueness of measure-valued solutions to these filtering
equations. We then establish the Feller property and Markov property of the
semigroups associated with the filtering equations and also prove existence
and uniqueness of invariant measures. Evolution of error covariance equation
for the nonlinear filter is derived. We also derive the nonlinear filtering
equations associated with finitely-additive white noise formulation due to
Kallianpur and Karandikar for the classical and quantum spin systems, and
study existence and uniqueness of measure-valued solutions.

Key words: quantum spin systems, quantum lattice systems, stochastic
processes, nonlinear filtering, invariant measures, Markov property,
transition semigroup, ergodicity.

Mathematics Subject Classification (2010): 60H30, 81T25, 82C10, 93E11

###### Contents

  1. 1 Introduction
  2. 2 Unified Mathematical Formulation of Spin Systems -Stochastic Dissipative Systems and Invariant Measures
  3. 3 Classical Spin Systems on Discrete Lattices
    1. 3.1 Diffusions on an Infinite Dimensional Torus
    2. 3.2 Unbounded Classical Spin Systems
    3. 3.3 Euclidean Field Theory and Spin Systems on Euclidean Spaces
  4. 4 Quantum Lattice Systems
  5. 5 Nonlinear Filtering Formulation and Filtering Equations
    1. 5.1 Stochastic Calculus Method of Nonlinear Filtering
    2. 5.2 Existence and Uniqueness of Measure-valued solutions to Filtering Equations
    3. 5.3 Evolution Equation for Error Covariance
  6. 6 White Noise Theory of Nonlinear Filtering

##  1\. Introduction

Classical and quantum spin systems are extensively studied in literature [46,
49, 50, 16, 17]. In this paper we will take a stochastic partial differential
equations approach as in [19] for spin systems and develop a classical
nonlinear filtering method for these models. For classical spin systems we
will consider the diffusions on infinite dimensional torus studied by Holley
and Stroock [28] and also the Euclidean and lattice systems studied by Da
Prato and Zabczyk [19]. For quantum spin systems we consider the models
studied by [1] and [19]. Nonlinear filtering for infinite dimensional problems
related to nonlinear partial differential equations of fluid dynamics was
initiated in [51] and later developed for several fluid dynamic models in [52,
53, 22]. Nonlinear filtering for nonlinear stochastic partial differential
equations for reaction diffusion type was originated in [27]. In these papers
the measure valued evolution equations of FKK type [23] and Zakai type[58]
have been derived and existense and uniqueness of measure valued nonlinear
filter was established. In this paper we will establish similar results for
classical and quantum spin systems [1, 19] and then prove convergence and
ergodicity type results similar to such results for finite dimensional
nonlinear filtering problems initiated by Kunita[38] and further developed by
several other authors [41, 54, 39, 42, 6, 13, 14, 18]. We will also consider
white noise nonlinear filtering equations developed by Kallianpur and
Karandikar [34, 35, 36] and derive the corresponding equations for the spin
systems and prove existence and uniqueness of measure-valued solutions. Other
works on nonlinear filtering of classical spin systems with different
perspective see [45]. We also note here that in this paper we develop
classical nonlinear filtering for quantum spin systems even though white noise
filtering in §6 has some attributes of quantum theory. Quantum nonlinear
filtering for open quantum systems has been developed in a series of papers by
V. P. Belavkin [8, 9, 10, 11, 12] (see also the introductory exposition [15])
for the Hudson-Parthasarathy equation [29, 43]. Belavkin type nonlinear
filtering theory for unbounded Hamiltonian and noise operators in the Hudson-
Parthasarathy equation is an open problem and will be addressed in the future.

##  2\. Unified Mathematical Formulation of Spin Systems -Stochastic
Dissipative Systems and Invariant Measures

In this section we will describe certain general results for infinite
dimensional stochastic equations driven by cylindrical Wiener process as
presented in [19, 20]. Let $H$ and $U$ be separable Hilbert spaces with norms
$\|\cdot\|,\|\cdot\|_{U}$ and scalar products $\langle\cdot,\cdot\rangle$,
$\langle\cdot,\cdot\rangle_{U}$. The spaces of all bounded operators from $U$
in to $H$ and from $U$ in to $U$ will be denoted by ${\mathcal{L}}(U,H)$ and
${\mathcal{L}}(U)$ respectively. Let
$(\Omega,{\mathcal{F}},{\mathcal{F}}_{t},m)$ be a complete filtered
probability space and $W(t),t\geq 0$ be an ${\mathcal{F}}_{t}$-adapted Wiener
process defined on $\Omega$ with values in $U$ and with the covariance
operator $Q\in{\mathcal{L}}(U)$. Hence for arbitary $\phi,\psi\in U$ and
$t,s\geq 0$:

$E\mathopen{}\mathclose{{}\left[\langle W(t),\phi\rangle_{U}\langle
W(s),\psi\rangle_{U}}\right]=t\wedge s\langle Q\phi,\psi\rangle_{U}.$

We will consider the general stochastic equation

$dX=(AX+F(X))dt+BdW,$ (2.1) $X(0)=x.$

Here $B\in{\mathcal{L}}(U,H)$, $A\in{\mathcal{L}}(D(A);H)$ is a linear
operator with $D(A)\subset H$, $F:D(F)\rightarrow H$ is a nonlinear operator
with $D(F)\subset H$ and both $A$ and $F$ are dissipative in the sense defined
below. An Important observation is that in the classes of problems studied in
this paper the covariance operator is not of trace class:
$\mbox{Tr}Q=\sum_{i}\langle Q\varphi_{i},\varphi_{i}\rangle_{U}=+\infty.$ In
particular $Q=I$ is often explicitly invoked.

Let $E,\|\cdot\|_{E}$ be a Banach space and $E^{*}$ its dual. For arbitrary
$x\in E$ the subdifferential $\partial\|x\|_{E}$ of the norm $\|\cdot\|_{E}$
at $x$ is given by:

$\partial\|x\|_{E}:=\mathopen{}\mathclose{{}\left\\{x^{*}\in
E^{*}:\|x+y\|_{E}-\|x\|\geq x^{*}(y),\forall y\in E}\right\\}.$

A mapping $G$ from $D(G)\subset E$ into $E$ is said to be dissipative in $E$
if for arbitrary $x,y\in D(G)$ there exists $z^{*}\in\partial\|x-y\|_{E}$ such
that

$z^{*}(G(x)-G(y))\leq 0.$ (2.2)

If in addition, for some $\alpha>0$ the mapping $(I-\alpha G):D(G)\rightarrow
E$ is surjective then $G$ is said to be $m$-dissipative.

Let $K\subset E$ is a Banach space embedded into $E$ then the part $G_{K}$.
Then the part $G_{K}$ of $G$ in $K$ is defined as follows:

$G_{K}(x)=G(x)\mbox{ for }x\in D(G_{K}),$

where

$D(G_{K})=\mathopen{}\mathclose{{}\left\\{x\in D(G)\cap K:G(x)\in
K}\right\\}.$

We will now state the two essential hypothesis on operators $A,F,B$, the
Wiener process $W(t),t\geq 0$ and on a Banach space $K,\|\cdot\|_{K}$
continuously and densely embedded into $H$.

######  Hypothesis 2.1.

  1. (1)

There exists $\eta\in\mathbb{R}$ such that the operators $A+\eta I$ and
$F+\eta I$ are $m$-dissipative on $H$.

  2. (2)

The parts in $K$ of $A+\eta I$ and $F+\eta I$ are $m$-dissipative on $K$.

  3. (3)

$K\subset D(F)$ and $F$ maps bounded sets in $K$ into bounded sets in $H$.

  4. (4)

$K$ is a reflective space.

  5. (5)

$B\in{\mathcal{L}}(U,H)$.

Let $W_{A}(t),t\geq 0$ be the solution to the linear equation

$dZ(t)=AZ(t)dt+BdW(t),$ (2.3) $Z(t)=W_{A}(t)=\int_{0}^{t}S(t-s)BdW(s),\hskip
7.22743ptt\geq 0,$

where $S(t),t\geq 0$, is the semigroup generated by $A$ on $H$.

######  Hypothesis 2.2.

The process $W_{A}(t),t\geq 0$, is continuous in $H$, takes values in the
domain $D(F_{K})$ of the part of $F$ in $K$ and for any $T>0$ we have

$\sup_{t\in[0,T]}\mathopen{}\mathclose{{}\left(\|W_{A}(t)\|_{K}+\|F_{K}(W_{A}(t))\|_{K}}\right)<+\infty,\hskip
7.22743ptm\mbox{ a.s. }$ (2.4)

######  Definition 2.1.

An $H$-valued continuous ${\mathcal{F}}_{t}$-adapted process $X(t),t\geq 0$,
is said to be a strong solution to 2.1 if it satisfies $m$-a.s. the equation

$X(t)=x+\int_{0}^{t}\mathopen{}\mathclose{{}\left(AX(s)+F(X(s))}\right)ds+BW(t),\hskip
7.22743ptt\geq 0,$ (2.5)

and it is a mild solution if it satisfies the following integral equation:

$X(t)=S(t)x+\int_{0}^{t}S(t-s)F(X(s))ds+W_{A}(t),\hskip 7.22743ptt\geq 0.$
(2.6)

If for a $H$-valued process $X$, there exists a sequence $X_{n}$, of mild
solutions to 2.1 such that $m$-a.s., $X_{n}\rightarrow X$ uniformly on any
interval $[0,T]$, then $X$ is said to be a generalized solution to 2.1.

Note that each strong solution is a mild solution and each mild solution is a
generalized solution.

######  Theorem 2.1.

[19, 20]Assume that Hypothesis 2.1 and 2.2 are satisfied. Then for arbitrary
$x\in K$ there exists a unique mild solution of 2.1 and for arbitrary $x\in H$
there exists a unique generalized solution $X(t,x),t\geq 0$ of 2.1. If the
operator $A$ and its part in $K$ are bounded then solutions for $x\in K$ are
strong.

The above theorem implies that

######  Theorem 2.2.

$X(t,x),t\geq 0$ is a Markov process with transition group $P_{t},t\geq 0$,
given by

$P_{t}\varphi(x)=E[\varphi(X(t,x))],\hskip 7.22743ptt\geq 0,x\in H,\varphi\in
C_{b}(H),$ (2.7)

where $C_{b}(H)$ denotes the space of all uniformly continuous and bounded
functions on $H$. Moreover, the transition semigroup is Feller:

$P_{t}:C_{b}([0,\infty];H)\rightarrow C_{b}([0,\infty];H).$

The next theorem concerns with existence and uniqueness of invariant measures
for 2.1 and asymptotic behavior of the transition semigroup $P_{t},t\geq 0$.

