# Onsager-Machlup action functional for stochastic partial differential
equations with Lévy noise 00footnotetext:
Jianyu Hu
School of Mathematics and Statistics & Center for Mathematical Sciences &
Hubei National Center for Applied Mathematics, Huazhong University of Science
and Technology, Wuhan 430074, China.
E-mail<EMAIL_ADDRESS>00footnotetext:
Jinqiao Duan
Department of Applied Mathematics, Illinois Institute of Technology, Chicago,
IL 60616, USA.
E-mail<EMAIL_ADDRESS>
∗ Corresponding author 00footnotetext:
Renming Song
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana,
IL 61801, USA.
E-mail<EMAIL_ADDRESS>
Jianyu Hu, Jinqiao Duan∗ and Renming Song

###### Abstract

This work is devoted to deriving the Onsager-Machlup action functional for
stochastic partial differential equations with (non-Gaussian) Lévy process as
well as Gaussian Brownian motion. This is achieved by applying the Girsanov
transformation for probability measures and then by a path representation.
This enables the investigation of the most probable transition path for
infinite dimensional stochastic dynamical systems modeled by stochastic
partial differential equations, by minimizing the Onsager-Machlup action
functional.

Key words: Onsager-Machlup action functional, stochastic partial differential
equations, Lévy process, Girsanov transformation, most probable transition
path.

##  1 Introduction

Stochastic dynamical systems are mathematical models for complex phenomena in
physical, chemical and biological sciences [2, 16, 17, 26]. In contrast to the
deterministic systems, noisy fluctuations result in the possibility of
transitions between two metastable states [15, 36]. The Onsager-Machlup action
functional is an essential tool to study these transition phenomena. We will
derive the Onsager-Machlup action functional for stochastic partial
differential equations in Hilbert space with non-Gaussian Lévy process and
this enables the study of the most probable transition pathway between two
metastable states.

Onsager and Machlup considered the probability of paths of a diffusion process
[33]. For stochastic differential equations, the Onsager-Machlup action
functional has been widely investigated. A rigorous mathematical derivation
was carried out by Ikeda and Watanabe [25], Takahashi and Watanabe [40], and
Fujita and Kotani [20]. Shepp and Zeitouni [38] derived the Onsager-Machlup
action functional in the Cameron-Martin space rather than twice continuously
differentiable space. Capitaine [8, 9] derived the Onsager-Machlup action
functional in the space of mean-square integrable functions. Hara and
Takahashi [22] further derived Onsager-Machlup action functional for Brownian
motion on a complete Riemannian manifold using a purely probabilistic method.
The key point is to express the transition probability of a diffusion process
in terms of a functional integral over paths of the process. Note that in the
case of finite dimensional diffusions, the Onsager-Machlup action functional
does not depend on the norm considered.

As in Dürr and Bach [18], Onsager-Machlup action functional may be regarded as
a Lagrangian for determining the most probable path of a diffusion process by
a variation principle. In quantum mechanics, Hochberg et.al [23] derived the
effective action for stochastic partial differential equations using the path
integral method. Cooper and Dawson [12] introduced an auxiliary field to
obtain the effective action for the KPZ equation. This effective action is
actually Onsager-Machlup action functional which is different from the
classical action. Thus, the Onsager-Machlup action functional serves as a tool
to study dynamical behaviors of stochastic dynamical systems, especially the
most probable transition paths connecting metastable states. Using the
Onsager-Machlup action functional, Yang et.al [44] studied the climate change
for global warming. Huang et.al [24] estimated the lower and upper bounds for
the most probable transition time by minimizing the Onsager-Machlup action
functional. In data science, it is possible to quantify most probable
transition trajectories from the Onsager-Machlup action functional [31, 37].
There are other works studying the transition phenomena but using the
Freidlin-Wentzell action functional via large deviation theory [6, 19, 42]. We
emphasize that large deviation theory only holds for small noise, but the
Onsager-Machlup action functional is for any noise.

Note that most existing works on Onsager-Machlup action functional are for
systems with Brownian motion. However, non-Gaussian random phenomena including
heavy-tailed distributions have been found to be suitable in modeling
biological evolution, climate change and other complex scientific and
engineering systems [5, 21, 28, 43]. As for deriving the Onsager-Machlup
action functional for systems with non-Gaussion noise, there are a few
attempts. Bardina, Rovira and Tindel [3] dealt with the asymptotic evaluation
of the Poisson measure for a tube. More recently, Chao and Duan [10] used the
Girsanov transformation to absorb the drift term and thus derive the Onsager-
Machlup action functional in one dimensional systems with Lévy process.

For infinite dimensional stochastic systems, the results for Onsager-Machlup
action functional are much fewer. The first result has been obtained by Dembo
and Zeitouni[14] for trace class elliptic stochastic partial differential
equations on a bounded domain of $\mathbb{R}^{d}$, and then by Mayer-Wolf and
Zeitouni [41] in the non-trace case. Bardina, Rovira and Tindel[4] derived the
Onsager-Machlup action functional for stochastic evolution equations in
Hilbert space. But for infinite dimensional systems with Lévy process, as far
as we know, the Onsager-Machlup action functional is not yet derived. This
paper solves the problem for a class of infinite dimensional systems with Lévy
process.

One difficulty in deriving Onsager-Machlup action functional for infinite
dimensional systems is that the normalizing factor cannot be a function of the
norm of the noise, unlike in finite dimensional systems. Moreover, the Lévy
process is much more complex (Brownian motion is a special case of Lévy
process) which is hard to handle. The methods in [4, 14, 41] encouter some new
difficulties here, because the random variables represented by Karhunen-Loève
expansion are not independent. Inspired by the works [10, 18], we are able to
overcome these difficulties by a path representation and derive the Onsager-
Machlup action functional for stochastic partial differential equations with
Lévy process as well as Brownian motion.

This paper is organized as follows. In section 2, we present the Lévy process
and stochastic partial differential equation in Hilbert space and define the
Onsager-Machlup action functional. In section 3, we apply the Girsanov
transformation to obtain the relation between the two probability measures and
then derive the Onsager-Machlup action functional by representing the Radon-
Nikodym derivative in terms of a path integral. We present an example in
section 4, and finally end with some discussions in section 5.

##  2 The framework

We now recall some basic facts about Lévy process in Hilbert space, then
introduce a class of stochastic partial differential equations [1, 13, 34],
and finally define the Onsager-Machlup action functional.

###  2.1 Lévy process in Hilbert space

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with a filtration
of right continuous $\sigma$-algebras $(\mathcal{F}_{t})_{t\geq 0}$. All of
the $\sigma$-algebras are assumed to be $\mathbb{P}$-completed. Given a real
separable Hilbert space $H$, a Lévy process on $H$ can be defined by the
formal series

$\begin{split}L(t)=\sum\limits_{j=1}^{\infty}L_{j}(t)e_{j},\end{split}$ (2.1)

where $\\{e_{j};j\geq 1\\}$ is an orthonormal basis in $H$,
$\\{L_{j}(t),t\in[0,1],j\geq 1\\}$ is a sequence of mutually independent Lévy
processes in $\mathbb{R}$ adapted to $\mathcal{F}_{t}$. Assume that each
$L_{j}(t)$ is a pure jump process which has the Lévy-Khinchin representation

$\displaystyle\mathbb{E}e^{i\xi L_{j}(t)}=e^{-t\psi_{j}(\xi)},\ \ 0\leq t\leq
1,\ \xi\in\mathbb{R},$

where

$\displaystyle\psi_{j}(\xi)=\int_{\mathbb{R}}(1-e^{i\xi
z}-\chi_{\\{|z|<1\\}}i\xi z)\nu_{j}(dz).$

Here, $\nu_{j}$ is the jump measure in the jth component. Then we have
$\sum\limits_{j}\int_{\mathbb{R}}(|z|^{2}\wedge 1)\nu_{j}(dz)<\infty$. In this
case, the series in (2.1) converges in probability, uniformly in $t$ on every
compact interval $[0,1]$. Moreover, it can be expressed by the Lévy-Itô
decomposition

$\begin{split}L_{j}(t)=\int_{|z|<1}z\tilde{N}_{j}(t,dz)+\int_{|z|\geq
1}zN_{j}(t,dz),\end{split}$ (2.2)

where $N_{j}(dt,dz)$ is the Poisson measure associated with $L_{j}(t)$, and
$\tilde{N}_{j}(dt,dz)=N_{j}(dt,dz)-\nu_{j}(dz)dt$ is the corresponding
compensated Poisson random measure. In this paper, our Lévy process $L(t)$ has
the following expression

$\begin{split}L(t)=\sum\limits_{j=1}^{\infty}L_{j}(t)=\sum\limits_{j=1}^{\infty}\int_{|z|<1}z\tilde{N}_{j}(t,dz)e_{j}+\sum\limits_{j=1}^{\infty}\int_{|z|\geq
1}zN_{j}(t,dz)e_{j}.\end{split}$ (2.3)

The following examples are of some important applications; see Cont and Tankov
[11].

