Linear Response Theory for Nonlinear Stochastic Differential Equations with \(\alpha \)-Stable Lévy Noises

Abstract

We consider a nonlinear stochastic differential equation driven by an \(\alpha \)-stable Lévy process (\(1<\alpha <2\)). We first prove existence and uniqueness of the invariant measure by the Bogoliubov-Krylov argument. Then we obtain some regularity results for the probability density of its invariant measure by establishing the a priori estimate of the corresponding stationary Fokker-Planck equation. Finally, by the a priori estimate of the Kolmogorov backward equation and the perturbation property of the Markov semigroup, we derive the response function and generalize the famous linear response theory in nonequilibrium statistical mechanics to non-Gaussian stochastic dynamic systems.

Introduction

We consider a stochastic dynamical system described by the following stochastic differential equation (SDE) on \(\mathbb {R}^n\):

$$\begin{aligned} dX_t = b\left( X_{t}\right) dt + \sigma \left( X_{t^{-}}\right) dL_t, \end{aligned}$$
(1.1)

where \(b(x):\mathbb {R}^n \rightarrow \mathbb {R}^n\) and \(\sigma (x)=(\sigma _{ij}(x))_{n\times n}:\mathbb {R}^n \rightarrow \mathbb {R}^{n \times n}\) are Borel measurable functions, and \(L_t\) is a n-dimensional \(\alpha \)-stable (non-Gaussian) Lévy process on the filtered probability space \((\Omega ,\mathcal {F},(\mathcal {F}_t)_{t\ge 0},\mathbb {P})\), with \(1<\alpha <2\). Note that the well-known (Gaussian) Brownian motion \(W_t\) corresponds to the special case \(\alpha =2\). Assume the SDE (1.1) is ergodic with a unique invariant distribution \(\mu \). If \(\mu \) is the initial distribution of \((X_t)_{t\ge 0}\), then \((X_t)_{t\ge 0}\) is a stationary Markov process. In physics, \((X_t)_{t\ge 0}\) being a stationary Markov process means that the corresponding stochastic dynamical system is in a steady state (including the equilibrium state and nonequilibrium steady state).

In recent years, signatures of Lévy noise and anomalous transport have been found ubiquitous in nature. As a result, stochastic dynamical systems with Lévy noises have attracted a lot of attention in many areas, and serve as appropriate models describing complex systems involve anomalous transport, including DNA-target search for binding sites [38], active transport within cells [24], transport in turbulent plasma [37], and the price variation in financial markets [25]. Unlike the Brownian motion for which the variances grows linearly in time \(( \langle W^{2}_t \rangle \sim t)\), the anomalous transport is characterized by a fractional moment \(( \langle L^{\alpha }_t \rangle \sim t)\). Since \(\alpha <2\), the variances is divergent, and the corresponding anomalous transport is superdiffusion ([30]). It is also called Lévy flight (\(\alpha \)-stable Lévy process). The particle undergoing Lévy flight is performing motion with random jumps following the \(\alpha \)-stable law \(L_t-L_s \sim (t-s)^{1/\alpha }\), which is a heavy-tailed distribution. Thus in the trajectories of Lévy flight, there is a significant probability of long jumps since the divergence of variances of Lévy flight. Moreover, a famous result by Mandelbrot [26] shows that the fractal dimension of Lévy flight is \(\alpha \), and Lévy flight satisfies the self-similarity property \(L_t/t^{1/\alpha } = X_1\) in distribution sense. As a comparison, Brownian motion satisfies the self-similarity property \(W_t/t^{1/2} = W_1\) in distribution sense, the fractal dimension of Brownian motion is 2, and there is no jump in the trajectories of Brownian motion.

In statistical physics, the fluctuation-dissipation theorem holds for dissipative systems near the equilibrium states, and is a useful tool in investigation of physical properties of systems at thermodynamic equilibrium. It connects the energy dissipation in an irreversible process to the thermal fluctuation in equilibrium through suitable correlation functions. This is illustrated in the following Langevin equation

$$\begin{aligned} \dot{x}=v, \quad m\dot{v}= - \gamma v+\sqrt{2D} \dot{W}(t), \end{aligned}$$

where x is the position of one particle with velocity v, \(\gamma \) (being positive) is friction coefficient, D is diffusion coefficient, and \(\dot{W}(t)\) is a Gaussian white noise which could be understood as a random force. The fluctuation-dissipation theorem [27, 45, 46] provides a precise connection between the dissipation term \(\gamma \) and the fluctuation term \(\sqrt{2D}\dot{W}(t)\):

$$\begin{aligned} D\mathbb {E}\left( \dot{W}(t)\dot{W}(s)\right) = k_BT \; \gamma \; \delta (t-s), \end{aligned}$$
(1.2)

where \(k_B\) is the Boltzmann constant and T is the absolute temperature, leading to the important Einstein relation \(D=k_B T\gamma \). By virtue of the fluctuation-dissipation theorem, measurable macroscopic physical quantities like the average kinetic energy or susceptibilities can be related to correlation functions of spontaneous fluctuations.

On the other hand, various works [7, 8, 15, 36, 46] indicate that for different random and deterministic dynamical systems, the fluctuation-dissipation theorem can be described by a linear response theory. The linear response theory can be viewed as a generalization of the well-known fluctuation-dissipation theorem when systems are near steady states. The famous Green-Kubo relation and Einstein can be obtained from the linear response theory of classical Langevin equations, as in [27, 45]. Moreover, it is valid under suitable general conditions with many variables, including positions, velocities, concentrations, and order parameters. When the system is near a steady state (invariant measure), and is subject to a certain perturbation, it is useful to know the response of the system to a perturbation of the system. If such response is smooth, the linear response to the perturbation can be described by the first order derivative of the response of a system to the perturbation. In this case, we say that the system exhibits linear response.