######  Theorem 2.3.

[19, 20] If in addition to Hypotheses 2.1, 2.2,

  1. (1)

there exist $\eta_{1},\eta_{2}\in\mathbb{R}$ such that
$\omega=\eta_{1}+\eta_{2}>0$ and operators $A+\eta_{1}I$, $F+\eta_{2}I$ are
dissipative in $H$,

  2. (2)

one has

$\sup_{t\geq
0}E\mathopen{}\mathclose{{}\left(\|W_{A}(t)\|+\|F(W_{A}(t))\|}\right)<+\infty.$

Then there exists a unique invariant measure $\mu$ for the semigroup
$P_{t},t\geq 0$ such that $P^{*}_{t}\mu=\mu$. Moreover, for all bounded and
Lipschitz continuous function $\varphi$ on $H$ one has

$|P_{t}\varphi(x)-\int_{H}\varphi(y)\mu(y)|\leq(C+2\|x\|)e^{-\omega
t}\|\varphi\|_{\mbox{Lip}}$ (2.8)

where

$C=\sup_{t\geq
0}E\mathopen{}\mathclose{{}\left(\|W_{A}(t)\|+\frac{1}{\omega}\|F(W_{A}(t))\|}\right).$

##  3\. Classical Spin Systems on Discrete Lattices

###  3.1. Diffusions on an Infinite Dimensional Torus

We will discuss one of the simplest classical spin systems studied by Holley
and Stroock [28]. Let $(\Omega,\Sigma,m)$ be a complete probability space and
let $x_{\gamma}(t)\in\mathbb{R}^{\mathbb{Z}^{d}}$ (resp. $x_{\gamma}(t)\in{\bf
T}^{\mathbb{Z}^{d}}$,) $\gamma\in\mathbb{Z}^{d}$ be a spin system that evolves
according to:

$x_{\gamma}(t,x)=x_{\gamma}+\int_{0}^{t}\sigma_{\gamma}(x(s,x))dB_{\gamma}(s)+\int_{0}^{t}b_{\gamma}(x(s,x))ds,\gamma\in\mathbb{Z}^{d},t>0,$
(3.1)

where for fixed $\gamma\in\mathbb{Z}^{d}$, $B_{\gamma}$ is a one dimensional
standard Brownian motion, $x\in\mathbb{R}^{\mathbb{Z}^{d}}$ (resp. $x\in{\bf
T}^{\mathbb{Z}^{d}})$, and
$\sigma_{\gamma}(\cdot):\mathbb{R}^{\mathbb{Z}^{d}}\rightarrow\mathbb{R}$,
$b_{\gamma}(\cdot):\mathbb{R}^{\mathbb{Z}^{d}}\rightarrow\mathbb{R}$ (resp.
defined on ${\bf T}^{\mathbb{Z}^{d}}$) and their derivatives are smooth
bounded functions such that $\sigma_{\gamma}(x)=\sigma_{\gamma}(y)$ if
$x_{l}=y_{l}$ for all $l\in Z^{d}$ satisfying $|l-\gamma|\leq L$. We will
start with a path-wise solvability:

######  Lemma 3.1.

[28] For each $x\in\mathbb{R}^{\mathbb{Z}^{d}}$ there exists a unique solution
$x_{\gamma}(\cdot,x)$ to 3.1. Moreover, if
$x,y\in\mathbb{R}^{\mathbb{Z}^{d}}$, then for each $T>0$:

$E\mathopen{}\mathclose{{}\left[\sum_{\gamma\in\mathbb{Z}^{d}}\sup_{0\leq
t\leq
T}\frac{1}{2^{|\gamma|}}|x_{\gamma}(t,x)-x_{\gamma}(t,y)|^{2}}\right]\leq
A_{T}\mathopen{}\mathclose{{}\left(\sum_{\gamma\in\mathbb{Z}^{d}}\frac{1}{2^{|\gamma|}}|x_{\gamma}-y_{\gamma}|^{2}}\right),$

where $A_{T}<\infty$ depends only on $T>0,L,$ and
$\max_{\gamma}\|\sigma_{\gamma}\|_{C^{1}(\mathbb{R}^{\mathbb{Z}^{d}})}$.
Finally, if $x^{N}(\cdot,x)_{\gamma}$ is defined by 3.1 with
$\sigma_{\gamma}(\cdot)=b_{\gamma}(\cdot)\equiv 0$ for $|\gamma|>N$ then for
all $T>0$,

$E\mathopen{}\mathclose{{}\left[\sum_{\gamma\in\mathbb{Z}^{d}}\sup_{0\leq
t\leq
T}\frac{1}{2^{|\gamma|}}|x_{\gamma}(t,x)-x^{N}_{\gamma}(t,x)|^{2}}\right]\rightarrow
0\mbox{ as }N\rightarrow\infty.$

Let us now recall the solvability of martingale problem given by Holley and
Stroock [28]. Let ${\bf
T}=\mathopen{}\mathclose{{}\left\\{z\in\mathbb{C};|z|=1}\right\\}$. For points
in ${\bf T}^{\mathbb{Z}}$ we will use $\eta$ to denote both the point in ${\bf
T}^{\mathbb{Z}}$ as well as the element $\alpha$ of $([0,2\pi))^{\mathbb{Z}}$
such that $\eta_{k}=e^{i\alpha_{k}},k\in\mathbb{Z}$. Let ${\mathcal{D}}$
denote the set of smooth functions on ${\bf T}^{\mathbb{Z}}$ (sometimes called
cylindrical test functions) which depend only on a finite number of
coordinates.

######  Theorem 3.1.

[28] Let $\Omega$ denote the Polish space $C([0,\infty);{\bf
T}^{\mathbb{Z}})$. For $\omega\in\Omega,\eta(t,\omega)\in{\bf T}^{\mathbb{Z}}$
is the position of $\omega$ at time $t\geq 0$. We set
$\mathbb{\mathcal{F}}_{t}:=\sigma\mathopen{}\mathclose{{}\left(\eta(s);0\leq
s\leq t}\right)$ to be the cannonical filtration and
$\mathbb{\mathcal{F}}=\sigma\mathopen{}\mathclose{{}\left(\cup_{t\geq
0}\mathbb{\mathcal{F}}_{t}}\right)$ and then $\mathbb{\mathcal{F}}$ coincides
with the Borel field over $\Omega$. Let $\sigma_{k},b_{k},k\in\mathbb{Z}$ be
smooth functions of ${\bf T}^{\mathbb{Z}}$ such that for a given
$k\in\mathbb{Z}$ the functions $\sigma_{k}$ and $b_{k}$ depend only on
$\mathopen{}\mathclose{{}\left\\{\eta_{l};|l-k|\leq L}\right\\}$. Also assume
that $\sigma_{k}$ and $b_{k}$ and each of their derivatives are bounded
independent of $k\in\mathbb{Z}$. For $f\in{\mathcal{D}}$, define

${\mathcal{L}}f(\eta)=\sum_{k\in\mathbb{Z}}\mathopen{}\mathclose{{}\left(\frac{1}{2}\sigma_{k}^{2}(\eta)\frac{\partial^{2}f}{\partial\eta_{k}^{2}}(\eta)+b_{k}(\eta)\frac{\partial
f}{\partial\eta_{k}}(\eta)}\right),\hskip 7.22743pt\eta\in{\bf
T}^{\mathbb{Z}}.$

Then for each $\eta\in{\bf T}^{\mathbb{Z}}$ there is exactly one probability
measure $P_{\eta}$ on $(\Omega,\mathbb{\mathcal{F}})$ such that
$P_{\eta}(\eta(0)=\eta)=1$ and
$\mathopen{}\mathclose{{}\left\\{f(\eta(t))-\int_{0}^{t}{\mathcal{L}}f(\eta(s))ds,{\mathcal{F}}_{t},P_{\eta}}\right\\}$
is a martingale for each $f\in{\mathcal{D}}$. Moreover, if
$\Phi:\mathbb{R}\rightarrow\mathbb{Z}$ is the map defined so that
$\Phi_{k}(x)=\eta_{k}$, where $\eta_{k}\in[0,2\pi)$ and $x_{k}=\eta_{k}$ mod
$(2\pi)$, then for any $n\geq 1$, $F\in B(({\bf T}^{\mathbb{Z}})^{n})$, and
$0\leq t_{1}<t_{1}<\cdots<t_{n}$:

$E^{P_{\eta}}\mathopen{}\mathclose{{}\left[F(\eta(t_{1}),\cdots,\eta(t_{n}))}\right]=E^{m}\mathopen{}\mathclose{{}\left[F\circ\Phi^{n}(x(t_{1},\eta),\cdots,x(t_{n},\eta)}\right],$

where $\Phi^{n}=\Phi\otimes\cdots\otimes\Phi$ ($n$-times ), and
$x(\cdot,\eta)$ is the solution to 3.1 with $x=\eta$ and the $\sigma_{k}$ and
$b_{k}$ in 3.1 replaced by $\sigma_{k}\circ\Phi$ and $b_{k}\circ\Phi$,
respectively. Finally, the family
$\mathopen{}\mathclose{{}\left\\{P_{\eta};\eta\in{\bf
T}^{\mathbb{Z}}}\right\\}$ is Feller continuous and strong Markov.

###  3.2. Unbounded Classical Spin Systems

Unbounded classical spin systems have been studied by several authors to
establish construction of Markov semigroup, existence and uniqueness of
invariant measures, ergodicity and exponential convergence of Markov
semigroups to equilibrium state characterized by the invariant measure [57,
19, 20]. We will recall some of the essential results below. We consider the
Markov process $X=\\{X_{\gamma}\\}_{\gamma\in\mathbb{Z}^{d}}$ satisfying an
infinite system of Ito’s equations

$dX_{\gamma}(t)=\mathopen{}\mathclose{{}\left(\sum_{j}a_{\gamma
j}X_{j}(t)+f(X_{\gamma}(t))}\right)dt+dW_{\gamma}(t),$ (3.2)
$X_{\gamma}(0)=x_{\gamma},\hskip 7.22743pt\gamma\in\mathbb{Z}^{d},t\geq 0,$

where $W_{\gamma}$, $\gamma\in\mathbb{Z}^{d}$ is a Wiener process on
$(\Omega,\Sigma,m)$ with values in $U=l^{2}(\mathbb{R}^{d})$ and with the
covariance operator $Q\in{\mathcal{L}}(U)$. In particular, if $Q=I$ then the
processes $W_{\gamma},\gamma\in\mathbb{Z}^{d}$, are independent standard real
valued Wiener processes. An interesting special case of the above system is:

$dX_{\gamma}(t)=\mathopen{}\mathclose{{}\left((\Delta_{d}-\alpha)X_{\gamma}(t)+f(X_{\gamma}(t))}\right)dt+dW_{\gamma}(t),$
(3.3) $X_{\gamma}(0)=x_{\gamma},\hskip 7.22743pt\gamma\in\mathbb{Z}^{d},t\geq
0,$

where $\Delta_{d}$ is the $d$-dimensional discrete Laplacian and $\alpha$ is a
constant.

In the light of general theory presented in Section 2, we define operators $A$
and $F$ as follows:

$A(X_{\gamma})=\mathopen{}\mathclose{{}\left(\sum_{j\in\mathbb{Z}^{d}}a_{j\gamma}X_{j}}\right),\hskip
7.22743ptX=(X_{\gamma})\in H,$ $F(X_{\gamma})=(f(X_{\gamma})),\hskip
7.22743ptX=(X_{\gamma})\in H.$

The following result [19, 20, 44] is useful in establishing the boundedness of
$A$.