######  Example 1.

(One sided exponentially tempered stable Lévy process)

As the Lévy jump measure $\nu_{j}$ of $L_{j}$, we take

$\begin{split}\nu_{j}(dr)=c_{j}\frac{e^{-\beta_{j}r}}{r^{1+\alpha_{j}}}\chi_{(0,\infty)}(r)dr.\end{split}$
(2.4)

Then $L=\sum_{j}L_{j}e_{j}$ is $H$-valued and square integrable if and only if
$\sum_{j}\frac{c_{j}}{\beta_{j}^{2-\alpha_{j}}}<\infty$ for all
$\alpha_{j}\in(0,2)$.

######  Example 2.

(Two sided exponentially tempered stable Lévy process)

As the Lévy jump measure $\nu_{j}$ of $L_{j}$, we take

$\begin{split}\nu_{j}(dr)=c_{j}^{-}\frac{e^{-\beta_{j}^{-}|r|}}{r^{1+\alpha_{j}^{-}}}\chi_{(\infty,0)}(r)dr+c_{j}^{+}\frac{e^{-\beta_{j}^{+}|r|}}{r^{1+\alpha_{j}^{+}}}\chi_{(0,\infty)}(r)dr.\end{split}$
(2.5)

Then $L=\sum_{j}L_{j}e_{j}$ is $H$-valued and square integrable if and only if
$\sum_{j}(\frac{c_{j}^{-}}{(\beta_{j}^{-})^{2-\alpha_{j}^{-}}}+\frac{c_{j}^{+}}{(\beta_{j}^{+})^{2-\alpha_{j}^{+}}})<\infty$
for all $\alpha_{j}\in(0,2)$.

###  2.2 Stochastic partial differential equations

Consider the following stochastic partial differential equation in $H$:

$\begin{split}&dX(t)=[AX(t)+F(t,X(t))]dt+BdW(t)+dL(t),t\in[0,1],\\\ &X(0)=x\in
H,\end{split}$ (2.6)

where $A:D(A)\subset H\rightarrow H$ generates a $C_{0}$-semigroup
$\\{e^{tA};t\geq 0\\}$, $F$ is a Lipschitz function defined on $[0,1]\times
H$, $B$ is non-negative bounded linear operators from $H$ to $H$, $W(t)$ is a
cylindrical Wiener process in $H$, and $L(t)$ is a Lévy process in $H$ defined
as (2.3). We will also denote by $X^{A}$ the stochastic convolution of $A$ by
$X$, that is, the solution to (2.1) with $F=0$ and $x=0$.

The norm in $H$ will be denoted by $|\cdot|_{H}$, and the scalar product by
$\langle\cdot,\cdot\rangle$. The $L^{2}$ norm in $L^{2}([0,1];H)$ is denoted
by $\|\cdot\|_{2}$. Let $\mathcal{L}(H)$ be the set of bounded linear
operators on $H$. The norm $\|\cdot\|$ will be the usual operator norm defined
on $\mathcal{L}(H)$. We shall make the following hypotheses:

$\mathbf{(H1)(Semigroup)}$ The operator A generates a self adjoint
$C_{0}$-semigroup $\\{e^{tA};t\geq 0\\}$ of negative type. Moreover, there
exists a complete orthonormal system $\\{e_{j};j\geq 1\\}$ which diagonalizes
$A$. We shall denote by $\\{-\lambda_{j};j\geq 1\\}$ the corresponding set of
eigenvalues and we assume that $\\{\lambda_{j};j\geq 1\\}$ is an increasing
sequence of real numbers such that $\lambda_{j}>0$ and
$\lim_{j\rightarrow\infty}\lambda_{j}=\infty$. Moreover, the semigroup
$\\{e^{tA};t\geq 0\\}$ is a contraction.

$\mathbf{(H2)(Lipschitz\ and\ Linear\ Growth)}$ There exists an integrable
function $L_{F}:(0,1]\rightarrow(0,\infty)$ such that for all $s$ with
$0<s\leq 1$ and $x,y\in H$,

$\begin{split}&|e^{sA}(F(t,x)-F(t,y))|_{H}\leq L_{F}(s)|x-y|_{H},\
t\in[0,1],\\\ &|e^{sA}F(t,x)|_{H}\leq L_{F}(s)(1+|x|_{H}),\
t\in[0,1].\end{split}$ (2.7)

$\mathbf{(H3)(Hilbert-Schmidt)}$ There exists an integrable function
$L_{B}:(0,1]\rightarrow(0,\infty)$ such that for all $s$ with $0<s\leq 1$ and
$x,y\in H$,

$\begin{split}&|e^{sA}B(x-y)|_{HS}\leq L_{B}(s)|x-y|_{H},\ t\in[0,1],\\\
&|e^{sA}Bx|_{HS}\leq L_{B}(s)(1+|x|_{H}),\ t\in[0,1].\end{split}$ (2.8)

$\mathbf{(H4)(Moment)}$ The Lévy jump measure satisfies

$\begin{split}\sum\limits_{j=1}^{\infty}\int_{|z|<1}|z|\nu_{j}(dz)<\infty.\end{split}$
(2.9)

######  Remark 2.1.

The contraction of the $C_{0}$-semigroup $e^{tA}$ guarantees that the solution
of (2.6) has a càdlàg version. The hypotheses $\mathbf{H2}$ and $\mathbf{H3}$
are usual conditions which ensure the existence and uniqueness of a solution
to (2.6). There are many works on the existence and uniqueness of a solution
to (2.6), as in [7, 34, 35, 39]

######  Remark 2.2.

The hypothesis $\mathbf{H4}$ is very important. On the one hand, it ensures
the availability of Girsanov’s transformation. On the other hand, it
guarantees that the process
$\int_{0}^{t}\int_{|y|_{H}<1}\exp((t-s)A)y\tilde{N}(dt,dy)+\int_{0}^{t}\int_{|y|_{H}\geq
1}\exp((t-s)A)yN(dt,dy)$ is finite variation. These will be shown latter.

Under the hypotheses $\mathbf{H1,H2}$ and $\mathbf{H3}$, there exists a unique
mild solution to (2.6). We say $X=\\{X(t),t\in[0,1]\\}$ is a mild solution to
(2.6) if it is an $H$-valued $\mathcal{F}_{t}$ -adapted square integrable
process such that

$\begin{split}X(t)&=\exp(tA)x+\int_{0}^{t}\exp((t-s)A)F(s,X(s))ds+\int_{0}^{t}\exp((t-s)A)BdW(s)\\\
&+\int_{0}^{t}\int_{|y|_{H}<1}\exp((t-s)A)y\tilde{N}(ds,dy)+\int_{0}^{t}\int_{|y|_{H}\geq
1}\exp((t-s)A)yN(ds,dy),\end{split}$ (2.10)

for all $t\in[0,1]$.

The following proposition is a special case of [34, Theorem 9.9].

######  Proposition 2.3.

Suppose that $\mathbf{H1,H2}$ and $\mathbf{H3}$ are satisfied. Then there
exists a unique solution X(t) to (2.8) such that $X\in
L^{2}(\Omega\times[0,1];H)$. Furthermore, the solution has a càdlàg version.