For deterministic dynamical systems, linear responses have been obtained by Ruelle [33,34,35], and further developed by Baladi [5], Gouëzel, Liverani [18, 19], and Lucarini [22]. In these works, the linear response of the invariant or SRB measure of the deterministic dynamical systems to small perturbations is derived. We remark that the classical fluctuation-dissipation theorem does not work for some deterministic dynamical systems because of the singularity of the invariant measure ([17]). And the link between deterministic and stochastic viewpoints can be found in [43].

Let us recall the linear response theory for stochastic dynamical systems introduced in [36, 46]. The theory only requires that the system \((X_t)_{t\ge 0}\) is a Markov process with an invariant measure. Consider a stochastic dynamical system in steady state, i.e. the initial distribution of \((X_t)_{t\ge 0}\) is an invariant measure of Markov semigroup, and \((X_t)_{t\ge 0}\) is a stationary Markov process. When a small external perturbation is applied to the system, let \(X^{ F}_t\) be the perturbed process. Given an arbitrary observable O(x), the response (evaluated to first order in the perturbation) can be written as

$$\begin{aligned} \mathbb {E}O\left( X^{F}_t\right) -\mathbb {E}O\left( X_{0}\right) \approx \int _0^t \mathcal {R}_{O}(t-s) F(s) ds, \end{aligned}$$
(1.3)

where \(\mathcal {R}_{O}(t)\) is the time-dependent susceptibility of variable O, and is called the response function in linear response theory. The linear response theory provides a relationship between the response function and a cross-correlation function

$$\begin{aligned} \mathcal {R}_{O}(t) =\frac{d}{dt}\mathbb {E}\left( O\left( X_t\right) U\left( X_{0}\right) \right) , \end{aligned}$$
(1.4)

where U(x) is, sometimes, called the variable conjugate with respect to the perturbation. The linear response theory reveals the susceptibility of every observable, when the stochastic dynamic system is close to a steady state and is under a small time-dependent perturbation.

The mathematical formulation of linear response theory for stochastic dynamical systems have attracted recent interest. Dembo and Deuschel [12] developed the mathematical theory of linear response theory for homogeneous Markov processes based on the strongly continuous semigroups and Dirichlet forms. This theory has been recently extended to a fully nonlinear case by Lucarini and Colangeli [23]. Chen and Jia [8] provided rigorous mathematical proofs of linear response theory and showed the Agarwal-type fluctuation-dissipation theorem for a stochastic differential equation driven by a Brownian motion with unbounded coefficients and a general perturbation. Hairer and Majda [20] developed a mathematically rigorous justification of linear response theory for forced dissipative stochastic dynamical systems. The Ruelle’s response formula for nonequilibrium systems is considered in [13].

Invariant measures play a central role in studying the linear response theory for stochastic dynamical systems. For SDEs driven by Lévy noise with some dissipative condition, it has been shown in [3, 4, 42, 44] that there is a unique invariant probability measure. The exponential ergodicity is also proved therein. In [2, 14], the authors also describe a class of explicit invariant measures for SDEs driven by Lévy noise.

In recent years, some physicists begin to consider the linear response theory for stochastic differential equations driven by Lévy processes. In [15], the authors established the linear response theory for linear stochastic differential equations with stable Lévy noise and constant coefficients. The linear response theory and Onsagers fluctuation theory for linear stochastic differential equations driven by a Gaussian noise and a Cauchy noise have been studied in [28, 29].

In this present paper, we study the linear response theory for nonlinear stochastic differential equations driven by an \(\alpha \)-stable Lévy process (\(1<\alpha <2\)), with rigorous mathematical formulation. We assume that there is a perturbation F(t)K(x) to the drift term, where \(F(t)\in L^{\infty }(\mathbb {R}^{+})\), \( K(x) \text{ and } \text {div}(K(x)) \in L^{\infty }(\mathbb {R}^n)\), and additionally \(\Vert F\Vert _{L^{\infty }} \ll 1\). Under the external perturbation F(t)K(x), the perturbed process \(X^{F}_t\) satisfies the following stochastic differential equation

$$\begin{aligned} \left\{ \begin{aligned}&dX^{ F}_t = \left( b\left( X^{ F}_t\right) + F(t)K\left( X^{ F}_{t}\right) \right) dt + \sigma \left( X^{F}_{t^{-}}\right) dL_t,\\&X^{ F}_0=X_{0}, \end{aligned} \right. \end{aligned}$$
(1.5)

where the distribution of \(X_{0}\) is an invariant measure of the unperturbed SDE (1.1). We prove that the fluctuation relation (1.4) is true for the SDE (1.1).

The main tools to establish the linear response theory is the Markov semigroup and Kolmogorov backward equation. We obtain the perturbation property of the corresponding Markov semigroup by the a priori estimate of the Kolmogorov backward equation. Then we establish the linear response theory and the Agarwal-type fluctuation-dissipation theorem for SDE (1.1). In the present paper, we also use nonlocal heat kernel estimates to prove some new results of existence of invariant measure and ergodicity of SDE (1.1), by the Bogoliubov-Krylov argument. Since the differentiation of invariant measure plays a crucial role in the linear response theory, we further derive a new form of Fokker-Planck equation associated with the SDE (1.1), and establish regularity for the density of invariant measure of SDE (1.1) by proving a new a priori estimate for the corresponding stationary Fokker-Planck equation. We believe this a priori estimate of stationary Fokker-Planck equation is of independent interest.

This paper is organized as follows. In Sect. 2, we recall some basic notations and definitions of the SDE driven by an \(\alpha \)-stable Lévy process, and introduce some well-posedness and ergodicity results for the SDE (1.1). In Sect. 3, we prove the ergodicity of SDE (1.1), and the existence and uniqueness of the invariant measure by the Bogoliubov-Krylov argument. Then we derive the Fokker-Planck equation associated with the SDE (1.1) and establish regularity results for the invariant measure, by proving the a priori estimate of the corresponding Fokker-Planck equation. In Sect. 4, we obtain the response function, and establish the linear response theory as Theorem 4.3. In addition, we further show the Agarwal-type fluctuation-dissipation theorem for SDE (1.1) as Theorem 4.2. This paper ends with some summary and discussion in Sect. 5.