######  Theorem 3.2.

Assume that

  1. (1)

$\sup_{\gamma\in\mathbb{Z}^{d}}\sum_{j\in\mathbb{Z}^{d}}|a_{\gamma
j}|=\alpha<+\infty.$

  2. (2)

There exists $\beta>0$ such that

$\sum_{\gamma\in\mathbb{Z}^{d}}|a_{\gamma
j}|\rho(\gamma)\leq\beta\rho(j),\hskip 7.22743ptj\in\mathbb{Z}^{d}.$

Then the formula

$A_{p}(X_{\gamma})=\mathopen{}\mathclose{{}\left(\sum_{j\in\mathbb{Z}^{d}}a_{j\gamma}X_{j}}\right),\hskip
7.22743ptX=(X_{\gamma})\in l^{p}_{\rho}(\mathbb{Z}^{d}),$

defines a bounded operator on $l^{p}_{\rho}(\mathbb{Z}^{d})$, (i.e.
$A\in{\mathcal{L}}(l^{p}_{\rho},l^{p}_{\rho})$) for all $p\in[1,+\infty]$,
with the norm not greater than

$\alpha^{1/q}\beta^{1/p},\hskip 7.22743pt\frac{1}{p}+\frac{1}{q}=1.$

In particular $A\in{\mathcal{L}}(l^{2}_{\rho},l^{2}_{\rho})$, with operator
norm less than or equal to $\sqrt{\alpha\beta}$.

######  Corollary 3.1.

If, for some $\alpha<\infty$,

$\sup_{k\in\mathbb{Z}^{d}}\sum_{j\in\mathbb{Z}^{d}}|a_{k,j}|\leq\alpha\mbox{
and }\sup_{j\in\mathbb{Z}^{d}}\sum_{k\in\mathbb{Z}^{d}}|a_{k,j}|\leq\alpha$

then $A\in{\mathcal{L}}(l^{p},l^{p})$ with norm $\leq\alpha$.

For example, the matrix coefficients $a_{\gamma,j}$ and positive weight $\rho$
can be assumed to satisfy the following conditions:

######  Definition 3.1.

The interactions $a_{\gamma,j}$ have bounded range if there are constants
$R,M>0$ such that

$a_{\gamma,j}=0\mbox{ if }|\gamma-j|>R\mbox{ and }|a_{\gamma,j}|\leq M\mbox{
for all }\gamma,j\in\mathbb{Z}^{d}.$ (3.4)

######  Lemma 3.2.

Assume bounded range and that

$|\frac{\rho(\gamma)}{\rho(j)}|\leq M\mbox{ if }|\gamma-j|\leq R,\mbox{ and
}\sum_{\gamma\in\mathbb{Z}^{d}}\rho(\gamma)<+\infty,$ (3.5)

then $A\in{\mathcal{L}}(l^{p}_{\rho},l^{p}_{\rho})$ for every $p\geq 1$.

Examples of weights $\rho$ that fit the conditions of Lemma 3.2 are:

$\rho_{\kappa}(k)=e^{-\kappa|k|}\mbox{ or
}\rho^{\kappa}(k)=\frac{1}{1+\kappa|k|^{r}},\kappa>0,r>d,k\in\mathbb{Z}^{d}.$

######  Definition 3.2.

The local interaction function $f:\mathbb{R}\rightarrow\mathbb{R}$ is such
that $f=f_{0}+f_{1}$ where $f_{1}$ is Lipchitz continuous and
$f_{0}+\eta\zeta,\zeta\in\mathbb{R}$ is continuous and decreasing for some
$\eta\in\mathbb{R}$ and for some $s\geq 1$ and $c_{0}>0$,

$|f_{0}(\zeta)|\leq c_{0}(1+|\zeta|^{s}),\zeta\in\mathbb{R}.$

For example with $s=2n+1$ and

$f_{0}(\zeta)=-\zeta^{2n+1}+\sum_{k=0}^{2n}b_{k}\zeta^{2n-k},\hskip
7.22743pt\zeta\in\mathbb{R},n\in\mathbb{N}\cup\\{0\\}.$

######  Theorem 3.3.

Let $H=l_{\rho}^{2}(\mathbb{Z}^{d})$, $K=l_{\rho}^{2s}(\mathbb{Z}^{d})$ and
let the operators $A$ and $F$ be as above. Then

  1. (1)

For arbitrary $x\in K=l_{\rho}^{2s}(\mathbb{Z}^{d})$ there exists a unique
strong solution $X(t,x),t\geq 0$.

  2. (2)

For arbitrary $x\in H=l_{\rho}^{2}(\mathbb{Z}^{d})$, there exists a unique
generalized solution $X(t,x),t\geq 0$, and the transition semigroup

$P_{t}\varphi(x)=E(\varphi(X(t,x))),\hskip 7.22743ptt\geq 0,x\in H,\varphi\in
C_{b}(H),$

is Feller.

######  Theorem 3.4.

Assume that conditions 3.4, 3.5 are satisfied. Let
$H=l_{\rho}^{2}(\mathbb{Z}^{d})$, $K=l_{\rho}^{2s}(\mathbb{Z}^{d})$ and let
the operators $A$ and $F$ be given as above. Then

  1. (1)

For arbitrary $x\in K=l_{\rho}^{2s}(\mathbb{Z}^{d})$ there exists a unique
strong solution $X(t,x),t\geq 0$ of 3.2.

  2. (2)

For arbitrary $x\in H=l_{\rho}^{2}(\mathbb{Z}^{d})$, there exists a unique
generalized solution $X(t,x),t\geq 0$ of 3.2 and the transition semigroup

$P_{t}\varphi(x)=E(\varphi(X(t,x))),\hskip 7.22743ptt\geq 0,x\in H,\varphi\in
C_{b}(H),$

is Feller.

######  Theorem 3.5.

Assume, in addition to conditions of 3.4 that for an $\eta>0$, operator
$A+I\eta$, restricted to $l^{2}(\mathbb{Z}^{d})$ is dissipative, that $f_{0}$
is decreasing and $\eta-\|f_{1}\|_{\mbox{Lip}}>\omega>0$. Then there exists
$\kappa_{0}>0$ such that in the spaces $l_{\rho}^{2}(\mathbb{Z}^{d})$ with
$\rho=\rho^{\kappa}$ or with $\rho=\rho_{\kappa}$, given as above, and
$\kappa\in]0,\kappa_{0}[$, equation 3.2 has unique generalized solutions. The
semigroup $P_{t},t\geq 0$ has a unique invariant measure $\mu$ on $H$ and
there exists $C>0$ such that, for any bounded and Lipschitz function $\varphi$
on $H$, all $t>0$ and all $x\in H$:

$|P_{t}\varphi(x)-\int_{H}\varphi(x)\mu(dx)|\leq(C+2|x|)e^{-\omega
t}\|\varphi\|_{\mbox{ Lip}}.$

These results have been extended to spin systems forced by Lévy noise in [44].

###  3.3. Euclidean Field Theory and Spin Systems on Euclidean Spaces

Spin systems on Euclidean spaces can be modeled by the following stochastic
partial differential equation:

$dX(t,\zeta)=[(\Delta-\alpha)X(t,\zeta)+f(X(t,\zeta))]dt+dW(t,\zeta),$ (3.6)
$X(0,\zeta)=x(\zeta),\hskip 7.22743pt\zeta\in\mathbb{R}^{d},t>0,$

where $\Delta$ is the Laplace operator, $\alpha$ is a nonnegative constant and
$W(t,\zeta),\zeta\in\mathbb{R}^{d}$, $t>0$, an infinite dimensional Wiener
process with covariance operator $Q$, defined on a probability space
$(\Omega,{\mathcal{F}},m)$.

######  Hypothesis 3.1.

  1. (1)

$Q=I\mbox{ on }U=L^{2}(\mathbb{R}^{d}),$ (3.7)

or

  2. (2)

$Q$ is a convolution operator on $U=L^{2}(\mathbb{R}^{d})$:

$Qu(\zeta)=q\ast u(\zeta),\zeta\in\mathbb{R}^{d},u\in L^{2}(\mathbb{R}^{d}),$
(3.8)

where
$q(\zeta)=\int_{\mathbb{R}^{d}}e^{i\zeta\cdot\eta}g(\eta)d\eta,\zeta\in\mathbb{R}^{d},$
with $g\geq 0$ and $q,g\in L^{1}(\mathbb{R}^{d})$.

######  Theorem 3.6.

Assume that $f$ is as in Definition 3.2 and either

  1. (1)

$d=1$ and 3.7 or 3.8 holds or

  2. (2)

$d>1$ and 3.8 holds.

Then the equation 3.6 has a unique generalized solution in
$L^{2}_{\rho}(\mathbb{R}^{d})$ where $\rho$ is given either by $\rho_{\kappa}$
or by $\rho^{\kappa}$. If $x\in L^{2s}_{\rho}(\mathbb{R}^{d})$ then the
generalized solution is mild.

######  Theorem 3.7.

In addition to conditions of Theorem 3.6 assume that the function $f_{0}$ is
decreasing and $\alpha-\|f_{1}\|_{\mbox{ Lip }}>\omega>0$. Then there exists
$\kappa_{0}>0$ such that the semigroup $P_{t},t\geq 0$ corresponding to
solution of equation 3.6 has a unique invariant measure both in
$H=L^{2}_{\rho^{\kappa}}(\mathbb{R}^{d})$ and
$H=L^{2}_{\rho_{\kappa}}(\mathbb{R}^{d})$, for any $\kappa\in]0,\kappa_{0}[.$
Moreover there exists $C>0$ such that for any bounded Lipschitz function
$\varphi$ on $H$, all $t>0$ and all $x\in H$

$|P_{t}\varphi(x)-\int_{H}\varphi(x)\mu(dx)|\leq(C+2)\|x\|e^{-\omega
t}\|\varphi\|_{\mbox{Lip }}.$ (3.9)

##  4\. Quantum Lattice Systems

One of the simplest quantum lattice system would be a combination of the
classical and Euclidean spin systems formulated in the previous sections (see
[19, 20]):

$dX_{\gamma}(t,\zeta)=\mathopen{}\mathclose{{}\left({\mathcal{A}}X_{\gamma}(t,\zeta)+\sum_{j\in\mathbb{Z}^{d}}a_{\gamma
j}X_{\gamma}(t,\zeta)+{\mathbb{\mathcal{F}}}(X_{\gamma}(t,\zeta))}\right)dt+dW_{\gamma}(t,\zeta),$
(4.1) $X_{\gamma}(0,\zeta)=x_{\gamma}(\zeta),\hskip
7.22743pt\zeta\in[0,1],t>0,$

where $\mathcal{A}$, $\mathbb{\mathcal{F}}$ are in general unbounded,
respectively linear and nonlinear operators on a Hilbert space
${\mathcal{H}}$, $(a_{\gamma j})_{\gamma,j\in\mathbb{Z}^{d}}$ is a given
matrix with real elements and $(W_{\gamma})_{\gamma\in\mathbb{Z}^{d}}$ is a
family of independent cylindrical Wiener processes on ${\mathcal{H}}$. We will
assume for simplicity that ${\mathcal{H}}=L^{2}(0,1)$,
${\mathcal{A}}=\frac{d^{2}}{d\zeta^{2}}-\alpha$,