Note that the solution process $X^{A}$ is given by

$\begin{split}X^{A}(t)&=\int_{0}^{t}\exp((t-s)A)BdW(s)+\int_{0}^{t}\int_{|y|_{H}<1}\exp((t-s)A)y\tilde{N}(ds,dy)\\\
&\ \ +\int_{0}^{t}\int_{|y|_{H}\geq 1}\exp((t-s)A)yN(ds,dy),\end{split}$
(2.11)

As shown in Proposition 2.3, this solution process has a càdlàg version.
Moreover, under the hypothesis $\mathbf{H4}$, we have the following lemma.

######  Lemma 2.4.

Let
$Z(t)=\int_{0}^{t}\int_{|y|_{H}<1}\exp((t-s)A)y\tilde{N}(ds,dy)+\int_{0}^{t}\int_{|y|_{H}\geq
1}\exp((t-s)A)yN(ds,dy)$, for $t\in[0,T]$. Then $Z(t)$ is finite variation
process if and only if
$\sum\limits_{j=1}^{\infty}\int_{|z|<1}|z|\nu_{j}(dz)<\infty$.

###### Proof.

Let $\Delta Z(t)=Z(t)-Z(t-)$, for $t\in[0,T]$. Then we have

$\displaystyle\mathbb{E}\exp\\{-\sum\limits_{0<t\leq T}|\Delta Z(t)|_{H}\\}$
$\displaystyle=\mathbb{E}\exp\\{-\sum\limits_{j=1}^{\infty}\int_{0}^{T}\int_{\mathbb{R}}|z|e^{-\lambda_{j}(T-t)}N_{j}(dt,dz)\\},$
$\displaystyle=\exp\\{-\sum\limits_{j=1}^{\infty}\frac{1}{\lambda_{j}}(1-e^{-\lambda_{j}T})\int_{\mathbb{R}}(1-e^{|z|})\nu_{j}(dz)\\}.$

Note that $1-e^{-\lambda_{j}T}\leq\lambda_{j}T$. Thus, if
$\sum\limits_{j=1}^{\infty}\int_{\mathbb{R}}(1-e^{|z|})\nu_{j}(dz)\\}=\infty$,
or equivalently if
$\sum\limits_{j=1}^{\infty}\int_{|z|<1}|z|\nu_{j}(dz)=\infty$, then
$\sum\limits_{0<t\leq T}|\Delta Z(t)|_{H}=\infty$, $\mathbb{P}$-a.s. And if
$\sum\limits_{j=1}^{\infty}\int_{\mathbb{R}}(1-e^{|z|})\nu_{j}(dz)\\}<\infty$,
or equivalently if
$\sum\limits_{j=1}^{\infty}\int_{|z|<1}|z|\nu_{j}(dz)<\infty$, then

$\displaystyle\mathbb{E}\sum\limits_{0<t\leq T}|\Delta
Z(t)|_{H}\chi_{\\{|Z(t)|_{H}\leq
M\\}}=\sum\limits_{j=1}^{\infty}\frac{1}{\lambda_{j}}(1-e^{-\lambda_{j}T})\int_{|z|<1}|z|\nu_{j}(dz)<\infty.$

Thus $Z(t)$ is of bounded variation. $\square$ ∎

This lemma is very important, which is able to control the path integral while
deriving the Onsager-Machlup action functional.

###  2.3 Definition of Onsager-Machlup action functional

Now, we define the Onsager-Machlup action functional [4] associated with the
stochastic partial differential equation (2.6). Our goal is to study the
limiting behaviour of ratios of the form

$\begin{split}\gamma_{\epsilon}(\phi)=\frac{\mathbb{P}(\|X-\phi\|\leq\epsilon)}{\mathbb{P}(\|X^{A}\|\leq\epsilon)},\end{split}$
(2.12)

when $\epsilon$ tends to $0$. Here $\phi$ is a deterministic function
satisfying some regularity conditions and $\|\cdot\|$ is a suitable norm
defined on the functions from $[0,1]$ to $H$.

When $\lim\limits_{\epsilon\rightarrow
0}\gamma_{\epsilon}(\phi)=\exp\\{J_{0}(\phi)\\}$ for all $\phi$ in a
reasonable class of functions, the functional $J_{0}$ is called the Onsager-
Machlup action functional associated to system (2.6). We will chose to work
with the Hilbert norm on $L^{2}([0,1];H)$ for reasons in [4].

As we can see, the Onsager-Machlup action functional quantifies the
probability of the solution path up to a given function on a small tube. If we
minimize this functional given two metastable states, we will obtain the most
probable transition pathway between two fixed points [37, 44]. This is an
important application to study the dynamics for stochastic partial
differential equations.

##  3 Onsager-Machlup action functional

In this section, we derive the Onsager-Machlup action functional for the
stochastic partial differential equation (2.6). This result covers the results
of both infinite case [4, 8] and finite case [10]. At first, in section 3.1,
we apply the Girsanov transformation in order to reduce our problem to the
evaluation of a functional of the stochastic convolution $X^{A}$. Then in
section 3.2, we use the Itô formula to express the Radon-Nikodym derivative in
terms of a path integral. Thus in section 3.3, we obtain the Onsager-Machlup
action functional by analyzing the limit behaviors of a path integral.

###  3.1 An application of Girsanov’s transformation

In this subsection, we will apply the Girsanov transformation to obtain the
relation between two probability measures. Actually, the Girsanov theorem for
Lévy process does not work here, because after the transformation, the
distribution of resulting noise in new probability space differs to the
original noise with respect to the original probability space, which results
in the failure of weak uniqueness. Benefiting from the independence of
Brownian motion and Poisson random measure, we can use the Brownian motion to
absorb the drift. This idea comes from the finite case in [10].

To do so, we first introduce an auxiliary system. We will take our
deterministic function $\phi$ in a so-called Cameron-Martin space. Fix a
function $h\in L^{2}([0,1];H)$. Let $\phi^{h}$ the solution of the infinite
dimensional equation

$\begin{split}&d\phi^{h}(t)=A\phi^{h}(t)dt+Bh(t)dt,t\in[0,1],\\\
&\phi^{h}(0)=x.\end{split}$ (3.1)

The assumption $h\in L^{2}([0,1];H)$ is required in order to apply Girsanov’s
transformation. We also assume that for each $t\in[0,1]$, $F(t,x)\in Im(B)$
a.s, and

$\begin{split}\mathbb{E}[\exp(\frac{1}{2}\int_{0}^{1}|B^{-1}F(t,X(t)|_{H}^{2})]<\infty.\end{split}$
(3.2)

Recall that we will call measures $\mu_{X}$ and $\mu_{Y}$ equivalent
($\mu_{X}\sim\mu_{Y}$) if $\mu_{X}$ is absolutely continuous with respect to
$\mu_{Y}$ ($\mu_{X}\ll\mu_{Y}$) and $\mu_{Y}\ll\mu_{X}$. Applying the Girsanov
theorem, we have the following lemma.