Preliminaries

In this section, we recall some basic notations and definitions. After making some assumptions, we discuss a well-posedness result of SDE (1.1) and the corresponding Kolmogorov equation. In the end, we recall some basic notions about invariant measure and ergodicity, and make the dissipativity assumption for SDE (1.1).

Basic Notations and Definitions

We first introduce some spaces and notations. The Fourier transform \(\mathcal {F}f\) and the inverse Fourier transform \(\mathcal {F}^{-1}g\) are defined by

$$\begin{aligned} \mathcal {F}f(\xi ):=\frac{1}{(2\pi )^{n/2}}\int _{\mathbb {R}^n} f(x)e^{-ix\cdot \xi } dx, \quad \mathcal {F}^{-1}g(x):=\frac{1}{(2\pi )^{n/2}}\int _{\mathbb {R}^n} g(\xi )e^{ix\cdot \xi } d\xi . \end{aligned}$$

For \(p\in [1,\infty ]\), let \(L^p(\mathbb {R}^n)\) be the usual Lebesgue space of functions on \(\mathbb {R}^n\) with \(L^p\) norm. For \(0<\alpha \le 2\) and \(1<p<\infty \), let \(H^{\alpha }_p(\mathbb {R}^n)\) be the usual Bessel potential space with the norm

$$\begin{aligned} \Vert f\Vert _{H^{\alpha }_p}=\Vert \left( (I-\Delta )^{\frac{\alpha }{2}}\right) ^{-1}f\Vert _{L^p}, \end{aligned}$$

where \((I-\Delta )^{\frac{\alpha }{2}}\) and \(\Delta ^{\frac{\alpha }{2}}\) are defined by

$$\begin{aligned} \left( I-\Delta \right) ^{\frac{\alpha }{2}}f:=\mathcal {F}^{-1}\left( \left( 1+|\cdot |^2\right) ^{\frac{\alpha }{2}}\mathcal {F}f\right) , \quad \left( -\Delta \right) ^{\frac{\alpha }{2}}f:=\mathcal {F}^{-1}\left( |\cdot |^{\alpha }\mathcal {F}f\right) . \end{aligned}$$

When \(p\in [1,\infty ]\) and \(\alpha \in \mathbb {N}\), \(H^{\alpha }_p(\mathbb {R}^n)\) is denoted for usual Sobolev space with the norm

$$\begin{aligned} \Vert f\Vert _{H^{\alpha }_p}=\left( \sum _{|\theta |\le \alpha }\Vert \partial ^{\theta }f \Vert ^p_{L^p}\right) ^{\frac{1}{p}}. \end{aligned}$$

We recall the following Sobolev embedding. Let \(0\le \theta \le \alpha \le 2\), and let \(1 \le p \le q \le \infty \) such that \(\theta -\frac{n}{q}<\alpha -\frac{n}{p}.\) Then \(H^{\alpha }_p(\mathbb {R}^n) \hookrightarrow H^{\theta }_q(\mathbb {R}^n)\).

Let X be a Banach space and L(X) be the space of linear bounded operators from X to X. For every \(p \in [1,\infty ]\) and \(0\le s<t \le \infty \), the space \(L^p(s,t;X)\) consists of all strongly measurable \(u:[s,t]\rightarrow X\) with

$$\begin{aligned} \Vert u\Vert _{L^p(s,t;X)}:=\left( \int ^t_s \Vert u(r)\Vert ^p_X dr \right) ^{\frac{1}{p}}<\infty \end{aligned}$$

for \(1\le p \le \infty \), and

$$\begin{aligned} \Vert u\Vert _{L^{\infty }(s,t;X)}:=\text {esssup}_{s\le r \le t}\Vert u(r)\Vert _X <\infty . \end{aligned}$$

Now we recall some basic facts for \(\alpha \)-stable Lévy processes [1, 10]. Let \((\Omega ,\mathcal {F},\mathbb {P})\) be a filtered probability space satisfying the usual conditions. Consider the n dimensional \(\alpha \)-stable Lévy process \(L_t\) on \((\Omega ,\mathcal {F},(\mathcal {F}_t)_{t\ge 0},\mathbb {P})\) with \(1<\alpha <2\). The characteristic function of \(L_t\) is

$$\begin{aligned} \varphi _{L_t}(\xi )= \mathbb {E}\exp \left( i\langle \xi , L_t \rangle \right) = e^{-t|\xi |^{\alpha }}. \end{aligned}$$

For the \(\alpha \)-stable Lévy process \(L_t\), the corresponding Lévy measure \(\nu (dy) = \frac{C_{\alpha }}{|y|^{n+\alpha }} dy\), where \(C_{\alpha }=2^{\alpha }\Gamma (n/2+\alpha /2)/[\pi ^{n/2}\Gamma (-\alpha /2)]\). We denote by \(N^L(dt,dy)\) the Poisson random measure associated to the pure jump-process \(\Delta L_t =L_{t}-L_{t^-}\) such that \(\mathbb {E}[N^L(dt,dy)] = dt\nu (dy)\), which is defined as

$$\begin{aligned} N^{L}(t,B)(\omega ):= \sharp \left\{ s\in [0,t]:\Delta L_{s}(\omega ) \in B\right\} , \quad t\ge 0, B\in \mathcal {B}\left( \mathbb {R}^{n}\backslash \{0\}\right) . \end{aligned}$$

And the corresponding compensated Poisson random measure \(\widetilde{N}^L\) is defined as

$$\begin{aligned} \widetilde{N}^{L}(dt,dy)=N^{L}(dt,dy)-dt\nu (dy) . \end{aligned}$$