$D({\mathcal{A}})=\mathopen{}\mathclose{{}\left\\{x\in
H^{2}(0,1):x(0)=x(1),x^{\prime}(0)=x^{\prime}(1)}\right\\},$
${\mathbb{\mathcal{F}}}(x)(\zeta)=f(x(\zeta)),\hskip 7.22743pt\zeta\in[0,1],$
$D({\mathbb{\mathcal{F}}})=\mathopen{}\mathclose{{}\left\\{x\in
L^{2}(0,1):f(x)\in L^{2}(0,1)}\right\\},$

and $f$ and the matrix $\\{a_{\gamma j}\\}_{\gamma,j\in\mathbb{Z}^{d}}$
satisfy conditions of Definition 3.2 and Definition 3.1 respectively. In order
to frame 4.1 in to the general form of equation 2.1 we define:

$H=l^{2}_{\rho}(L^{2}(0,1))=\mathopen{}\mathclose{{}\left\\{(x_{\gamma})\in{\mathcal{H}}^{(\mathbb{Z}^{d})}:\sum_{\gamma\in\mathbb{Z}^{d}}\rho(\gamma)\|x_{\gamma}\|^{2}_{\mathcal{H}}<\infty}\right\\},$
$K=l^{2s}_{\rho}(L^{2s}(0,1))=\mathopen{}\mathclose{{}\left\\{(x_{\gamma})\in(L^{2s}(0,1))^{(\mathbb{Z}^{d})}:\sum_{\gamma\in\mathbb{Z}^{d}}\rho(\gamma)\|x_{\gamma}\|^{2}_{L^{2s}(0,1)}<\infty}\right\\},$

where $\rho=\rho^{\kappa}$ or $\rho=\rho_{\kappa}$, $\kappa>0$ as in Section
3.2. Let $A=A_{0}+A_{1}$ where $A_{1}$ is a bounded linear operator on $H$
given by

$A_{1}(x_{\gamma})=\mathopen{}\mathclose{{}\left(\sum_{j\in\mathbb{Z}^{d}}a_{\gamma
j}X_{\gamma}}\right),x\in D(A_{1})=H,$

and $A_{0}(x_{\gamma})=({\mathcal{A}}x_{\gamma}),$ $x=(x_{\gamma})\in
D(A_{0})$,

$D(A_{0})=\mathopen{}\mathclose{{}\left\\{(x_{\gamma})\in
H:\sum_{\gamma\mathbb{Z}^{d}}\rho(\gamma)\|{\mathcal{A}}x_{\gamma}\|^{2}_{\mathcal{H}}<\infty}\right\\}.$

$A_{1}$ is bounded by a generalization of Theorem 3.2. and $A_{0}+\eta$ on $H$
and its restriction $A_{0\rho}$ to $K$ are $m$-dissipative for sufficiently
small $\eta$. Let us define

$F(x_{\gamma})=({\mathbb{\mathcal{F}}}x_{\gamma}),\hskip
7.22743ptx=(x_{\gamma})\in D(F),$
$D(F)=\mathopen{}\mathclose{{}\left\\{(x_{\gamma})\in
H:\sum_{\gamma\in\mathbb{Z}^{d}}\rho(\gamma)\|{\mathbb{\mathcal{F}}}x_{\gamma}\|^{2}_{\mathcal{H}}<\infty}\right\\}.$

######  Theorem 4.1.

Assume that conditions of Definition 3.2 hold. Then for arbitrary $x\in H$
equation 4.1 has a unique generalized solution $X(\cdot,x)$. If $x\in K$ the
solution is mild.

######  Theorem 4.2.

In addition to conditions of Theorem 4.1 assume that the function $f_{0}$ is
decreasing and $\alpha-\|f_{1}\|_{\mbox{ Lip }}>\omega>0$. Then there exists
$\kappa_{0}>0$ such that the semigroup $P_{t},t\geq 0$ corresponding to
solution of equation 4.1 has a unique invariant measure $\mu$ both in
$H=L^{2}_{\rho^{\kappa}}(\mathbb{R}^{d})$ and
$H=L^{2}_{\rho_{\kappa}}(\mathbb{R}^{d})$, for any $\kappa\in]0,\kappa_{0}[$.
Moreover there exists $C>0$ such that for any bounded Lipschitz function
$\varphi$ on $H$, all $t>0$ and all $x\in H$

$|P_{t}\varphi(x)-\int_{H}\varphi(x)\mu(dx)|\leq(C+2)\|x\|e^{-\omega
t}\|\varphi\|_{\mbox{Lip }}.$ (4.2)

##  5\. Nonlinear Filtering Formulation and Filtering Equations

###  5.1. Stochastic Calculus Method of Nonlinear Filtering

Nonlinear filtering theory for nonlinear stochastic partial differential
equations of fluid mechanics and reacting and diffusing systems was initiated
in [51, 27] and further developed for stochastic models of fluid mechanics
with Lévy noise in [52, 53, 22]. The nonlinear filtering problem for the class
of spin systems studied in this paper is formulated as follows. The key
mathematical equations of stochastic calculus method of nonlinear filtering
are the Kallianpur-Striebel formula, the Fujisaki-Kallianpur-Kunita (FKK)
equation, the Zakai equation and the Kunita’s semigroup versions of FKK and
the Zakai equation, all will be developed below. We will also establish the
equivalency of FKK, Zakai with their Kunita counter parts in the spirit of
Szpirglas [55]. We call the process $X(t)\in H,t\geq 0$ the signal process and
it is governed by the evolution formulated in the earlier sections as:

$dX=(AX+F(X))dt+BdW,$ (5.1) $X(0)=x.$

We note here that $H$ is compact for the finitely truncated classical spin
system, locally compact for the classical spin system and general Hilbert
space for the Euclidean fields and also quantum spin lattice system as
formulated in Section 2, 3 and 4.

The measurement process $Y(t)\in\mathbb{R}^{N},t\geq 0$ is defined as

$dY=h(X)dt+dZ,$ (5.2)

where $Z(t)\in\mathbb{R}^{N}$ is a $N$-dimensional Wiener process which may or
may not be correlated with $W(t)$. Let us denote by ${\mathcal{F}}^{Y}_{t}$
the filtration generated by the sensor data over the time period $0\leq s\leq
t$ (sigma algebra generated by the back measurement):

${\mathcal{F}}^{Y}_{t}=\sigma\mathopen{}\mathclose{{}\left\\{Y_{s},0\leq s\leq
t}\right\\}.$

Our goal is to study the time evolution of the conditional expectation
$E[f(X(t))|{\mathcal{F}}^{Y}_{t}],t\geq 0$ called the nonlinear filter, where
$f$ is some measurable function. It is also the least square best estimate of
$f(X(t))$ given the back measurements. We will first consider a special case
where the signal (spin system dynamics) noise and observation noise are
independent. The following Bayes formula known as the Kallianpur-Striebel
formula can be derived following [30]: Let $g$ be
${\mathcal{F}}^{X}_{t}$-measurable function $g(\cdot):H\rightarrow\mathbb{R}$
integrable on $(\Omega,\Sigma,m)$, $0\leq t\leq T$. We may assume that
processes in 5.2 be defined on the product space
$(\Omega\times\Omega_{0},\Sigma\times\Sigma_{0},m\times m_{0})$ where $Z$ is a
Wiener process on $(\Omega_{0},\Sigma_{0},m_{0})$. Then

$E_{m}[g|{\mathcal{F}}^{Y}_{t}]=\frac{\vartheta^{Y}(g)}{\vartheta^{Y}(1)}$
(5.3)

where

$\vartheta^{Y}_{t}(g)=\int_{\Omega}gq_{t}dm$

with

$q_{t}=\exp\mathopen{}\mathclose{{}\left\\{\int_{0}^{t}h(X(s))\cdot
dY(s)-\frac{1}{2}\int_{0}^{t}|h(X(s))|^{2}ds}\right\\}.$

From the Kallianpur-Striebel formula we can derive the nonlinear filtering
equations of Zakai and FKK type using Itó formula for the special case of
signal and noise independent. However we can proceed as follows to get the
filtering equations for the general case.

Let us define the formal infinitesimal generator ${\mathcal{A}}$ of the
process $X(t)$ using the transition semigroup $P_{t}$ of $X(t)$ in the
following way: for a function $f(\cdot):H\rightarrow\mathbb{R}$, we set
$P_{t}f(x)=E[f(X(t))|X(0)=x]$.

Let us consider the measure space $(H,{\mathcal{B}}(H))$ where
${\mathcal{B}}(H)$ is the Borel algebra generated by closed/open subsets of
$H$. Let ${\mathcal{B}}_{b}(H)$ be the class of bounded
${\mathcal{B}}(H)$-measurable functions on $H$. For
$f_{n},f\in{\mathcal{B}}_{b}(H)$ we say that $f_{n}\rightarrow f$ weakly if
$f_{n}\rightarrow f$ pointwise and the sequence $f_{n}$ is uniformly bounded.

Let ${\mathcal{D}}({\mathcal{A}})$ be the class of functions $f$ in
${\mathcal{B}}_{b}(H)$ such that there exists $f_{0}\in{\mathcal{B}}_{b}(H)$
satisfying

$(P_{t}f)(x)=f(x)+\int_{0}^{t}(P_{s}f_{0})(x)ds,\hskip 7.22743pt\forall x\in
H.$ (5.4)

Note that $f_{0}$ is uniquely determined by the above equation. Hence we set
${\mathcal{A}}f=f_{0}$ and formally define the extended generator
${\mathcal{A}}$ as:

${\mathcal{A}}f(x):=\mbox{ weak-limit}_{t\downarrow
0^{+}}\frac{P_{t}f(x)-f(x)}{t},\hskip 7.22743ptf\in D({\mathcal{A}}),$

where

$D({\mathcal{A}})=\mathopen{}\mathclose{{}\left\\{f\in{\mathcal{B}}_{b}(H):\mbox{
weak-limit }_{t\downarrow
0^{+}}\frac{P_{t}f(x)-f(x)}{t}={\mathcal{A}}f(x)\mbox{ exists}}\right\\}.$

Since the domain $D({\mathcal{A}})$ cannot easily be defined explicitly, we
will also work with a sub-class of functions known as cylindrical test
functions ${\mathcal{D}}_{\mbox{cyl}}$ defined as follows:

${\mathcal{D}}_{\mbox{cyl}}:=\mathopen{}\mathclose{{}\left\\{f\in
C_{b}(H):f(X)=\varphi(\langle X,e_{1}\rangle_{H},\cdots,\langle
X,e_{N}\rangle_{H}),e_{i}\in D(A),i=1,\cdots,N,\varphi\in
C^{\infty}_{0}(\mathbb{R}^{N})}\right\\}.$

Then $D_{x}f(x)\in D(A)$ and also ${\mathcal{D}}_{\mbox{cyl}}\subset
D({\mathcal{A}})$. The formal generator of the stochastic process $X(t)$ is
given by

${\mathcal{A}}\Phi(x)=\frac{1}{2}\mbox{Tr}(B^{*}QBD^{2}_{x}\Phi(x))+\langle
Ax+F(x),D_{x}\Phi\rangle,\hskip 7.22743pt\Phi\in{\mathcal{D}}_{\mbox{cyl}}.$
(5.5)

We now note that the solvability theorems established in the previous section
imply by Itó’s formula the following result.