######  Lemma 3.1.

Let $X_{t}$ and $Y_{t}$ be two solution processes defined by the stochastic
partial differential equations with respect
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$

$\begin{split}dX(t)&=[AX(t)+F(t,X(t))]dt+BdW(t)+dL(t),\\\
dY(t)&=[AY(t)+G(t,Y(t))]dt+BdW(t)+dL(t),\end{split}$ (3.3)

where the driven Lévy process is defined as (2.3) satisfying the hypothesis
$\mathbf{H4}$. Here, $X_{0}=Y_{0}=x\in H$, $t\in[0,1]$, $A,B,F$ satisfy the
previous condition, and $G$ is required the same condition as $F$. Then, we
have $\mu_{X}\sim\mu_{Y}$ and the Radon-Nikodym derivative of $\mu_{X}$ with
respect to $\mu_{Y}$ is given by

$\begin{split}\frac{d\mu_{X}}{d\mu_{Y}}[Y_{t}(\omega)]&=\exp\\{\int_{0}^{t}\langle
B^{-1}(F(s,Y(s))-G(s,Y(s))),dW(s)\rangle\\\ &\ \ \ \ \
-\frac{1}{2}\int_{0}^{t}|B^{-1}(F(s,Y(s))-G(s,Y(s)))|_{H}^{2}ds\\}.\end{split}$
(3.4)

###### Proof.

Define

$\displaystyle M_{t}=\exp$ $\displaystyle\\{\int_{0}^{1}\langle
B^{-t}(F(s,Y(s))-G(s,Y(s))),dW(s)\rangle$
$\displaystyle-\frac{1}{2}\int_{0}^{1}|B^{-t}(F(s,Y(s))-G(s,Y(s)))|_{H}^{2}ds\\}.$

According to the Girsanov transformation [27, Theorem 3.17][35, Theorem 2.1],
$\widehat{\mathbb{P}}$ defined as
$d\widehat{\mathbb{P}}(\omega)=M_{t}(\omega)d\mathbb{P}(\omega)$ is a
probability measure, and the process
$\widehat{W}_{t}:=W_{t}+\int_{0}^{t}B^{-1}(F(s,Y(s))-G(s,Y(s)))ds$ is a
cylindrical Brownian motion under the new filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\widehat{\mathbb{P}})$.
Moreover, $\widetilde{N}(dt,dy)$ is still the compensated martingale measure
with jump measure $\nu$ with respect to $\widehat{\mathbb{P}}$. Thus the
process $Y_{t}$ respect to $\widehat{\mathbb{P}}$ has the stochastic
differential representation

$\begin{split}dY(t)=[AY(t)+F(t,Y(t))]dt+Bd\widehat{W}(t)+dL(t).\end{split}$
(3.5)

Due to the uniqueness in distribution, we have the equality of $\mu_{X}$ and
$\mu_{Y}^{Q}$. Then we obtain

$\displaystyle\frac{d\mu_{X}}{d\mu_{Y}}[Y_{t}(\omega)]$
$\displaystyle=\frac{d\widehat{\mathbb{P}}}{d\mathbb{P}}(\omega)$
$\displaystyle=\exp\\{\int_{0}^{t}\langle
B^{-1}(F(s,Y(s))-G(s,Y(s))),dW(s)\rangle$ $\displaystyle\ \ \ \ \
-\frac{1}{2}\int_{0}^{t}|B^{-1}(F(s,Y(s))-G(s,Y(s)))|_{H}^{2}ds\\}.$

The proof is complete. $\square$ ∎

Now, let us consider the function $\gamma_{\epsilon}(\phi)$ defined as (2.10).
For every $\epsilon>0$, applying lemma 3.1 with $G(s)=Bh(s)$ and noting that
$Y(t)=\phi^{h}(t)+X^{A}(t)$, we have

$\begin{split}\mathbb{P}(\|X-\phi^{h}\|_{2}\leq\epsilon)=\widehat{\mathbb{P}}(\|Y-\phi^{h}\|_{2}\leq\epsilon)=\mathbb{E}[\exp\\{\Lambda\\}\chi_{\|X^{A}\|_{2}\leq\epsilon}],\end{split}$
(3.6)

where

$\begin{split}\Lambda&=\int_{0}^{1}\langle
B^{-1}F(s,X^{A}(s_{-})+\phi^{h}(s))-h(s),dW(s)\rangle\\\ &\ \
-\frac{1}{2}\int_{0}^{1}|B^{-1}F(s,X^{A}(s_{-})+\phi^{h}(s))-h(s)|^{2}ds.\end{split}$
(3.7)

######  Remark 3.2.

Because we use the Girsanov theorem of Brownian motion to absorb the drift
term by renorming the probability space, the only difference to Brownian
motion is the solution $X^{A}$ of linear system (F=0 and x=0) which is now

$\begin{split}X^{A}(t)&=\int_{0}^{t}e^{(t-s)A}BdW(s)+\int_{0}^{t}\int_{|y|_{H}<1}e^{(t-s)A}y\widetilde{N}(dt,dy)\\\
&+\int_{0}^{t}\int_{|y|_{H}\geq 1}e^{(t-s)A}yN(dt,dy).\end{split}$ (3.8)

But this is just the essential difference to the Brownian case. Bardina,
Rovira and Tindel [4] use the $L^{2}$ techniques to analyze the limiting
behaviors through representing the process by random variables. But now, this
method will meet new difficulties, because the representation of the process
$X^{A}$ by Karhunen-Loève expansion can’t obtain the independence of random
variables. Inspired of by the works[10, 18], we could use Itô’s formula to
express the Radon-Nikodym derivative in terms of a path integral.

######  Remark 3.3.

By weak uniqueness, the transformed $Y(t)=\phi^{h}(t)+X^{A}(t)$ is the
solution of the following SPDEs

$dY(t)=[AY(t)+Bh(t)]dt+BdW(t)+dL(t)$ (3.9)

We will apply Itô’s formula to this process in next subsection. Furthermore,
the norm of $X^{A}(t)$ is less than $\epsilon$, thus $y(t)$ can be considered
as a perturbation of $\phi^{h}(t)$. Then we could take the Taylor’s expansion
at $\phi^{h}(t)$ for every $C^{2}$ function.

###  3.2 Representation by a path integral

In this subsection, we will use Itô’s formula to represent the Radon-Nikodym
derivative in terms of a path integral. This method works because the Poisson
stochastic integral can also be expressed by path integral and it can be
controlled by bounded variation. Thus, we are able to derive the Onsager-
Machlup action functional.

Let $V:H\rightarrow R$ be $C^{2}$ function. Assume that for each $x$,
$DV(x)=(B^{-1})^{*}(B^{-1}F(x)-h)$, $D^{2}V(x)\in L_{(HS)}(H,H)$ and that the
mapping $x\mapsto D^{2}V(x)$ is uniformly continuous on any bounded subset of
$H$. Applying Itô’s formula [34, Appendix 4] [32, Theorem 27.1] to $Y(t)$
driven by (3.9) in Hilbert space, we have

$\begin{split}V(y(t))-V(y(0))&=\int_{0}^{t}\langle
B^{-1}F(s,y(s_{-}))-h(s),B^{-1}(Ay(s_{-})+Bh(s)\rangle ds\\\
&+\int_{0}^{t}\langle B^{-1}F(s,y(s_{-}))-h(s),dW(s)\rangle\\\
&+\frac{1}{2}\int_{0}^{t}\sum\limits_{J=1}^{\infty}D^{2}_{jj}V(y(s_{-}))Be_{j}\otimes
Be_{j}ds\\\ &+\sum\limits_{0\leq s\leq t}[V(y(s))-V(y(s_{-}))]\\\
&+\int_{0}^{t}\int_{|\xi|_{H}<1}\langle
B^{-1}F(s,y(s_{-}))-h(s),B^{-1}\xi\rangle\nu(d\xi)ds.\end{split}$ (3.10)

Then the exponent part of Radon-Nikodym derivative can be represented as

$\begin{split}\Lambda&=V(y(1))-V(y(0))+\sum\limits_{0\leq t\leq
1}[V(y(t))-V(y(t_{-}))]\\\
&-\int_{0}^{1}b(y(t_{-}))dt+\int_{0}^{1}\int_{|\xi|_{H}<1}\langle
B^{-1}F(s,y(t_{-}))-h(t),B^{-1}\xi\rangle\nu(d\xi)dt,\end{split}$ (3.11)

where $b$ is defined as

$\begin{split}b(y(t_{-})))&=|B^{-1}F(t,y(t_{-})))-h(s)|_{H}^{2}+\sum\limits_{J=1}^{\infty}D^{2}_{jj}V(y(t_{-})))Be_{j}\otimes
Be_{j}\\\ &+2\langle
B^{-1}F(t,y(t_{-})))-h(t),B^{-1}(Ay(t_{-}))+Bh(t)\rangle.\end{split}$ (3.12)

We see that $\Lambda$ is now the functional of the path $y(t)$. But we can not
always find the the primitive function $V$ given a function F. We will apply
the Poincaré lemma to prove the the exitance of primitive function $V$.
Actually in one dimension [10, 18], the primitive function $V$ can be defined
as a potential function

$\begin{split}V(x)=\frac{1}{B}\int^{x}[\frac{1}{B}F(y)-h]dy.\end{split}$
(3.13)

But in infinite dimensional case or even two dimensional case, this potential
is not always available. We have to use the Poincaré lemma which states the
relation between closed form and exact form. It provides a sufficient
condition for the existence of a potential function. Hence, we have the
following lemma.