Then by Lévy-Itô decomposition theorem, we have following path-wise description of \(L_t\)

$$\begin{aligned} L_t = \int _0^t \int _{0<|y|<1}y\widetilde{N}^L(dt,dy)+ \int _0^t \int _{|y|\ge 1}yN^L(dt,dy). \end{aligned}$$

SDE Driven by \(\alpha \)-Stable Lévy Process

Consider the following stochastic differential equation on \(\mathbb {R}^n\):

$$\begin{aligned} dX_t = \hat{b}\left( t,X_{t}\right) dt + \sigma \left( X_{t^{-}}\right) dL_t, \end{aligned}$$
(2.1)

where \(\hat{b}(t,x):\mathbb {R}^{+}\times \mathbb {R}^n \rightarrow \mathbb {R}^n\) and \(\sigma (x)=(\sigma _{ij}(x))_{n\times n}:\mathbb {R}^n \rightarrow \mathbb {R}^{n \times n}\) are Borel measurable functions, and \(L_t\) is a n-dimensional \(\alpha \)-stable Levy process on \((\Omega ,\mathcal {F},(\mathcal {F}_t)_{t\ge 0},\mathbb {P})\) with \(1<\alpha <2\). Note that here the drift term \(\hat{b}(t,x)\) is dependent of t and x, and the form of (2.1) includes the SDE (1.1) and the perturbed SDE (1.5). So the following results in this subsection are hold for these two SDEs.

The SDE (2.1) is equivalent to

$$\begin{aligned} dX_t = \hat{b}\left( t,X_{t}\right) dt + \int _{|y|<1}\sigma \left( X_{t^{-}}\right) y\widetilde{N}^L(dt,dy)+\int _{|y|\ge 1}\sigma \left( X_{t^{-}}\right) yN^L(dt,dy). \end{aligned}$$
(2.2)

We make the following assumptions on the drift coefficient b and the diffusion coefficient \(\sigma \).

(A):

(Hölder continuity) For all \( x \in \mathbb {R}^n\), there are constants \(c_0 >0\) and \(\beta \in (0, 1)\) such that

$$\begin{aligned} \Vert \sigma (x)-\sigma (y)\Vert \le c_0 |x-y|^{\beta }. \end{aligned}$$

Here and below, \(\Vert \cdot \Vert \) denotes the Hilbert-Schmidt norm of a matrix, and \(|\cdot |\) denotes the Euclidean norm.

(B):

(Uniform ellipticity) There exists a constant \(\Lambda >0\) such that for all \(x \in \mathbb {R}^n\),

$$\begin{aligned} \Lambda ^{-1}|\xi | \le |\sigma (x)\xi )| \le \Lambda |\xi |, \quad \forall \xi \in \mathbb {R}^n. \end{aligned}$$
(C):

(Uniform boundedness) There exists a constant \(C_0>0\) such that for all \((t,x) \in \mathbb {R}^{+}\times \mathbb {R}^n\),

$$\begin{aligned} |\nabla \hat{b}(t,x)|,|\hat{b}(t,x)|,|\nabla \sigma (x)|,|\nabla \sigma ^{-1}(x)|< C_0. \end{aligned}$$

The following well-posedness result of the SDE (2.1) is proved by Xie and Zhang ([44], Theorem 2.4 and Remark 2.5).

Theorem 2.1

Suppose that \((\mathbf{A})\)-\((\mathbf{C})\) hold. Then for each \(X_0=x \in \mathbb {R}^n\), the SDE (2.1) admits a unique strong solution \(X_{t}\).

The stochastic process \((X_t)_{t\ge 0}\) is a Markov process with a Markov transition kernel

$$\begin{aligned} \pi _{s,t}(x,B):=\mathbb {P}\left( X_{t}\in B|X_{s}=x\right) , \quad 0\le s \le t, x\in \mathbb {R}^n, B\in \mathcal {B}\left( \mathbb {R}^n\right) . \end{aligned}$$

We denote by \((P_{s,t})_{t-s\ge 0}\) the associated Feller semigroup of \((X_t)_{t\ge 0}\), i.e.

$$\begin{aligned} P_{s,t}f(x):=\int _{\mathbb {R}^n}f(y)\pi _{s,t}(x,dy)=\mathbb {E}\left( f\left( X_t\right) |X_s =x\right) , \quad 0\le s\le t, \ x\in \mathbb {R}^n, \end{aligned}$$

where \(f\in \mathcal {B}_b(\mathbb {R}^n)\). If the Markov process is time-homogeneous, we denote \(P_{s,t}=P_{t-s}\) for all \(0\le s\le t\). If the transition probability densities p(sxty) exists, then

$$\begin{aligned} P_{s,t}f(x):=\int _{\mathbb {R}^n}f(y)p(s,x;t,y)dy. \end{aligned}$$

The generator A(s) of \(P_{s,t}\) is the following integro-differential operator

$$\begin{aligned} A(s)u(x) := \hat{b}(s,x) \cdot \nabla u + \int _{\mathbb {R}^{n}\setminus \{0\}} \left[ u(x+\sigma (x)y)-u(x)\right] \nu (dy), \end{aligned}$$

where \(u \in Dom(A)\subset L^2(\mathbb {R}^n) \).

By Itô’s formula, for each \(f\in \mathcal {B}_b(\mathbb {R}^n)\) and \(0\le s\le t\), the function \(u(t,x):=\mathbb {E}[f(X_t)|B_s =x]\) satisfies the following Kolmogorov backward equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s u(s,x) = -A_x(s)u(s,x),\\&u(t,x)=f(x). \end{aligned} \right. \end{aligned}$$
(2.3)

for all \(s\in [0,t]\). Moreover, the transition density p(sxty) of Markov process \((X_t)_{t\ge 0}\) is the fundamental solution of following Kolmogorov backward equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s u(s,x) = -A_x(s) u(s,x),\\&u(t,x)=\delta (x-y). \end{aligned} \right. \end{aligned}$$
(2.4)

Here and below, \(A_x\) denotes the operator A acting on functions of x.