######  Proposition 5.1.

Define

$M_{t}(f)=f(X(t))-f(x)-\int_{0}^{t}{\mathcal{A}}f(X(s))ds,\hskip
7.22743ptf\in{\mathcal{D}}_{\mbox{cyl}},$ (5.6)

where $X(t)$ is the spin process defined by the stochastic partial
differential equation 5.1. Then $(M_{t}(f),{\mathcal{F}}_{t},m)$ is a
martingale.

In fact applying Itó formula [21] to the stochastic process defined by 5.1

$f(X(t))=f(x)+\int_{0}^{t}{\mathcal{A}}f(X(s))ds+\int_{0}^{t}(\frac{\partial
f}{\partial x}(X(s),BdW(s)),\hskip 7.22743ptf\in{\mathcal{D}}_{\mbox{cyl}}.$

Let us now state a general martingale lemma and then specialize it to the
filtering problem.

######  Lemma 5.1.

Let ${\mathcal{F}}_{t}$ and ${\mathcal{G}}_{t}$ be filtrations with
${\mathcal{G}}_{t}\subset{\mathcal{F}}_{t}$. Suppose that

$M_{t}^{\mathcal{F}}=U(t)-U(0)-\int_{0}^{t}V(s)ds$

is an ${\mathcal{F}}_{t}$-martingale. Then

$M^{\mathcal{G}}_{t}=E[U(t)|{\mathcal{G}}_{t}]-E[U(0)|{\mathcal{G}}_{0}]-\int_{0}^{t}E[V(s)|{\mathcal{G}}_{s}]ds$

is a ${\mathcal{G}}_{t}$-martingale.

We now state this property in the nonlinear filtering context:

######  Theorem 5.1.

Define

$M_{t}^{Y}(f)=E[f(X(t))|{\mathcal{F}}_{t}^{Y}]-f(x)-\int_{0}^{t}E[{\mathcal{A}}f(X(s))|{\mathcal{F}}_{s}^{Y}]ds,\hskip
7.22743ptf\in{\mathcal{D}}_{\mbox{cyl}}.$ (5.7)

Then $(M_{t}^{Y}(f),{\mathcal{F}}_{t}^{Y},m)$ is a martingale.

Proof: First note that

$E\mathopen{}\mathclose{{}\left[M^{Y}_{f}(f)|{\mathcal{F}}^{Y}_{s}}\right]-M^{Y}_{s}(f)=E\mathopen{}\mathclose{{}\left[M^{Y}_{f}(f)-M^{Y}_{s}(f)|{\mathcal{F}}^{Y}_{s}}\right].$

Noting that ${\mathcal{F}}_{t}^{Y}\subset{\mathcal{F}}_{t}$, we consider, for
$s<t$,

$E\mathopen{}\mathclose{{}\left[M^{Y}_{f}(f)-M^{Y}_{s}(f)|{\mathcal{F}}^{Y}_{s}}\right]$
$=E\mathopen{}\mathclose{{}\left[E[f(X(t))|{\mathcal{F}}_{t}^{Y}]-E[f(X(s))|{\mathcal{F}}_{s}^{Y}]|{\mathcal{F}}^{Y}_{s}}\right]-E\mathopen{}\mathclose{{}\left[\int_{s}^{t}E[{\mathcal{A}}f(X(r))|{\mathcal{F}}_{r}^{Y}]dr|{\mathcal{F}}^{Y}_{s}}\right].$

Using the tower property of the conditional expectation we conclude that

$E\mathopen{}\mathclose{{}\left[M^{Y}_{f}(f)-M^{Y}_{s}(f)|{\mathcal{F}}^{Y}_{s}}\right]=E[f(X(t))|{\mathcal{F}}_{s}^{Y}]-E[f(X(s))|{\mathcal{F}}_{s}^{Y}]-\int_{s}^{t}E[{\mathcal{A}}f(X(r))|{\mathcal{F}}_{s}^{Y}]dr$
$=E\mathopen{}\mathclose{{}\left[f(X(t))-f(X(s))-\int_{s}^{t}{\mathcal{A}}f(X(r))dr|{\mathcal{F}}_{s}^{Y}}\right]$
$=E\mathopen{}\mathclose{{}\left[M_{t}(f)-M_{s}(f)|{\mathcal{F}}^{Y}_{s}}\right]=E\mathopen{}\mathclose{{}\left[E\mathopen{}\mathclose{{}\left[M_{t}(f)-M_{s}|{\mathcal{F}}_{s}}\right]|{\mathcal{F}}^{Y}_{s}}\right]=0,$

since $M_{t}(f)$ is a $({\mathcal{F}}_{t},m)$\- martingale. Hence
$E\mathopen{}\mathclose{{}\left[M^{Y}_{f}(f)|{\mathcal{F}}^{Y}_{s}}\right]=M^{Y}_{s}(f)$.

Using Fujisaki-Kallianpur-Kunita (1972) [23] (in particular Theorem 3.1, Lemma
4.2, Theorem 4.1 and equation 6.11 in that paper) we will characterize
$M_{t}^{Y}$ explicitly using the next set of arguments.

######  Definition 5.1.

Innovation process is defined as:

$d\nu^{Y}(t)=dY(t)-E[h(X(t))|{\mathcal{F}}_{t}^{Y}]dt,\hskip
7.22743ptt\in[0,T].$ (5.8)

######  Lemma 5.2.

[23] The innovation process $(\nu^{Y},{\mathcal{F}}^{Y}_{t},m)$ is an
$N$-vector standard Wiener process. Furthermore ${\mathcal{F}}^{Y}_{s}$ and
$\sigma\mathopen{}\mathclose{{}\left\\{\nu_{v}^{Y}-\nu_{u}^{Y};s\leq u\leq
v\leq T}\right\\}$ are independent.

Let us now state an important martingale representation theorem (Theorem 3.1
[23]):

######  Theorem 5.2.

Every square integrable martingale $(M^{Y}_{t},{\mathcal{F}}^{Y}_{t},m)$ is
sample continuous and has the representation

$M_{t}^{Y}-M^{Y}_{0}=\int_{0}^{t}\Xi(s)\cdot d\nu^{Y}(s).$ (5.9)

where $\nu(t)$ is the innovation process and
$\Xi(s)=(\Xi_{s}^{1},\cdots,\Xi_{s}^{N})$ is jointly measurable and adapted to
${\mathcal{F}}^{Y}_{s}$.

######  Lemma 5.3.

[23] Let $(M_{t}(f),{\mathcal{F}}_{t},m)$ be the square integrable martingale
5.6. Then there exists unique sample continuous process $\langle
M(f),Z^{i}\rangle,(i=1,\cdots,N)$ adapted to ${\mathcal{F}}_{t}$ such that
almost all sample functions are of bounded variation and $M_{t}(f)-\langle
M(f),Z^{i}\rangle_{t}$ are ${\mathcal{F}}_{s}$-martingales. Furthermore each
$\langle M(f),Z^{i}\rangle_{t}$ has the following properties: it is absolutely
continuous with respect to Lebesgue measure in $[0,T]$. There exists a
modification of the Radon-Nikodym derivative which is $(t,\omega)$-measurable
and adapted to ${\mathcal{F}}_{t}$ and which shall denote by
$\tilde{D}_{t}^{i}f(\omega)$. Then using the vector notation
$\tilde{D}_{t}f=(\tilde{D}_{t}^{1}f,\cdots,\tilde{D}_{t}^{N}f)$,

$\langle M(f),Z\rangle_{t}=\int_{0}^{t}\tilde{D}_{s}fds,\hskip 7.22743pt\mbox{
a.s},$

where

$\int_{0}^{T}E|\tilde{D}_{s}f|^{2}ds<\infty.$

If the process $X(t)$ and $Z(t)$ are independent then

$\langle M(f),Z\rangle_{t}=0,\mbox{ a.s}.$

The following theorem characterizes $\Xi$ explicitly:

######  Theorem 5.3.

Let $f\in D({\mathcal{A}})$ and

$\int_{0}^{T}|h(X(t))|^{2}dt<\infty,\hskip 7.22743pt\mbox{ a.s.}$ (5.10)

Then the evolution of the conditional expectation
$E[f(X(t))|{\mathcal{F}}_{t}^{Y}]$ (the nonlinear filter) is characterized by
the Fujisaki-Kallianpur-Kunita equation:

$E[f(X(t))|{\mathcal{F}}_{t}^{Y}]=E[f(X(0))|{\mathcal{F}}_{0}^{Y}]+\int_{0}^{t}E[{\mathcal{A}}f(X(s))|{\mathcal{F}}_{s}^{Y}]ds$
$+\int_{0}^{t}\mathopen{}\mathclose{{}\left\\{E[f(X(s))h(X(s))|{\mathcal{F}}_{s}^{Y}]-E[f(X(s))|{\mathcal{F}}_{s}^{Y}]E[h(X(s))|{\mathcal{F}}_{s}^{Y}]+E[\tilde{D}_{s}f(X(s))|{\mathcal{F}}_{s}^{Y}]}\right\\}\cdot
d\nu^{Y}(s)$ (5.11)

where $\tilde{D}f(X(s))$ is given by

$\langle M(f),Z\rangle=\int_{0}^{t}\tilde{D}_{s}fds.$

In particular $\tilde{D}_{t}f=0$ if $W$ and $Z$ are independent. Moreover, if

$BW=\sigma_{1}W_{1}+\sigma_{2}Z,$

then

$\tilde{D}_{t}f=\sigma_{2}^{*}\frac{\partial f}{\partial x}.$

Note that if $h$ has a growth condition

$|h(x)|\leq C||x||^{p},\mbox{ for some }p\geq 1,\hskip 7.22743ptx\in H,$

then by theorem 2.1 the condition 5.10 is satisfied.