######  Lemma 3.4.

Let H be a separable Hilbert space with basis $\\{e_{j}\\}_{j\geq 1}$ and G be
a mapping from $H$ to $H$. For each $x\in H$, assume $DF(x)$ is symmetric
operator from $H$ to $H$. Then there exists a smooth function
$V:H\rightarrow\mathbb{R}$, such that for all $x\in H$, DV(x)=G(x).

###### Proof.

The dual space of Hilbert space is itself. So for each $x\in H$, $F(x)$ could
be considered as a bounded linear functional in $H$. Thus, it defines a 1-form
as

$w=\langle F(x),de\rangle=\sum\limits_{j}F_{j}(x)de_{j}.$ (3.14)

Set $G_{ij}(x)=\langle DF(x)e_{i},e_{j}\rangle$. Since $DF(x)$ is symmetric,
i.e. $\langle DF(x)e_{i},e_{j}\rangle=\langle e_{i},DF(x)e_{j}\rangle$, we
have $G_{ij}(x)=G_{ji}(x)$. Hence, the 2-form $dw$ is

$\begin{split}dw=\sum\limits_{ij}\langle e_{i},DF(x)e_{j}\rangle de_{i}\wedge
de_{j}=\sum\limits_{ij}G_{ij}(x)de_{i}\wedge de_{j},\end{split}$ (3.15)

where $de_{i}\wedge de_{j}$ is the natural wedge product which is bilinear and
antisymmetric. Thus, we obtain

$\begin{split}dw=\sum\limits_{1\leq i<j}(G_{ij}(x)-G_{ji}(x))de_{i}\wedge
de_{j}=0.\end{split}$ (3.16)

This implies that the mapping F is a closed 1-form. Then by the Poincaré lemma
[30, Page 137 Theorem 4.1] [29, Page 350 Theorem 33.20], it is an exact form,
i.e. there exists a smooth function $V:H\rightarrow\mathbb{R}$, such that for
all $x\in H$, $DV(x)=G(x)$. $\square$ ∎

######  Remark 3.5.

Note that our method will additionally need the symmetry of $DF(x)$, while
$L^{2}$ technique doesn’t. Bardina, Rovira and Tindel [4, Proposition 3.7]
uses a symmetrization technique by
$S_{QR^{*}}=\frac{1}{2}(QR^{*}+(QR^{*})^{*})$ in the analysis of random
variables. But they also need to assume that the symmetry of $\nabla_{x}F$,
that is, $\nabla_{x}F$ can be diagonalized in the same basis as the operators
A and B, when they further express the operator $S_{QR^{*}}$ by $\nabla_{x}F$
and $\phi^{h}$. That is, if for every $s\in[0,1]$, the gradient
$\nabla_{x}F(s,\phi^{h}(s))$ is a trace class operator, they obtain

$\begin{split}S_{QR^{*}}=\frac{1}{2}\int_{0}^{1}Tr[\nabla_{x}F(s,\phi^{h}(s))]ds.\end{split}$
(3.17)

Actually, we will see that the trace part in Onsager-Machlup action functional
comes from the Itô correction in stochastic analysis.

###  3.3 Derivation of Onsager-Machlup action functional

Now, we are ready to derive the Onsager-Machlup action functional for the
stochastic partial differential equation

$\begin{split}dX(t)=[AX(t)+F(t,X(t))]dt+BdW(t)+dL(t),\end{split}$ (3.18)

with initial data $X(0)=x\in H$.

Our main result for this stochastic partial differential equation is in the
following theorem.

######  Theorem 3.6.

$\mathbf{(Onsager-Machlup\ action\ functional)}$ Assume that
$\mathbf{H1,H2,H3}$ and $\mathbf{H4}$ are satisfied, $h\in L^{2}([0,1])$,
$\phi^{h}$ is defined by (3.1) and $B^{-1}F$ is $C_{b}^{2}$ in $x$ uniformly
in $s\in[0,1]$. If for every $s\in[0,1]$, the gradient
$\nabla_{x}F(s,\phi^{h}(s))$ is symmetric and trace class, and
$\int_{0}^{1}Tr[\nabla_{x}F(s,\phi^{h}(s))]ds<\infty$, then the Onsager-
Machlup action functional of system (3.18) is given by

$\begin{split}J_{0}(\phi^{h},\dot{\phi}^{h})&=-\frac{1}{2}\int_{0}^{1}|B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)-\eta]|_{H}^{2}dt\\\
&\ \ \ -\frac{1}{2}\int_{0}^{1}Tr[\nabla_{x}F(s,\phi^{h}(s))]ds,\end{split}$
(3.19)

where $\eta=\int_{|\xi|_{H}<1}\xi\nu(d\xi)$.

######  Remark 3.7.

Up to a constant, the Onsager-Machlup action functional can also be written as

$\begin{split}J_{0}(\phi^{h},\dot{\phi}^{h})&=-\frac{1}{2}\int_{0}^{1}|B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)]|_{H}^{2}dt\\\
&\ \ \ -\frac{1}{2}\int_{0}^{1}Tr[\nabla_{x}F(s,\phi^{h}(s))]ds\\\ &\ \ \
-\frac{1}{2}\int_{0}^{1}\langle
B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)],B^{-1}\int_{|\xi|_{H}<1}\xi\nu(d\xi)\rangle
ds.\end{split}$ (3.20)

As we can see, the first term is the main term, while the second term comes
from the Itô correction of Brownian motion. The Lévy noise results in the
third term.

It is interesting to see that only small jumps contribute to the Onsager-
Machlup action functional. Moreover, the effect is similar to adding the mean
value of small jumps to the drift. Furthermore, if the integral
$\eta=\int_{|\xi|_{H}<1}\xi\nu(d\xi)$ is the zero element in $H$, then the
third term will vanish, in this case, the Lévy noise will have no effect to
the Onsager-Machlup action functional.

######  Remark 3.8.

We derived the Onsager-Machlup action functional for the stochastic partial
differential equation

$\begin{split}dX(t)=[AX(t)+F(t,X(t))]dt+BdW(t)+dL(t),\end{split}$ (3.21)

with initial data $X(0)=x\in H$. This theorem covers several present works. If
this system does not have Lévy process $L(t)$(i.e., $\nu=0$), then the
Onsager-Machlup action functional will reduce to that in the Brownian case
[4].

Moreover, our results hold for the Lévy process with both small and large
jumps, while Chao and Duan [10] only dealt with small jumps. Our derivation is
available for two reasons. One is that the Girsanov theorem still holds for
large jumps. The other is that we can still use the bounded variation to
control the summation of the jumps.

######  Remark 3.9.

The proof remains the same for a function $F$ satisfying

1) $F:[0,1]\times L^{2}([0,1];H)\rightarrow L^{2}([0,1];H)$.

2) $B^{-1}F(t,\cdot):L^{2}([0,1];H)\rightarrow L^{2}([0,1];H)$ is $C_{b}^{2}$
uniformly in $t\in[0,1]$.

3) For all $t\in[0,1]$ and $\xi\in L^{2}([0,1];H)$, $F(t,\xi)=F(t,\xi
1_{[0,1]})$.

Now, we prove Theorem 3.6.