Furthermore, for \(0 \le s < t\le T\), if the probability density of \(X_s\) is \(p_s(x)\), then the probability density \(p_t\) of the Markov process \((X_t)_{t\ge 0}\) is the solution of the following Kolmogorov forward equation, or Fokker-Planck equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u(t,x) = A_x^{*}(t) u(t,x),\\&u(s,x)=p_s(x), \end{aligned} \right. \end{aligned}$$
(2.5)

where the operator \(A^{*}(t)\) is the adjoint operator of A defined through \(\langle Af,g \rangle _{L^2}=\langle f,A^{*}g \rangle _{L^2}\). And the probability density \(p_t\) of \(X_t\) is given by

$$\begin{aligned} p_{t}(y)=\int _{\mathbb {R}^n}p_s(x)p(s,x;t,y)dx. \end{aligned}$$

Moreover, since b(tx) is uniformly bounded, from [9], Theorem 1.5 or or [44] Theorem 2.4, the transition probability densities p(sxty) of \(X_t\) exists, and it enjoys the following estimates.

Theorem 2.2

Under \(\mathbf{(A)}\)-\(\mathbf{(C)}\), there is a unique continuous function p(sxty) satisfying (2.4), i.e. the transition probability densities p(sxty) of \(X_t\) exists. Moreover, p(sxty) enjoys the following properties.

  1. (i)

    (Two-sides estimate) For every \(T > 0\), there are two positive constants \(c_1,c_2\) such that for \(0\le s < t \le T\) and \(x,y\in \mathbb {R}^n\),

    $$\begin{aligned} c_1 (t-s)((t-s)^{\frac{1}{\alpha }}+|x-y|)^{-n-\alpha }\le p(s,x;t,y) \le c_2 (t-s)((t-s)^{\frac{1}{\alpha }}+|x-y|)^{-n-\alpha }. \end{aligned}$$
    (2.6)
  2. (ii)

    (Gradient estimate) For every \(T > 0\), there exists a positive constant \(c_3\) such that for \(0\le s < t \le T\) and \(x,y\in \mathbb {R}^n\),

    $$\begin{aligned} |\nabla _x p(s,x;t,y)|\le c_3 (t-s)^{-\frac{1}{\alpha }}p(s,x;t,y). \end{aligned}$$
    (2.7)
  3. (iii)

    (Fractional derivative estimate) For every \(\theta \in [0,\alpha )\), there exists a positive constant \(c_4\) such that for \(0\le s < t \le T\) and \(x,y\in \mathbb {R}^n\),

    $$\begin{aligned} |\Delta ^{\theta /2}_x p(s,x;t,y)|\le c_4 (t-s)^{1-\frac{\theta }{\alpha }}((t-s)^{\frac{1}{\alpha }}+|x-y|)^{-n-\alpha }. \end{aligned}$$
    (2.8)
  4. (vi)

    (Continuity) For every bounded and uniformly continuous function f(x), we have

    $$\begin{aligned} \lim _{|t-s|\rightarrow 0} \Vert \int _{\mathbb {R}^n}p(s,x;t,y)f(y)dy-f(x)\Vert _{L^{\infty }}=0. \end{aligned}$$
    (2.9)

Remark 2.1

In the proof of well-posedness of the SDE (2.1) and corresponding heat kernel estimates, we need to use the regularization effect of the jump noise to cancel the singularity from drift. For this, we restrict \(1<\alpha <2\), such that the nonlocal part plays a dominant role in the generator of corresponding Markov semigroup.

Now from above estimates of the transition probability density, we have the following results of solvability and regularity of corresponding Kolomogrov equation. By Minkowski’s inequality for integral, the following result is a direct consequence of two-sides estimate (2.6) and fractional derivative estimate (2.8).

Lemma 2.1

Assume that condition \((\mathbf{A})\)-\((\mathbf{C})\) hold. Assume \(f(x)\in L^p\) with some \(1\le p \le \infty \). Let \(\theta \in (0,\alpha )\). Then for \(0\le s \le t\), the function \(u(s,t):=P_{s,t}f(x) \in H^{\theta }_p\) is the unique solution to the Kolmogorov backward equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s u(s,x) = -A(s)u(s,x),\quad (s,x)\in [0,t)\times \mathbb {R}^n\\&u(t,x)=f(x),\quad x\in \mathbb {R}^n, \end{aligned} \right. \end{aligned}$$

where A(s) is the generator of SDE (2.1). Moreover, there is a constant \(C>0\) such that for all s with \(0\le s \le t\),

$$\begin{aligned} \Vert u(s,x)\Vert _{L^p} \le C \Vert f\Vert _{L^p} \end{aligned}$$

and

$$\begin{aligned} \Vert u(s,x)\Vert _{H^{\theta }_p} \le C (t-s)^{-\frac{\theta }{\alpha }}\Vert f\Vert _{L^p}. \end{aligned}$$

Proof

From Theorem 2.2, the unique solution u(sx) is given by

$$\begin{aligned} u(s,x)=\int _{\mathbb {R}^n} p(s,x,t,y)f(y)dy. \end{aligned}$$

The two-sides estimate (2.6) yields that

$$\begin{aligned} |u(s,x)|&\lesssim \int _{\mathbb {R}^n}(t-s)\left[ (t-s)^{\frac{1}{\alpha }}+|y|\right] ^{-n-\alpha }|f(x-y)|dy \end{aligned}$$

Then for \(1 \le p<\infty \), by Minkowski’s inequality for integral, we have

$$\begin{aligned} \Vert u(s,x)\Vert _{L^p}&\lesssim \int _{\mathbb {R}^n}(t-s)\left[ (t-s)^{\frac{1}{\alpha }}+|y|\right] ^{-n-\alpha }\Vert f(y)\Vert _{L^p}dy\\&\lesssim \Vert f\Vert _{L^p}\int _{\mathbb {R}^n}(t-s)\left[ (t-s)^{\frac{1}{\alpha }}+|y|\right] ^{-n-\alpha }dy\\&\le C \Vert f\Vert _{L^p}. \end{aligned}$$

This estimate is obvious when \(p=\infty \).