###  5.2. Existence and Uniqueness of Measure-valued solutions to Filtering
Equations

A theorem of Getoor [24] provides the existence of a random measure
$\pi_{t}^{Y}$ that is measurable with respect to ${\mathcal{F}}_{t}^{Y}$ such
that

$E[f(X(t))|{\mathcal{F}}_{t}^{Y}]=\pi_{t}^{Y}[f]=\int_{H}f(\zeta)\pi_{t}^{Y}(d\zeta).$

Substituting in 5.11, the probability measure-valued process
$\pi_{t}^{Y}[\cdot]$ satisfies the Fujisaki-Kallianpur-Kunita (FKK) type
equation:

$d\pi_{t}^{Y}[f]=\pi_{t}^{Y}[{\mathcal{A}}f]dt+\mathopen{}\mathclose{{}\left(\pi_{t}^{Y}[hf+\tilde{D}_{t}f]-\pi_{t}^{Y}[h]\pi_{t}^{Y}[f]}\right)\cdot
d\nu^{Y}(t),\hskip 7.22743pt\mbox{ for }f\in{\mathcal{D}}_{\mbox{cyl}}.$
(5.12)

If we set

$\vartheta_{t}^{Y}[f]:=\pi_{t}^{Y}[f]\exp\mathopen{}\mathclose{{}\left\\{\int_{0}^{t}\pi_{t}^{Y}[h]\cdot
dY(s)-\frac{1}{2}\int_{0}^{t}|\pi_{t}^{Y}[h]|^{2}ds}\right\\},$ (5.13)

then the measure-valued process $\vartheta_{t}^{y}[\cdot]$ satisfies the Zakai
type equation:

$d\vartheta_{t}^{Y}[f]=\vartheta_{t}^{Y}[{\mathcal{A}}f]dt+\vartheta_{t}^{Y}[hf+\tilde{D}_{t}f]\cdot
dY(t),\mbox{ for }f\in{\mathcal{D}}_{\mbox{cyl}}.$ (5.14)

Such measure-valued evolutions were first derived by Kunita [38] in the
context of compact space valued signal processes.

One method of proving uniqueness of measure-valued solutions to filtering
equations is to start with the unique solution of the backward Kolmogorov
equation similar to the papers in the case of nonlinear filtering of
stochastic Navier-Stokes equation [51] or for the case of stochastic reaction
diffusion equation [27]. Here we use the solution $\Phi(t)$ of the backward
Kolmogorov equation with initial data $\Phi$ and express the Zakai equation
as:

$\vartheta_{t}^{Y}[\Psi]=\vartheta_{0}^{Y}[\Phi(t)]+\int_{0}^{t}\vartheta_{s}^{Y}[(\frac{\partial}{\partial
s}+{\mathcal{A}})\Phi(s)]ds+\int_{0}^{t}\mathopen{}\mathclose{{}\left(\vartheta_{s}^{Y}[(h+\tilde{D}_{t})\Phi(s)]}\right)\cdot
d\nu_{s},$ (5.15)

and utilize the fact the $\Phi$ satisfies $(\frac{\partial}{\partial
s}+{\mathcal{A}})\Phi(s)=0$ in the above equation (eliminating the integral
involving the generator ${\mathcal{A}})$ and then proceed to prove the
uniqueness of the random measures $\vartheta_{t}^{Y}$ as well as
$\pi_{t}^{Y}$.

We now specialize to the case of signal noise $W$ and observation noise $Z$
being independent. In this setting we can use the results of Kunita [38] and
Szpirglas[55] to obtain equivalent nonlinear filtering equation that do not
explicitly involve the generator ${\mathcal{A}}$ of the signal Markov process
$X$.

Uniqueness theorem relies on the following lemma [55, 32]:

######  Lemma 5.4.

Let us define a subclass of ${\mathcal{D}}({\mathcal{A}})$ as:

${\mathcal{D}}_{2}({\mathcal{A}}):=\mathopen{}\mathclose{{}\left\\{f\in{\mathcal{D}}({\mathcal{A}}):{\mathcal{A}}f\in{\mathcal{D}}({\mathcal{A}})}\right\\}.$

Let $\mu_{1},\mu_{2}\in{\mathcal{M}}(H)$ be finite measures on
$(H,{\mathcal{B}}(H))$ such that

$\langle\mu_{1},f\rangle=\langle\mu_{2},f\rangle,\hskip 14.45377pt\forall
f\in{\mathcal{D}}_{2}({\mathcal{A}}),$ (5.16)

where $\langle\mu,f\rangle=\int_{H}f(x)d\mu(x).$ Then
$\mu_{1}=\mu_{2}\in{\mathcal{M}}(H)$.

The uniqueness of measure-valued solutions for the nonlinear filtering
equations and equivalence of two different forms, one involving the formal
generator and other using Feller semigroup of the system Markov process
eliminating the term involving the generator started with Kunita[38] and
Szpirglass [55] and the following series of theorems can be proven by the same
methods as in those original works for signal state space compact, locally
compact and general Hilbert space cases as in [38, 54, 14, 18, 22]:

######  Theorem 5.4.

The random probability measure valued process $\pi^{Y}_{t}\in{\mathcal{P}}(H)$
uniquely solves the evolution equation:

$\pi_{t}^{Y}[f]=\pi_{0}^{Y}[f]+\int_{0}^{t}\pi_{s}^{Y}[{\mathcal{A}}f]ds+\int_{0}^{t}\mathopen{}\mathclose{{}\left(\pi_{s}^{Y}[hf]-\pi_{s}^{Y}[h]\pi_{s}^{Y}[f]}\right)\cdot
d\nu^{Y}(s),\hskip 7.22743pt\mbox{ for }f\in D({\mathcal{A}}).$ (5.17)

Equivalently, $\pi^{Y}_{t}$ uniquely solves the evolution equation:

$\pi_{t}^{Y}[f]=\pi_{0}[P_{t}f]+\int_{0}^{t}\mathopen{}\mathclose{{}\left(\pi_{s}^{Y}((P_{t-s}f)h)-\pi_{s}^{Y}(P_{t-s}f)\pi_{s}^{Y}(h)}\right)\cdot
d\nu^{Y}_{s},\hskip 7.22743pt\mbox{ for }f\in{\mathcal{B}}_{b}(H).$ (5.18)

######  Theorem 5.5.

the random measure valued process $\vartheta^{Y}_{t}\in{\mathcal{M}}(H)$
uniquely solves the evolution equation:

$\vartheta_{t}^{Y}[f]=\vartheta_{0}^{Y}[f]+\int_{0}^{t}\vartheta_{s}^{Y}[{\mathcal{A}}f]ds+\int_{0}^{t}\vartheta_{s}^{Y}[hf]\cdot
dY(s),\hskip 7.22743pt\mbox{ for }f\in D({\mathcal{A}}).$ (5.19)

Equivalently, $\vartheta^{Y}_{t}\in{\mathcal{M}}(H)$ uniquely solves the
evolution equation:

$\vartheta_{t}^{Y}[f]=\vartheta_{0}[P_{t}f]+\int_{0}^{t}\vartheta_{s}^{Y}((P_{t-s}f)h)\cdot
dY({s}),\hskip 7.22743pt\mbox{ for }f\in{\mathcal{B}}_{b}(H).$ (5.20)

######  Theorem 5.6.

Let $\pi^{Y}_{t}$ be the unique solution of the FKK equation for arbitrary
initial condition $\pi_{0}$. Furthermore,

  1. (1)

The solution $\pi^{Y}_{t}$ is $\sigma(Y(s)-Y(0);0\leq s\leq
t)\vee\sigma(\pi_{0})$ -measurable.

  2. (2)

Let $\pi_{t}^{Y(\nu)}$ and $\pi_{t}^{Y(\mu)}$ be solutions with initial
conditions $\pi_{0}=\nu$ and $\pi_{0}=\mu$ respectively, where
$\mu,\nu\in{\mathcal{M}}(H)$. Then for every $t>0$

$\lim_{\nu\rightarrow\mu}E\mathopen{}\mathclose{{}\left[|\pi_{t}^{Y(\nu)}(f)-\pi_{t}^{Y(\mu)}(f)|}\right]=0,\hskip
7.22743ptf\in C_{b}(H).$ (5.21)

######  Theorem 5.7.

The filtering process $(\pi^{Y},{\mathcal{F}}_{t}^{Y},P_{\mu})$,
$\mu\in{\mathcal{M}}(H)$ are Markov process associated with the transition
probabilities $\Pi_{t}(\nu,\Gamma)$ defined by

$\Pi_{t}(\nu,\Gamma)=P(\pi_{t}^{Y}\in\Gamma:\hskip
7.22743pt\Gamma\in{\mathcal{B}}({\mathcal{M}}(H)),$

where ${\mathcal{B}}({\mathcal{M}}(H))$ is the Borel algebra generated by the
open (or closed) sets in ${\mathcal{M}}(H)$.

Furthermore, the transition probabilities $\Pi_{t}(\nu,\Gamma)$ define a
Feller semigroup in $C_{b}({\mathcal{M}}(H))$, where $C_{b}({\mathcal{M}}(H))$
is the space of all real continuous functions over ${\mathcal{M}}(H)$.

Systematic study of time asymptotic analysis of nonlinear filter was initiated
by H. Kunita [38] and further developed by several authors [41, 54, 39, 42, 6,
13, 14, 18]. We will further develop this subject to be applicable to
classical and quantum systems studied in this paper.

######  Definition 5.2.

A probability measure $\nu\in{\mathcal{P}}(H)$ on the Borel algebra
${\mathcal{B}}(H)$ is called the barycenter of the probability measure
$\Phi\in{\mathcal{P}}({\mathcal{P}}(H))$ on the Borel algebra
${\mathcal{B}}({\mathcal{P}}(H))$ if and only if for every $\phi\in C_{b}(H)$

$\nu(\phi)=\int_{{\mathcal{P}}(H)}\nu^{\prime}(\phi)\Phi(d\nu^{\prime}).$
(5.22)

The following theorem is inspired by the original paper of Kunita[38] which
was in turn technically corrected by several authors [7, 18, 56].

######  Theorem 5.8.

Let us assume that the following condition regarding the sigma fields is
satisfied:

$\bigcap_{t\leq
0}\mathopen{}\mathclose{{}\left({\mathcal{F}}_{-\infty,0}^{Y}\vee{\mathcal{F}}^{X}_{-\infty,t}}\right)={\mathcal{F}}_{-\infty,0}^{Y}\vee{\mathcal{F}}^{X}_{-\infty,-\infty},\hskip
7.22743ptm\mbox{ a.s}.$ (5.23)

The existence and uniqueness of invariant measures for the transition
probabilites $P_{t}(x,A),A\in{\mathcal{B}}(H)$, associated with the classical
(Theorem 2.3, Theorem 3.5, Theorem 3.7) and quantum spin systems (Theorem 4.2)
imply that the transition probabilities $\Pi_{t}(\nu,\Gamma)$ associated with
the filtering process for these classican and quantum spin systems have unique
invariant measures. The invariant measure $\mu$ for $P_{t}(x,A)$ is the
barrycenter of the invariant measure $\Phi$ of $\Pi_{t}(\nu,\Gamma)$:

$\mu(f)=\int_{{\mathcal{M}}(H)}\nu(f)\Phi(d\nu),\hskip 7.22743pt\forall f\in
C_{b}(H).$ (5.24)

###  5.3. Evolution Equation for Error Covariance

The seminal paper of Kalman and Bucy [37] also gives a characterization of the
evolution of error covariance using a Riccati equation. This concept was
generalized by Kunita[39] for nonlinear filters. Let us describe the time
evolution of error covariance slightly generalizing the results of Kunita[39].
The error covariance of general moments $f(X(t))$ and $g(X(t))$ is given by