###### Proof.

As calculated in (3.11) and (3.12), the exponent part of Radon-Nikodym
derivative can be represented as

$\begin{split}\Lambda&=V(y(1))-V(y(0))+\sum\limits_{0\leq t\leq
1}[V(y(t))-V(y(t_{-}))]\\\
&-\int_{0}^{1}b(y(t_{-}))dt+\int_{0}^{1}\int_{|\xi|_{H}<1}\langle
B^{-1}F(s,y(t_{-}))-h(t),B^{-1}\xi\rangle\nu(d\xi)dt,\end{split}$ (3.22)

where $b$ is defined as

$\begin{split}b(y(t_{-})))&=|B^{-1}F(t,y(t_{-})))-h(s)|_{H}^{2}+\sum\limits_{J=1}^{\infty}D^{2}_{jj}V(y(t_{-})))Be_{j}\otimes
Be_{j}\\\ &+2\langle
B^{-1}F(t,y(t_{-})))-h(t),B^{-1}(Ay(t_{-}))+Bh(t)\rangle.\end{split}$ (3.23)

The integrals with respect to time variable $t$ in (3.23) are Riemann
integrals. Now we expand the exponent of (3.23) into a Taylor series around
$y(t)=\phi^{h}(t)$ due to $|X^{A}(t)|\leq\epsilon$, and split the terms of
zero order. If we choose $\epsilon$ small enough, the remaining terms can be
made arbitrarily small.

Since $B^{-1}F$ is a $C_{b}^{2}$ in $x$ uniformly in $t\in[0,1]$, and $V$ is
at least $C^{2}$ function, we obtain

$\begin{split}V(y(1))&=V(\phi^{h}(1))+\langle
DV(\phi^{h}(1)),X^{A}(1)\rangle+o(\epsilon),\\\
b(y(t-))&=|B^{-1}F(t,\phi^{h}(t))-h(t)|^{2}+2\langle
B^{-1}F(s,\phi^{h}(t))-h(t),B^{-1}(A\phi^{h}(t)+Bh(t))\rangle\\\ &\ \
+Tr(D_{x}F(t,\phi^{h}(t)))+Tr(\langle
D_{x}^{2}F(t,\phi^{h}(t)),X^{A}(t)\rangle)\\\ &\ \ +2\langle
B^{-1}F(s,\phi^{h}(t))-h(t),B^{-1}AX^{A}(t)\rangle\\\ &\ \ +2\langle
B^{-1}F(s,\phi^{h}(t))-h(t),\langle
D_{x}F(t,\phi^{h}(t)),X^{A}(t)\rangle\rangle+o(\epsilon),\\\ \end{split}$
(3.24)

and

$\begin{split}\langle B^{-1}F(s,&y(t-))-h(t),B^{-1}\eta\rangle=\langle
B^{-1}F(s,\phi^{h}(t))-h(t),B^{-1}\eta\rangle\\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ +\langle B^{-1}\langle
D_{x}F(s,\phi^{h}(t)),X^{A}(t)\rangle,B^{-1}\eta\rangle+o(\epsilon),\end{split}$
(3.25)

where $\eta=\int_{|\xi|_{H}<1}\xi\nu(d\xi)$. Note that

$\begin{split}V(\phi^{h}(1))-V(\phi^{h}(o))=\int_{0}^{1}\langle\dot{\phi}^{h}(t),(B^{-1})^{*}(B^{-1}F(\phi^{h}(t),t)-h(t))\rangle
dt.\end{split}$ (3.26)

Since $|X^{A}(t)|\leq\epsilon$, by Hölder inequality and by the fact of
$\dot{\phi}^{h}(t)=A\phi^{h}(t)-Bh(t)$, we obtain

$\begin{split}\Lambda&=V(\phi^{h}(1))-V(\phi^{h}(1))\\\ &\ \
-\int_{0}^{1}Tr(D_{x}F(t,\phi^{h}(t)))dt-\int_{0}^{1}|B^{-1}F(t,\phi^{h}(t))-h(t)|^{2}dt\\\
&\ \ -2\int_{0}^{1}\langle B^{-1}F(s,\phi^{h}(t))-h(t),B^{-1}\eta\rangle dt\\\
&\ \ +\sum\limits_{0\leq t\leq 1}[V(y(t))-V(y(t-))]+O(\epsilon)\\\
&=-\frac{1}{2}\int_{0}^{1}|B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)]|_{H}^{2}dt\\\
&\ \ \ -\frac{1}{2}\int_{0}^{1}Tr[\nabla_{x}F(s,\phi^{h}(s))]ds\\\ &\ \ \
-\frac{1}{2}\int_{0}^{1}\langle
B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)],B^{-1}\eta\rangle
dt\\\ &\ \ \ +\sum\limits_{0\leq t\leq
1}[V(y(t))-V(y(t-))]+O(\epsilon),\end{split}$ (3.27)

where $\eta=\int_{|\xi|_{H}<1}\xi\nu(d\xi)$. As for the remaining term, we
have the following control

$\begin{split}\sum_{0\leq t\leq 1}[V(y(t))-V(y(t-))]\leq|DV|\sum_{0\leq t\leq
1}|y(t)-y(t-)|,\end{split}$ (3.28)

where the operator norm $|DV|$ is finite since the operator
$D_{x}V(x,t)=(B^{-1})^{*}(B^{-1}F(x,t)-h(t))$ is $C_{b}^{2}$ in $x$ uniformly
in $t\in[0,1]$. And $y(t)$ is given by

$\begin{split}y(t)&=\phi^{h}(t)+\int_{0}^{t}\exp((t-s)A)BdW(s)+\int_{0}^{t}\int_{|y|_{H}<1}\exp((t-s)A)y\tilde{N}(dt,dy)\\\
&\ \ +\int_{0}^{t}\int_{|y|_{H}\geq 1}\exp((t-s)A)yN(dt,dy).\end{split}$
(3.29)

Since $\phi^{h}(t)+\int_{0}^{t}\exp((t-s)A)BdW(s)$ is continuous in
$t\in[0,1]$, we have

$\begin{split}\sum_{0\leq t\leq 1}|y(t)-y(t-)|=\sum_{0\leq t\leq
1}|z(t)-z(t-)|,\end{split}$ (3.30)

where
$z(t)=\int_{0}^{t}\int_{|y|_{H}<1}\exp((t-s)A)y\tilde{N}(dt,dy)+\int_{0}^{t}\int_{|y|_{H}\geq
1}\exp((t-s)A)yN(dt,dy)$. Then by lemma 2.4, we obtain

$\begin{split}\sum_{0\leq t\leq
1}|z(t)-z(t-)|=\sum\limits_{j=1}^{\infty}\frac{1}{\alpha_{j}}(1-e^{-\alpha_{j}t})\int_{|z|<1}|z|\nu_{j}(dz)<\infty.\end{split}$
(3.31)

Combining (3.8), (3.9), (3.27) and (3.32), we have

$\begin{split}&\mathbf{P}(\|X-\phi^{h}\|_{2}\leq\varepsilon)\propto\mathbf{E}[\exp\\{J_{0}(\phi^{h},\dot{\phi}^{h})-C+O(\epsilon)\\}\mathbf{1}_{\|X^{A}\|_{2}\leq\varepsilon}]\end{split}$
(3.32)

where
$C=\sum\limits_{j=1}^{\infty}\frac{1}{\alpha_{j}}(1-e^{-\alpha_{j}t})\int_{|z|<1}|z|\nu_{j}(dz)$
and $J_{0}(\phi^{h},\dot{\phi}^{h})$ is given by

$\begin{split}J_{0}(\phi^{h},\dot{\phi}^{h})&=-\frac{1}{2}\int_{0}^{1}|B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)]|_{H}^{2}dt\\\
&\ \ \ -\frac{1}{2}\int_{0}^{1}Tr[\nabla_{x}F(t,\phi^{h}(t))]dt\\\ &\ \ \
-\frac{1}{2}\int_{0}^{1}\langle
B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)],B^{-1}\eta\rangle
dt.\end{split}$ (3.33)

Hence, we obtain

$\begin{split}\lim\limits_{\epsilon\rightarrow
0}\gamma_{\epsilon}(\phi)=\lim\limits_{\epsilon\rightarrow
0}\frac{P(\|X-\phi^{h}\|_{2}\leq\epsilon)}{P(\|X^{A}\|_{2}\leq\epsilon)}\propto\exp\\{J_{0}(\phi^{h},\dot{\phi}^{h})\\}.\end{split}$
(3.34)

Thus, $J_{0}(\phi^{h},\dot{\phi}^{h})$ is the desired Onsager-Machlup action
functional by the definition in section 2.3. And up to a constant, it can also
be written as

$\begin{split}J_{0}(\phi^{h},\dot{\phi}^{h})&=-\frac{1}{2}\int_{0}^{1}|B^{-1}[(A\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)-\eta]|_{H}^{2}dt\\\
&\ \ \ -\frac{1}{2}\int_{0}^{1}Tr[\nabla_{x}F(s,\phi^{h}(s))]ds,\end{split}$
(3.35)

where $\eta=\int_{|\xi|_{H}<1}\xi\nu(d\xi)$. $\square$ ∎

##  4 An example

In this section, we present an example to illustrate the Onsager-Machlup
action functional.