By the fractional derivative estimate (2.7) and two-sides estimate (2.6), we have

$$\begin{aligned} |\Delta ^{\theta /2}_x u(s,x)|&\le \int _{\mathbb {R}^n} |\Delta ^{\theta /2}_x p(s,x;t,y)||f(y)|dy \\&\lesssim (t-s)^{-\frac{\theta }{\alpha }}\int _{\mathbb {R}^n} p(s,x;t,y)|f(y)|dy. \end{aligned}$$

Then we get

$$\begin{aligned} \Vert \Delta ^{\theta /2} u(s,x)\Vert _{L^p}\le C (t-s)^{-\frac{\theta }{\alpha }} \Vert f\Vert _{L^p}. \end{aligned}$$

The proof is complete. \(\square \)

We now consider the following nonlocal parabolic equation corresponding to SDE (2.1):

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s w(s,x) = -A(s)w(s,x)+\lambda w(s,x)- g(s,x), \quad (s,x)\in [0,t]\times \mathbb {R}^n,\\&w(t,x)=0, \quad x\in \mathbb {R}^n \end{aligned} \right. \end{aligned}$$
(2.10)

where A(s) is the generator of SDE (2.1), and \(\lambda \ge 0\).

As in the proof of Theorem 4.5 in [44], by fractional derivative estimate (2.7), two- sides estimate (2.6) and Young’s convolution inequality, we have the following solvability and \(L^p\)-estimate of (2.10).

Lemma 2.2

Assume that condition \((\mathbf{A})\)-\((\mathbf{C})\) hold. Let \(p,q\in (1,\infty )\) and \(p'\in [p,\infty ]\), \(q'\in [q,\infty ]\), \(\vartheta \in [1,\alpha )\) with

$$\begin{aligned} \frac{n}{p}+\frac{\alpha }{q}<\alpha -\vartheta +\frac{n}{p'}+\frac{\alpha }{q'}. \end{aligned}$$

Then for every \(t>0\), \(g\in L^q((0,t);L^p)\), there are constants \( c>1\) and unique mild solution \(u(s,x)=\int ^t_s P_{s,v}g(v,x)dv\) to (2.10) such that for all \(\lambda \ge 0\) and \(s\in [0,t]\),

$$\begin{aligned} \left( 1\vee \lambda \right) ^{\frac{1}{\alpha }(\alpha -\vartheta +\frac{n}{p'}+\frac{\alpha }{q'}-\frac{n}{p}-\frac{\alpha }{q})} \Vert w\Vert _{L^{q'}(s,t;H^{\vartheta }_{p'})}\le c\Vert g\Vert _{L^q(s,t;L^p)}. \end{aligned}$$

Steady States, Invariant Measures and Ergodicity

We recall some basic notions about the invariant measure and ergodicity. Now we assume that the drift term b and diffusion term \(\sigma \) of SDE (1.1) is independent of time t. Thus the solution \((X_t)_{t\ge 0}\) is a homogeneous Markov process with Markov Feller semigroup \(P_t\).

Definition 2.1

A probability measure \(\mu \) on \((\mathbb {R}^{n}, \mathcal {B}(\mathbb {R}^n))\) is called an invariant measure under Markov semigroup \(P_t\) if it satisfies

$$\begin{aligned} \int _{\mathbb {R}^n} Af(x)\mu (dx)=0, \ \forall f\in C^{\infty }_0\left( \mathbb {R}^n\right) , \end{aligned}$$

where A is the generator of Markov semigroup \(P_t\).

This means \(\int _{\mathbb {R}^n}(P_tf)(x)\mu (dx)=\int _{\mathbb {R}^n}f(x)\mu (dx)\) for all \(t\ge 0\) and \(f\in C^{\infty }_0(\mathbb {R}^n)\). If the invariant measure \(\mu \) has probability density \(p_{ss}(x)\), then \(p_{ss}(x)\) is a solution of stationary Fokker-Planck equation (2.5), i.e. \(A^{*}p_{ss}(x)=0,\) or \(P^{*}_{t}p_{ss}(x)=p_{ss}(x)\) for all \(t>0\). And if initial distribution of \((X_{t})_{t\ge 0}\) is above invariant measure \(\mu \), then \((X_{t})_{t\ge 0}\) is a stationary Markov process, which satisfies that for every \(f\in \mathcal {B}_b(\mathbb {R}^n)\) and \(t>0\), \(P_t f(X_{0})=f(X_{0})\).

It is known that a stochastic dynamic system \((X_{t})_{t\ge 0}\) is said in a steady state if its initial distribution is the invariant measure \(\mu \) of the corresponding SDE and \((X_{t})_{t\ge 0}\) is a stationary Markov process ([6]).

Definition 2.2

A Markov semigroup \(P_t\) is ergodic if \(P_t\) admits a unique invariant probability measure \(\mu \), or equivalently,

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{1}{t}\int _0^t P_s f(x)ds = \int _{\mathbb {R}^n}f(x)\mu (dx), \quad \forall f\in \mathcal {B}_b\left( \mathbb {R}^n\right) . \end{aligned}$$

To present the ergodicity result, we need a dissipativity assumption on the drift term b(x).