$\mathcal{P}_{t}[f,g]=E\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(f(X(t))-\pi_{t}(f)}\right)\mathopen{}\mathclose{{}\left(g(X(t))-\pi_{t}(g)}\right)}\right].$
(5.25)

######  Theorem 5.9.

Let $f,g\in D({\mathcal{A}})$. Then $\mathcal{P}_{t}[f,g]$ is differentiable
with respect to $t$ and the derivative satisfies

$\frac{d}{dt}\mathcal{P}_{t}[f,g]=\mathcal{P}_{t}[f,{\mathcal{A}}g]+\mathcal{P}_{t}[{\mathcal{A}}f,g]+\mathcal{Q}_{t}[f,g]$
$-\sum_{i=1}^{N}E\mathopen{}\mathclose{{}\left[\pi_{t}\mathopen{}\mathclose{{}\left\\{(f-\pi_{t}(f))(h^{i}-\pi_{t}(h^{i}))+\tilde{D}f}\right\\}\pi_{t}\mathopen{}\mathclose{{}\left\\{(g-\pi_{t}(g))(h^{i}-\pi_{t}(h^{i}))+\tilde{D}g}\right\\}}\right],$
(5.26)

where

$\mathcal{Q}_{t}[f,g]=E\mathopen{}\mathclose{{}\left[\mbox{
tr}(B^{*}QBD_{x}f\otimes D_{x}g)}\right].$ (5.27)

Proof:

Note that we have:

${\mathcal{A}}(fg)=f{\mathcal{A}}g+g{\mathcal{A}}f+\mbox{
tr}(B^{*}QBD_{x}f\otimes D_{x}g).$

We first write the FKK equation for $\pi_{t}(fg)$ as

$\pi_{t}(fg)=\pi_{0}(fg)+\int_{0}^{t}\pi_{s}(f{\mathcal{A}}g)ds+\int_{0}^{t}\pi_{s}(g{\mathcal{A}}f)ds+\int_{0}^{s}\pi_{s}(\mbox{
tr}(B^{*}QBD_{x}f\otimes D_{x}g))ds+M^{\pi}_{t},$

where $M^{\pi}_{t}$ is a ${\mathcal{F}}^{Y}_{t}$-martingale with mean zero and
$f,g\in D({\mathcal{A}})$. Similarly since $\pi_{t}(f)$ and $\pi_{t}(g)$
satisfy FKK equation we have by Ito formula for the product
$\pi_{t}(f)\pi_{t}(g)$:

$\pi_{t}(f)\pi_{t}(g)=\pi_{0}(f)\pi_{0}(g)+\int_{0}^{t}\pi_{s}(f)\pi_{s}({\mathcal{A}}g)ds+\int_{0}^{t}\pi_{s}(g)\pi_{s}({\mathcal{A}}f)ds$
$+\sum_{i}\int_{0}^{t}\mathopen{}\mathclose{{}\left\\{\pi_{s}(fh^{i})-\pi_{s}(f)\pi_{s}(h^{i})+\pi_{s}(\tilde{D}f)}\right\\}\mathopen{}\mathclose{{}\left\\{\pi_{s}(gh^{i})-\pi_{s}(g)\pi_{s}(h^{i})+\pi_{s}(\tilde{D}g)}\right\\}ds+\tilde{M}^{\pi}_{t},$

where $\tilde{M}^{\pi}_{t}$ is a ${\mathcal{F}}^{Y}_{t}$-martingale with mean
zero. Now noting that

$\pi_{s}(fg)-\pi_{s}(f)\pi_{s}(g)=\pi_{s}\mathopen{}\mathclose{{}\left\\{(f-\pi_{s}(f))(g-\pi_{s}(g))}\right\\},$

we get

$\mathcal{P}_{t}(f,g)=E[\pi_{t}(fg)-\pi_{t}(f)\pi_{t}(g)].$

Substituting for $\pi_{t}(fg)$ and $\pi_{t}(f)\pi_{t}(g)$ from the previous
two equations and taking expectation we arrive at

$\mathcal{P}_{t}(f,g)=\mathcal{P}_{0}(f,g)+\int_{0}^{t}\mathopen{}\mathclose{{}\left\\{\mathcal{P}_{s}[f,{\mathcal{A}}g]+\mathcal{P}_{s}[{\mathcal{A}}f,g]+\mathcal{Q}_{s}[f,g]}\right\\}ds$
$-\sum_{i=1}^{N}\int_{0}^{t}E\mathopen{}\mathclose{{}\left[\pi_{s}\mathopen{}\mathclose{{}\left\\{(f-\pi_{s}(f))(h^{i}-\pi_{s}(h^{i}))+\tilde{D}f}\right\\}\pi_{s}\mathopen{}\mathclose{{}\left\\{(g-\pi_{s}(g))(h^{i}-\pi_{s}(h^{i}))+\tilde{D}g}\right\\}}\right]ds.$

##  6\. White Noise Theory of Nonlinear Filtering

White noise theory of nonlinear filtering was initiated by several papers of
A. V. Balakrishnan [5, 4, 5] and systematically developed by G. Kallianpur and
R. Karandikar [31, 32, 33, 34, 35] (see also Kallianpur’s review [36]). We
will further develop this method to be applicable to classical and quantum
spin systems discussed in this paper. Key mathematical equations of white
noise nonlinear filtering theory are essentially parallel to stochastic
calculus counterpart but of deterministic in nature. The Kallianpur-Striebel
formula, the Fujisaki-Kallianpur-Kunita equation, the Zakai equation and the
Kunita’s semigroup versions of FKK and the Zakai equation all have white noise
counterparts and will be developed below for the spin system models. We will
also establish the equivalency of FKK, Zakai with their Kunita counter parts
in the spirit of Szpirglas[55].

We will now describe the theory of Segal [48] and Gross [25, 26] of finitely
additive cylindrical measures on separable Hilbert spaces. Let ${\mathcal{H}}$
be a separable Hilbert space and ${\mathcal{P}}$ be the set of orthogonal
projections on ${\mathcal{H}}$ having finite dimensional range. For
$P\in{\mathcal{P}}$, let ${\mathcal{C}}_{P}=\\{P^{-1}B:B\mbox{ a Borel set in
the range of }P\\}$. Let
${\mathcal{C}}=\cup_{P\in{\mathcal{P}}}{\mathcal{C}}_{P}$. A cylinder measure
${\bf n}$ on ${\mathcal{H}}$ is a finitely additive measure on
$({\mathcal{H}},{\mathcal{C}})$ such that its restriction to
${\mathcal{C}}_{P}$ is countably additive for each $P\in{\mathcal{P}}$.

Let $L$ be a representative of the weak-distribution corresponding to the
cylinder measure ${\bf n}$. This means that $L$ is a linear map from
${\mathcal{H}^{*}}$ (identified with ${\mathcal{H}}$) in to
${\mathcal{L}}_{0}(\Omega_{1},\Sigma_{1},m_{1})$ -the space of all random
variables on a $\sigma$-additive probability space
$(\Omega_{1},\Sigma_{1},m_{1})$ such that

${\bf n}(h\in{\mathcal{H}}:((h,h_{1}),(h,h_{2}),\cdots,(h,h_{k}))\in B)$
$=m_{1}(\omega\in\Omega_{1};\mathopen{}\mathclose{{}\left(L(h_{1})(\omega),L(h_{2})(\omega),\cdots,L(h_{k})(\omega)}\right)\in
B)$ (6.1)

for all Borel sets $B$ in $\mathbb{R}^{k}$,
$h_{1},\cdots,h_{k}\in{\mathcal{H}}$, $k\geq 1$. Two maps $L,L^{\prime}$ are
said to be equivalent if both satisfy 6.1 and the equivalence class of such
maps is the weak distribution corresponding to the cylindrical measure ${\bf
n}$.

A function $f$ on ${\mathcal{H}}$ is called a tame function if it is of the
form

$f(y)=\varphi((y,h_{1}),\cdots,(y,h_{k})),\mbox{ for some }k\geq
1,h_{1},\cdots,h_{k}\in{\mathcal{H}}$ (6.2)

and a Borel function $\varphi:\mathbb{R}^{k}\rightarrow\mathbb{R}$. For a tame
function $f$ given by 6.2, we associate the random variable
$\varphi(L(h_{1}),\cdots,L(h_{k}))$ on $(\Omega_{1},\Sigma_{1},m_{1})$ and
denote it by $\tilde{f}$. We can extend this map $f\rightarrow\tilde{f}$ to a
larger class of functions as follows [26]:

######  Definition 6.1.

Let ${\mathcal{L}}({\mathcal{H}},{\mathcal{C}},{\bf n})$ be the class of
continuous functions $f$ on ${\mathcal{H}}$ such that the net
$\\{\tilde{(f\circ P)}:P\in{\mathcal{P}}\\}$ is Cauchy in $m_{1}$-measure.
Here $P_{1}<P_{2}$ if the range $P_{1}\subseteq$ range $P_{2}$. For
$f\in{\mathcal{L}}({\mathcal{H}},{\mathcal{C}},{\bf n})$, we define:

$\tilde{f}={\mbox{Lim in Prob}}_{P\in{\mathcal{P}}}\tilde{(f\circ P)}.$

The map $f\rightarrow\tilde{f}$ is linear, multiplicative and depends only on
$f$ and ${\bf n}$ and is independent of the representative of the weak
distribution.

######  Definition 6.2.

The function $f\in{\mathcal{L}}({\mathcal{H}},{\mathcal{C}},{\bf n})$ is
integrable with respect to ${\bf n}$ if
$\int_{\Omega_{1}}|\tilde{f}|dm_{1}<\infty$ and then for $C\in{\mathcal{C}}$
define the integral $f$ with respect to ${\bf n}$ over $C$ denoted by
$\int_{C}fd{\bf n}$ by

$\int_{C}fd{\bf n}=\int_{\Omega_{1}}\tilde{1_{C}}\tilde{f}dm_{1}.$

Let us now consider the class of all Gauss measures on ${\mathcal{H}}$ defined
by

$\mu\\{y\in{\mathcal{H}}:(y,h)\leq a\\}=\frac{1}{\sqrt{2\pi
v(h)}}\int_{-\infty}^{a}\exp{(-\frac{x^{2}}{2v(h)})}dx,\hskip 7.22743pt\forall
h\in{\mathcal{H}}.$

The special case of $v(h)=\|h\|^{2}$ is called the canonical Gauss measure
${\bf m}$. The following Lemma from Sato [47] (in particular, Lemma 6 in that
paper)clarifies the $\sigma$-additivity of Gauss measures on separable Hilbert
spaces.

######  Lemma 6.1.

let ${\mathcal{H}}$ be a separable Hilbert space and let $\mu$ be a Gaussian
cylinder measure on $({\mathcal{H}},{\mathcal{C}})$ with variance $v(h)$. Then
$\mu$ has a $\sigma$-additive extension to $({\mathcal{H}},\bar{\mathcal{C}})$
(here $\bar{\mathcal{C}}$ is the minimal $sigma$-algebra containing
${\mathcal{C}})$ if and only if the characteristic functional of $\mu$ is of
the form:

$\int_{\mathcal{H}}e^{i(h,x)}d\mu(x)=\exp{\mathopen{}\mathclose{{}\left[i<h,m_{e}>-\frac{1}{2}\|Sh\|^{2}}\right]},\hskip
7.22743pt\forall h\in{\mathcal{H}},$

where $m_{e}$ is an element of ${\mathcal{H}}$, and $S$ is a nonnegative
selfadjoint Hilbert-Schmidt operator on ${\mathcal{H}}$.