######  Example 3.

Suppose that $H=L^{2}([0,1])$ with Neumann boundary conditions, that is
$u^{\prime}(0)=u^{\prime}(1)=0,A=\triangle,B=Id$, and $F:[0,1]\times
L^{2}([0,1];H)\rightarrow L^{2}([0,1];H)$ is given by

$\begin{split}[F(t,\xi)](t,x)=f(0)\chi_{[0,1]}(t)\chi(x)+f(\int_{0}^{\frac{1}{2}}dv\int_{0}^{1}dy\xi(v,y))\chi_{[\frac{1}{2},1](t)}\chi(x)\end{split}$
(4.1)

where $f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C_{b}^{2}$ function. In
addition, the Lévy process is given by (2.3) with one sided exponentially
tempered stable Lévy process in Example 1. Then the conditions of in Remark
3.9 are satisfied. The Onsager-Machlup action functional for the stochastic
partial differential equation $dX(t)=[\Delta X(t)+F(t,X(t))]dt+dW(t)+dL(t)$ is
given by

$\displaystyle J_{0}(\phi^{h},\dot{\phi}^{h})$
$\displaystyle=-\frac{1}{2}\int_{0}^{1}|\Delta\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)-\eta|_{H}^{2}dt,$

where
$\eta=\sum_{j}e_{j}\int_{-1}^{1}\frac{c_{j}e^{-\beta_{j}x}}{x^{1+\alpha_{j}}}dx$.

In fact, for each $\xi\in L^{2}([0,1];H)$, define

$\displaystyle l(\xi)=\int_{0}^{\frac{1}{2}}ds\int_{0}^{1}\xi(s,t)dt,$

which defines a linear functional $l$ on $L^{2}([0,1];H)$. Hence

$\displaystyle[F(t,\xi)](t,x)=f(0)\chi_{[0,1]}(t)\chi(x)+f(l(\xi))\chi_{[\frac{1}{2},1]}(t)\chi(x).$

Since $f:\mathbb{R}\rightarrow\mathbb{R}$ is a $C_{b}^{2}$ function, we obtain
that $F(t,\cdot)$ is a $C_{b}^{2}(L^{2}[0,1];H)$ uniformly in $t\in[0,1]$.
Moreover, for each $h\in L^{2}([0,1];H)$, the Frechét derivative of
$F(t,\cdot)$ is given by

$\displaystyle DF(t,\xi)(h)$ $\displaystyle=\lim\limits_{\tau\rightarrow
0}\frac{F(t,\xi+\tau h)-F(t,\xi)}{\tau}$
$\displaystyle=f^{\prime}(l(\xi))l(h)\chi_{[\frac{1}{2},1]}(t)\chi(x).$

Thus the conditions of in Theorem 3.6 Remark 3.9 are satisfied. And we have
$e_{j}(y)=\cos(2\pi jy)$ for all $y\in[0,1]$ since $H=L^{2}[0,1]$. Hence,
${e_{j}\otimes e_{k}}$ is the complete orthonormal basis of $L^{2}([0,1],H)$.
Then, we obtain

$\displaystyle Tr[\nabla_{x}F(s,\phi^{h}(s))]$
$\displaystyle=\sum\limits_{j=1}^{\infty}\sum\limits_{k=1}^{\infty}\langle\nabla_{x}F(s,\phi^{h}(s))e_{j}\otimes
e_{k},e_{j}\otimes e_{k}\rangle_{L^{2}([0,1];H)}$
$\displaystyle=\sum\limits_{j=1}^{\infty}\sum\limits_{k=1}^{\infty}f^{\prime}(l(\phi^{h}))\int_{0}^{\frac{1}{2}}\int_{0}^{1}cos(2\pi
jy)\cos(2\pi kz)dydz\int_{\frac{1}{2}}^{1}\int_{0}^{1}\cos(2\pi jy)cos(2\pi
kz)dydz$ $\displaystyle=0.$

Therefore, the Onsager-Machlup action functional is

$\displaystyle J_{0}(\phi^{h},\dot{\phi}^{h})$
$\displaystyle=-\frac{1}{2}\int_{0}^{1}|\Delta\phi^{h}(t)+F(t,\phi^{h}(t))-\dot{\phi}^{h}(t)-\eta|_{H}^{2}dt,$

where
$\eta=\sum_{j}e_{j}\int_{-1}^{1}\frac{c_{j}e^{-\beta_{j}x}}{x^{1+\alpha_{j}}}dx$.

##  5 Conclusion and discussion

In this paper, we derive the Onsager-Machlup action functional for stochastic
partial differential equations with Lévy process. As far as we know, it is the
first attempt for infinite systems with non-Gaussion Lévy process. Moreover,
our derivation for the Onsager-Machlup action functional is available for the
Lévy process with both small and large jumps, while Chao and Duan [10] only
dealt with small jumps. Furthermore, we emphasize that the path representing
method is an effective attempt for Lévy process while the $L^{2}$ techniques
will encouter a new difficulty, that is, the representation of the solution
process of linear system by Karhunen-Loève expansion can not result in the
independence of random variables.

With the Onsager-Machlup action functional, it becomes possible to examine the
most probable pathway for systems modeled by stochastic partial differential
equations with non-Gaussian Lévy noise, by studying the associated Euler-
Lagrange equations [10, 18].

However, as the similar problem occurs in [10], we can only deal with the
noise having both Brownian motion and Lévy process. In our methods, we use the
Brownian motion to absorb the drift as the Girsanov theorem permits. It is
still open to derive the Onsager-Machlup action functional for stochastic
dynamical systems with pure Lévy process both in finite and infinite
dimensional systems, because the Girsanov theorem fails to normalize two
probability measures. Bardina et.al [3] tried to deal with jump functions
directly rather than using the Girsanov theorem to obtain the asymptotic
evaluation of the Poisson measure for a tube. But Lévy process is much more
complex than Poisson process. A different approach may be needed to attack
this open problem.

## Acknowledgements

We would like to thank Ying Chao, Yanjie Zhang, Qi Zhang, Pingyuan Wei,
Yuanfei Huang, Qiao Huang, Wei Wei and Ao Zhang for helpful discussions. This
work was partly supported by NSFC grants 11771449 and 11531006.