\((\mathbf{D})\)(Dissipativity) For all \(x\in \mathbb {R}^n\), there exits a constant \(k_1>0\), such that

$$\begin{aligned} \langle x,b(x) \rangle \le -k_1|x| \end{aligned}$$

Moreover, the constant \(k_1\) satisfies

$$\begin{aligned} \sqrt{2} k_1 > \Lambda ^2 \int _{0<|y|<1}|y|^2 \nu (dy)+\Lambda \int _{|y|\ge 1}|y| \nu (dy), \end{aligned}$$

where \(\Lambda \) is the constant in uniform ellipticity assumption \((\mathbf{B})\).

A usual method for proving the existence of invariant measures of Markov processes is the Bogoliubov-Krylov argument, which is based on Lyapunov functions (e.g. [11], Theorem 7.1 and Proposition 7.10).

Theorem 2.3

Let \(X_t(x_0)\) be a Markov process with initial value \(X_0=x_0 \in \mathbb {R}^n\), and \(P_t\) be the corresponding Markov Feller semigroup of \(X_t\). Let \(V:\mathbb {R}^n\rightarrow \mathbb {R}^{+}\) be a Borel measurable function whose level sets

$$\begin{aligned} K_a:= \left\{ x\in \mathbb {R}^n:\ V(x)\le a\right\} ,\quad a>0, \end{aligned}$$

are compact for every \(a>0\). Assume that there exists \(x_0\in \mathbb {R}^n\) and \(C(x_0)>0\) such that

$$\begin{aligned} \mathbb {E}\left( V\left( X_t\left( x_0\right) \right) \right) <C\left( x_0\right) , \quad \forall t\ge 0. \end{aligned}$$

Then there is an invariant measure for \(P_t\).

Remark 2.2

In Theorem 2.3, the Borel measurable function V(x) is called a Lyapunov function for \(P_t\).

We recall the following notations for Markov Feller semigroup \(P_t\).

\(\mathbf {(Strong \ Feller)}\):

\(P_t\) has the strong Feller property if for all \(f \in \mathcal {B}_b(\mathbb {R}^n)\), \(P_t f\in C_b(\mathbb {R}^n)\).

\(\mathbf {(Irreduciblility)}\):

\(P_t\) is irreducible if for each open ball B and for all \(t>0\), \(x\in \mathbb {R}^n\), \(P_t 1_{B_1}(x)>0\).

We have a sufficient condition for the uniqueness of invariant measure of Markov semigroup \(P_t\).

Lemma 2.3

If \(P_t\) is strong Feller and irreducible, then it possesses at most one invariant measure.

The following ergodicity result is standard.

Theorem 2.4

If a Markov process has a unique invariant measure, then it is ergodic.

Existence, Uniqueness and Regularity of Invariant Measure

Existence of Invariant Measure and Ergodic Property

We show the following moment estimate of the unique strong solution to (1.5).

Lemma 3.1

Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold. Let \(X_0\) be the initial value of SDE (1.1) with initial distribution \(\mu _0\). Suppose that \(X_0\) has a finite first moment, i.e. \(\mathbb {E}|X_0|<\infty \). Then for solution \((X_t)_{t \ge 0}\) to SDE (1.5) with initial value \(X_0\), there exits a positive constant C such that

$$\begin{aligned} \mathbb {E}|X_t| \le \mathbb {E}\sqrt{1+|X_0|^2}+C, \quad \forall t>0. \end{aligned}$$
(3.1)

Proof

Define \(r(x):=\sqrt{1+|x|^2}\). Then by Itô’s formula, for \(\forall t>0\) we have

$$\begin{aligned}&r\left( X_{t}\right) -r\left( X_{0}\right) \\&\quad = \int _{0}^{ t} b\left( X_{s^{-}}\right) \nabla r\left( X_{s^{-}}\right) ds\\&\qquad + \int _{0}^{t} \int _{\mathbb {R}^N\setminus \{0\}} r\left( X_{s^{-}}+\sigma (X_{s^{-}})y\right) -rh\left( X_{s^{-}}\right) \widetilde{N}^L(ds,dy)\\&\qquad +\int _{0}^{t} \int _{\mathbb {R}^N\setminus \{0\}}\left[ r\left( X_{s^{-}}+\sigma \left( X_{s^{-}}\right) y\right) -r\left( X_{s^{-}}\right) -1_{B_1(0)}(y) \sigma \left( X_{s^{-}}\right) y\nabla r\left( X_{s^{-}}\right) \right] \nu (dy)ds. \end{aligned}$$

Take expectation for two sides, and we have

$$\begin{aligned} \mathbb {E}r\left( X_{t}\right) =\mathbb {E}r\left( X_{0}\right) + \mathbb {E}\int _{0}^{t}Ar\left( X_s\right) ds. \end{aligned}$$

The stochastic Fubini theorem implies that

$$\begin{aligned} \mathbb {E}r\left( X_{t}\right) =\mathbb {E}r\left( X_{0}\right) +\int _{0}^{ t}\mathbb {E}Ar\left( X_s\right) ds. \end{aligned}$$

By the dissipativity assumption \((\mathbf {D})\), we have

$$\begin{aligned} b(x)\cdot \nabla r(x)=(b(x)\cdot x)(1+|x|^2)^{-\frac{1}{2}}\le&-k_1 |x|(1+|x|^2)^{-\frac{1}{2}} \le -\frac{k_1}{\sqrt{2}}1_{\{ |x|\ge 1\} }, \end{aligned}$$
(3.2)

Note that

$$\begin{aligned} |r(x+y)-r(x)|\le |y|\int _0^1 |\nabla r(x+sy)|ds \le \frac{|y|}{2}, \end{aligned}$$

and

$$\begin{aligned} r(x+y)-r(x)-y\cdot \nabla r(x) \le \frac{|y|^2}{2}. \end{aligned}$$