The canonical Gauss measure ${\bf m}$ corresponds to $m_{e}=0$ and $S=I$ and
since the identity operator $I$ is not Hilbert-Schmidt, the cannonical Gauss
measure ${\bf m}$ is only finitely additive (See also Chapter-I, Proposition
4.1 of Kuo[40]).

The identity map $e$ on ${\mathcal{H}}$, considered as a map from
$({\mathcal{H}},{\mathcal{C}},{\bf n})$ in to $({\mathcal{H}},{\mathcal{C}})$
is called the Gaussian white noise.

Let us now start with the abstract version of the white noise filtering model
as formulated by Kallianpur and Karandikar [35]:

$y=\zeta+e$ (6.3)

where $\zeta$ is an ${\mathcal{H}}$-valued random variable defined on a
countably additive probability space $(\Omega,{\mathcal{F}},m)$, independent
of $e$. To formulate this mathematicially precisely let
${\mathcal{E}}={\mathcal{H}}\times\Omega$ and

$\Sigma=\cup_{P\in{\mathcal{P}}}{\mathcal{C}}_{P}\otimes{\mathcal{F}}\mbox{
where }{\mathcal{C}}_{P}\otimes{\mathcal{F}}\mbox{ is the product sigma
field}.$

For $P\in{\mathcal{P}}$, let $\alpha_{P}$ be the product measure $({\bf
m}|{\mathcal{C}}_{P})\otimes m$ which is countably additive. This defines a
unique finitely additive probability measure $\alpha$ on
$({\mathcal{E}},\Sigma)$ such that $\alpha=\alpha_{P}$ on
${\mathcal{C}}_{P}\otimes{\mathcal{F}}$.

let $e,\zeta,y$ be $H$-valued maps on ${\mathcal{E}}$ defined by

$e(h,\omega)=h,$ $\zeta(h,\omega)=\zeta(\omega),$
$y(h,\omega)=e(h,\omega)+\zeta(h,\omega),\hskip
7.22743pt(h,\omega)\in{\mathcal{H}}\times\Omega.$ (6.4)

The model 6.4 is the abstract version of the white noise filtering on
$({\mathcal{E}},\Sigma,\alpha)$. Our goal is to first characterize the
conditional expectation $E[g|y]$ in this finitely additive setting.

######  Definition 6.3.

If there exists a $v\in{\mathcal{L}}({\mathcal{H}},{\mathcal{C}},{\bf n})$
such that

$\int_{{\mathcal{H}}\times\Omega}g(\omega)1_{C}(y(h,\omega))d\alpha(h,\omega)=\int_{C}v(y)d{\bf
n}(y),\hskip 7.22743pt\forall C\in{\mathcal{C}},$ (6.5)

then we define $v$ to be the conditional expectation of $g$ given $y$ and
express it as $E[g|y]=v.$

Note that the integrand $g(\omega)1_{C}(y(h,\omega))$ is
${\mathcal{C}}_{P}\otimes{\mathcal{F}}$-measurable for $C\in{\mathcal{C}}_{P}$
and hence the integrals in 6.5 are well-defined. Let us now state the elegant
Bayes formula for the finitely additive white noise framework proved in [31]:

######  Theorem 6.1.

Let $y,\zeta$ be as in 6.4. Let $g$ be an integrable function on
$(\Omega,{\mathcal{F}},m)$. Then

$E[g|y]=\frac{\int_{\Omega}g(\omega)exp{\mathopen{}\mathclose{{}\left\\{(y,\zeta(\omega))-\frac{1}{2}\|\zeta(\omega)\|^{2}}\right\\}}dm(\omega)}{\int_{\Omega}exp{\mathopen{}\mathclose{{}\left\\{(y,\zeta(\omega))-\frac{1}{2}\|\zeta(\omega)\|^{2}}\right\\}}dm(\omega)}.$
(6.6)

We now return to our spin system models where the state variable $X(t)$ is an
$(H,{\mathcal{B}}(H))$-valued stochastic process (where ${\mathcal{B}}(H)$ is
the Borel algebra generated by the closed or open subsets of $H$) that is
defined on the probability space $(\Omega,{\mathcal{F}},m)$. Let
${\mathcal{K}}$ be a separable Hilbert space (including $\mathbb{R}^{N}$) and
let the observation vector $\zeta=\zeta(X(t))$ be a measurable function from
$H\rightarrow{\mathcal{K}}$ such that

$\int_{0}^{T}\|\zeta(X(t))\|_{\mathcal{K}}^{2}dt<\infty,\mbox{ a.s.}.$ (6.7)

The Hilbert space ${\mathcal{H}}$ we started with in the abstract white noise
formulation will be in this case ${\mathcal{H}}=L^{2}(0,T;{\mathcal{K}})$.

The white noise sensor measurement model:

$Y(t)=\zeta(X(t))+e(t)\in{\mathcal{K}},\hskip 7.22743ptt>0,$ (6.8)

where $e(t)\in{\mathcal{K}}$ is a finite or infinite dimensional white noise.

Let us define a Borel measure $\rho^{Y}_{t}(\cdot)\in{\mathcal{M}}(H)$ and a
probability measure $\pi_{t}^{Y}(\cdot)\in{\mathcal{P}}(H)$ as follows:

$\rho^{Y}_{t}(B)=E\mathopen{}\mathclose{{}\left[1_{B}(X(t))\exp{\int_{0}^{t}\mathopen{}\mathclose{{}\left\\{(\zeta(X(s)),Y(s))_{\mathcal{K}}-\frac{1}{2}\|\zeta(X(s))\|^{2}_{\mathcal{K}}}\right\\}ds}}\right],$

for $B\in{\mathcal{B}}(H)$, and

$\pi^{Y}_{t}(B)=\frac{\rho^{Y}_{t}(B)}{\rho^{Y}_{t}(H)}.$

Then the measures $\rho^{Y}_{t}\in{\mathcal{M}}(H)$ and
$\pi_{t}^{Y}\in{\mathcal{P}}(H)$ satisfy:

$<\pi_{t}^{Y},f>=\int_{H}f(x)\pi^{Y}_{t}(dz)=E\mathopen{}\mathclose{{}\left[f(X(t))|Y(s),0\leq
s\leq t}\right],$ (6.9)

$<\pi_{t}^{Y},f>=\frac{<\rho^{Y}_{t},f>}{<\rho_{t}^{Y},1>},$

$<\rho_{t}^{Y},f>=E\mathopen{}\mathclose{{}\left\\{f(X(t))\exp{\int_{0}^{t}{\mathfrak{C}}_{s}^{Y}(X(s))ds}}\right\\},$

where

${\mathfrak{C}}_{s}^{Y}(X)=(\zeta(X(s)),Y(s))_{\mathcal{K}}-\frac{1}{2}\|\zeta(X(s))\|^{2}_{\mathcal{K}}.$

The following series of theorems on equivalence of finitely additive nonlinear
filtering equations, uniqueness of measure-valued filter, Markov property and
robustness are proven by methods similar to those of Kallianpur and Karandikar
[31, 32, 33, 35, 36]:

######  Theorem 6.2.

For $Y\in C([0,T];{\mathcal{H}})$ and the class of measures
$\rho_{t}^{Y}\in{\mathcal{M}}(H)$ uniquely solves

$<\rho_{t}^{Y},f>=<\rho_{0},f>+\int_{0}^{t}<\rho_{s}^{Y},{\mathcal{A}}f+{\mathfrak{C}}_{s}^{Y}f>ds,\hskip
7.22743ptf\in D({\mathcal{A}}),0\leq t\leq T,$ (6.10)

or equivalently, $\rho_{t}^{Y}\in{\mathcal{M}}(H)$ uniquely solves

$<\rho_{t}^{Y},f>=<\rho_{0},P_{t}f>+\int_{0}^{t}<\rho_{s}^{Y},{\mathfrak{C}}_{s}^{Y}(P_{t-s}f)>ds,\hskip
7.22743ptf\in{\mathcal{B}}_{b}(H),0\leq t\leq T.$ (6.11)

######  Theorem 6.3.

For $Y\in C([0,T];{\mathcal{H}})$ the probability measure valued process
$\pi_{t}^{Y}\in{\mathcal{P}}(H)$, $0\leq t\leq T$, uniquely solves

$<\pi_{t}^{Y},f>=<\pi_{0},f>+\int_{0}^{t}\mathopen{}\mathclose{{}\left[<\pi_{s}^{Y},{\mathcal{A}}f+{\mathfrak{C}}_{s}^{Y}f>-<\pi_{s}^{Y},{\mathfrak{C}}_{s}^{Y}><\pi_{s}^{Y},f>}\right]ds,f\in
D({\mathcal{A}}),$ (6.12)

or equivalently $\pi_{t}^{Y}\in{\mathcal{P}}(H)$ uniquely solves

$<\pi_{t}^{Y},f>=<\pi_{0},P_{t}f>+\int_{0}^{t}\mathopen{}\mathclose{{}\left[<\pi_{s}^{Y},{\mathfrak{C}}_{s}^{Y}(P_{t-s}f)>-<\pi_{s}^{Y},{\mathfrak{C}}_{s}^{Y}><\pi_{s}^{Y},(P_{t-s}f)>}\right]ds,f\in{\mathcal{B}}_{b}(H).$
(6.13)

Let

${\mathcal{H}}_{t}=\mathopen{}\mathclose{{}\left\\{\eta\in{\mathcal{H}}:\int_{t}^{\infty}\|\eta(r)\|^{2}_{\mathcal{K}}dr=0}\right\\}.$

Note that ${\mathcal{H}}_{t}$ is a closed subspace of ${\mathcal{H}}$. Let
$Q_{t}$ be the orthogonal projection onto ${\mathcal{H}}_{t}$ and let
${\mathcal{C}}_{Q_{t}}={\mathcal{C}}({\mathcal{H}}_{t})$.

######  Theorem 6.4.

$\pi^{Y}_{t}$ and $\rho^{Y}_{t}$ are respectively ${\mathcal{P}}(H)$ and
${\mathcal{M}}(H)$ -valued $\\{{\mathcal{C}}_{Q_{t}}\\}$ Markov processes on
$({\mathcal{H}},{\mathcal{C}},{\bf n})$.

######  Theorem 6.5.

If $Y_{n}\rightarrow Y$ in ${\mathcal{H}}$ then
$\pi_{t}^{Y_{n}}\rightarrow\pi_{t}^{Y}$ and
$\rho_{t}^{Y_{n}}\rightarrow\rho_{t}^{Y}$ in total variation norm.

Acknowledgment: The first author’s research has been supported by the U. S.
Air Force Research Laboratory through the National Research Council Senior
Research Fellowship of the National Academies of Science, Engineering and
Medicine.

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