## References

  * [1] D. Applebaum.  Lévy processes and stochastic calculus.  Cambridge university press, 2009.
  * [2] L. Arnold.  Random Dynamical Systems.  Springer Science & Business Media, 2013.
  * [3] X. Bardina, C. Rovira, and S. Tindel.  Asymptotic evaluation of the poisson measures for tubes around jump curves.  Applicationes Mathematicae, 29:145–156, 2002.
  * [4] X. Bardina, C. Rovira, and S. Tindel.  Onsager–machlup functional for stochastic evolution equations.  In Annales de l’Institut Henri Poincare (B) Probability and Statistics, volume 39, pages 69–93. Elsevier, 2003.
  * [5] B. Böttcher, R. Schilling, and J. Wang.  Lévy matters. iii.  Lecture Notes in Mathematics, 2099, 2013.
  * [6] F. Bouchet, J. Laurie, and O. Zaboronski.  Langevin dynamics, large deviations and instantons for the quasi-geostrophic model and two-dimensional euler equations.  Journal of Statistical Physics, 156(6):1066–1092, 2014.
  * [7] Z. Brzeźniak, W. Liu, and J. Zhu.  Strong solutions for spde with locally monotone coefficients driven by lévy noise.  Nonlinear Analysis: Real World Applications, 17:283–310, 2014.
  * [8] M. Capitaine.  Onsager-machlup functional for some smooth norms on wiener space.  Probability theory and related fields, 102(2):189–201, 1995.
  * [9] M. Capitaine.  On the onsager-machlup functional for elliptic diffusion processes.  In Séminaire de Probabilités XXXIV, pages 313–328. Springer, 2000.
  * [10] Y. Chao and J. Duan.  The onsager-machlup function as lagrangian for the most probable path of a jump-diffusion process.  Nonlinearity, 32(10):3715, 2019.
  * [11] R. Cont and P. Tankov.  Financial modelling with jump processes, 2004.  Chapman &amp, 1975.
  * [12] F. Cooper and J. F. Dawson.  Auxiliary field loop expansion of the effective action for a class of stochastic partial differential equations.  Annals of Physics, 365:118–154, 2016.
  * [13] G. Da Prato and J. Zabczyk.  Stochastic equations in infinite dimensions.  Cambridge university press, 2014.
  * [14] A. Dembo and O. Zeitouni.  Onsager-machlup functionals and maximum a posteriori estimation for a class of non-gaussian random fields.  Journal of multivariate analysis, 36(2):243–262, 1991.
  * [15] P. D. Ditlevsen.  Observation of $\alpha$-stable noise induced millennial climate changes from an ice-core record.  Geophysical Research Letters, 26(10):1441–1444, 1999.
  * [16] J. Duan.  An introduction to stochastic dynamics.  Cambridge University Press, 2015.
  * [17] J. Duan and W. Wei.  Effective dynamics of stochastic partial differential equations.  Elsevier, 2014.
  * [18] D. Dürr and A. Bach.  The onsager-machlup function as lagrangian for the most probable path of a diffusion process.  Communications in Mathematical Physics, 60(2):153–170, 1978.
  * [19] M. I. Freidlin and A. D. Wentzell.  Random perturbations of hamiltonian systems.  In Random Perturbations of Dynamical Systems, pages 258–354. Springer, 2012.
  * [20] T. Fujita and S.-i. Kotani.  The onsager-machlup function for diffusion processes.  Journal of Mathematics of Kyoto University, 22(1):115–130, 1982\.
  * [21] M. Hao, J. Duan, R. Song, and W. Xu.  Asymmetric non-gaussian effects in a tumor growth model with immunization.  Applied Mathematical Modelling, 38(17-18):4428–4444, 2014.
  * [22] K. Hara and Y. Takahashi.  Stochastic analysis in a tubular neighborhood or onsager-machlup functions revisited.  arXiv preprint arXiv:1610.06670, 2016.
  * [23] D. Hochberg, C. Molina-Paris, J. Perez-Mercader, and M. Visser.  Effective action for stochastic partial differential equations.  Physical Review E, 60(6):6343, 1999.
  * [24] Y. Huang, Y. Chao, W. Wei, and J. Duan.  Estimating the most probable transition time for stochastic dynamical systems.  arXiv:2006.10979, 2020.
  * [25] N. Ikeda and S. Watanabe.  Stochastic differential equations and diffusion processes.  Elsevier, 2014.
  * [26] P. Imkeller and J.-S. Von Storch.  Stochastic climate models, volume 49.  Birkhäuser, 2012.
  * [27] J. Jacod and A. N. Shiryaev.  Limit Theorems for Stochastic Processes.  Springer Science & Business Media, 2013.
  * [28] B. Jourdain, S. Méléard, and W. A. Woyczynski.  Lévy flights in evolutionary ecology.  Journal of mathematical biology, 65(4):677–707, 2012.
  * [29] A. Kriegl and P. W. Michor.  The convenient setting of global analysis, volume 53.  American Mathematical Soc., 1997.
  * [30] S. Lang.  Fundamentals of differential geometry, volume 191.  Springer Science & Business Media, 2012.
  * [31] Y. Li, J. Duan, and X. Liu.  A machine learning framework for computing the most probable paths of stochastic dynamical systems.  arXiv:2010.04114, 2020.
  * [32] M. Metivier.  Semimartingales: A course on stochastic processes (de gruyter, berlin, new york).  1982\.
  * [33] L. Onsager and S. Machlup.  Fluctuations and irreversible processes.  Physical Review, 91(6):1505, 1953.
  * [34] S. Peszat and J. Zabczyk.  Stochastic partial differential equations with Lévy noise: An evolution equation approach.  Cambridge University Press, 2007.
  * [35] H. Qiao and J. Wu.  On the path-independence of the girsanov transformation for stochastic evolution equations with jumps in hilbert spaces.  arXiv:1707.07828, 2017.
  * [36] B. Qiu and W. Miao.  Kuroshio path variations south of japan: Bimodality as a self-sustained internal oscillation.  Journal of Physical Oceanography, 30(8):2124–2137, 2000.
  * [37] J. Ren and J. Duan.  Identifying stochastic governing equations from data of the most probable transition trajectories.  arXiv:2002.10251, 2020.
  * [38] L. A. Shepp and O. Zeitouni.  A note on conditional exponential moments and onsager-machlup functionals.  The Annals of Probability, pages 652–654, 1992.
  * [39] X. Sun, L. Xie, and Y. Xie.  Pathwise uniqueness for a class of spdes driven by cylindrical $\alpha$-stable processes.  Potential Analysis, pages 1–17, 2019.
  * [40] Y. Takahashi and S. Watanabe.  The probability functionals (onsager-machlup functions) of diffusion processes.  In Stochastic Integrals, pages 433–463. Springer, 1981.
  * [41] E. M. Wolf and O. Zeitouni.  Onsager machlup functionals for non trace class spde’s.  Probability theory and related fields, 95(2):199–216, 1993.
  * [42] X. Zhang, B. Yang, C. Wei, and M. Luo.  The transition of energy and bound states in the continuum of fractional schrödinger equation in gravitational field and the effect of the minimal length.  Communications in Nonlinear Science and Numerical Simulation, 67:290–302, 2019.
  * [43] Y. Zheng, L. Serdukova, J. Duan, and J. Kurths.  Transitions in a genetic transcriptional regulatory system under lévy motion.  Scientific reports, 6:29274, 2016.
  * [44] Y. Zheng, F. Yang, J. Duan, X. Sun, L. Fu, and J. Kurths.  The maximum likelihood climate change for global warming under the influence of greenhouse effect and lévy noise.  Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(1):013132, 2020.


* (112) D. Chowdhury and A. Hait, _Thermalization in the presence of a time-dependent dissipation and its impact on dark matter production_ , 2302.06654.
  * (113) N. Bernal, F. Elahi, C. Maldonado and J. Unwin, _Ultraviolet Freeze-in and Non-Standard Cosmologies_ , _JCAP_ 1911 (2019) 026 [1909.07992].
  * (114) K. Kaneta, Y. Mambrini and K.A. Olive, _Radiative production of nonthermal dark matter_ , _Phys. Rev. D_ 99 (2019) 063508 [1901.04449].
  * (115) B. Barman, D. Borah and R. Roshan, _Effective Theory of Freeze-in Dark Matter_ , 2007.08768.
  * (116) A. Chakraborty, M.R. Haque, D. Maity and R. Mondal, _Inflaton phenomenology via reheating in the light of PGWs and latest BICEP/ $Keck$ data_, 2304.13637.
  * (117) Y. Shtanov, J.H. Traschen and R.H. Brandenberger, _Universe reheating after inflation_ , _Phys. Rev. D_ 51 (1995) 5438 [hep-ph/9407247].
  * (118) Planck collaboration, _Planck 2018 results. X. Constraints on inflation_ , _Astron. Astrophys._ 641 (2020) A10 [1807.06211].
  * (119) BICEP, Keck collaboration, _Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season_ , _Phys. Rev. Lett._ 127 (2021) 151301 [2110.00483].
  * (120) M. Tristram et al., _Improved limits on the tensor-to-scalar ratio using BICEP and Planck data_ , _Phys. Rev. D_ 105 (2022) 083524 [2112.07961].