We have

$$\begin{aligned}&\int _{\mathbb {R}^{n}\setminus \{0\}} \left[ r(x+\sigma (x)y)-r(x)-1_{B_{1}(0)}(y)\sigma (x)y \cdot \nabla r(x)\right] \nu (dy) \nonumber \\&\quad \le \frac{1}{2}\int _{0<|y|<1}|\sigma (x)y|^2 \nu (dy)+\frac{1}{2}\int _{|y|\ge 1}|\sigma (x)y| \nu (dy) \nonumber \\&\quad \le \frac{1}{2}\left( \Lambda ^2 \int _{0<|y|<1}|y|^2 \nu (dy)+\Lambda \int _{|y|\ge 1}|y| \nu (dy)\right) < \infty . \end{aligned}$$
(3.3)

Denote \(C_1=\Lambda ^2 \int _{0<|y|<1}|y|^2 \nu (dy)+\Lambda \int _{|y|\ge 1}|y| \nu (dy)>0\). Combining with (3.2) and (3.3), we have

$$\begin{aligned} Ar(x) =&b(x) \cdot \nabla r(x) + \int _{\mathbb {R}^{n}\setminus \{0\}} [r(x+\sigma (x)y)-r(x)-1_{B_{1}(0)}(y)\sigma (x)y \cdot \nabla r(x)] \nu (dy) \\ \le&-\frac{k_1}{\sqrt{2}}1_{\left\{ |x|\ge 1\right\} }+ \frac{C_1}{2}, \end{aligned}$$

Then we get

$$\begin{aligned} \mathbb {E}r\left( X_t\right) =&\mathbb {E}r\left( X_{0}\right) +\int _{0}^{ t}\mathbb {E}Ar\left( X_s\right) ds \\ \le&\mathbb {E}r\left( X_{0}\right) -\frac{k_1}{\sqrt{2}}\int _{0}^{t} P\left( |X_s|\ge 1\right) ds + \frac{t C_1}{2} \\ =&\mathbb {E}r\left( X_{0}\right) +\left( \frac{C_1}{2}-\frac{k_1}{\sqrt{2}}\right) t + \frac{k}{\sqrt{2}}\int _{0}^{t}P\left( |X_s|<1\right) ds. \end{aligned}$$

By two-sides estimate (2.6) and Fubini theorem, there exits a constant \(C_2\) such that

$$\begin{aligned} P\left( |X_s|<1\right) =&\int _{|y|<1}\int _{\mathbb {R}^n}p(0,x;s,y)\mu _0(dx)dy \\<&c_2\int _{\mathbb {R}^n}\int _{|y|<1}s\left( (s)^{\frac{1}{\alpha }}+|x-y|\right) ^{-n-\alpha }dy\mu _0(dx)\\ <&\frac{c_2 \pi ^{n/2}}{\Gamma \left( \frac{n}{2}+1\right) } s^{-\frac{n}{\alpha }}. \end{aligned}$$

Then for each \(t>1\), we have

$$\begin{aligned} \int _{0}^{t}P\left( |X_s|<1\right) ds\le 1+\int _{1}^{t}P\left( |X_s|<1\right) ds=1-\frac{\alpha }{\alpha -n}\left( t^{1-\frac{n}{\alpha }}-1\right) \frac{c_2 \pi ^{n/2}}{\Gamma \left( \frac{n}{2}+1\right) }. \end{aligned}$$

By dissipativity condition \(\mathbf {(D})\), \(-\sqrt{2} k_1+C_1 <0\). Thus there exists a constant \(C>0\) such that

$$\begin{aligned} \mathbb {E}|X_t| \le \mathbb {E}r\left( X_t\right) \le \mathbb {E}r\left( X_0\right) +C, \quad \forall t>0. \end{aligned}$$

The proof is complete. \(\square \)

Now we prove the following ergodicity result for SDE (1.1).

Lemma 3.2

Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold. Then there exists a unique invariant measure \(\mu \) for SDE (1.1), and the SDE (1.1) is ergodic. Moreover, if \(X_{ss}\) is a random variable with invariant distribution \(\mu \), then \(X_{ss}\) has finite first moment, i.e. \(\int _{\mathbb {R}^n}|x|\mu (dx)<\infty \).

Proof

Assume \(X_0=x_0\) for some \( x_0 \in \mathbb {R}^n\). Then Lemma 3.1 implies \(\mathbb {E}|X_t|<|x_0|+C\) for each \(t>0\). If we choose \(V(x)=|x|\) as the Lyapunov function for \(P_t\), then by Theorem 2.3, there is an invariant measure for the Markov Feller semigroup \(P_t\).

Now we prove the uniqueness of invariant measure. By the two-side estimates (2.6) for the transition probability density p(sxty), the Markov semigroup \(P_t\) of \(X_t\) is irreducible. Moreover, since the transition probability density p(sxty) is unique continuous, for each \(f\in \mathcal {B}_b(\mathbb {R}^n)\),

$$\begin{aligned} P_t f(x)=\int _{\mathbb {R}^n}p(0,x;t,y)f(y)dy\in C_b\left( \mathbb {R}^n\right) . \end{aligned}$$

So the Markov semigroup \(P_t\) is strong Feller. Thus there exists a unique invariant measure for SDE (1.1), and the SDE (1.1) is ergodic.

For each \(m \in \mathbb {N}\), consider the bounded measurable function \(x1_{B_{m}(0)}(x)\in \mathcal {B}_b(\mathbb {R}^n)\). Then from the definition of ergodic property, we have

$$\begin{aligned} \mathbb {E}\left( |X_{ss}1_{\{|X_{ss}|<m\}}|\right) =&\int _{\mathbb {R}^n}|x|1_{B_{m}(0)}(x)\mu (dx) \\ =&\lim _{t\rightarrow \infty }\frac{1}{t}\int _0^t P_s |x|1_{B_{m}(0)}(x)ds \\ =&\lim _{t\rightarrow \infty }\frac{1}{t}\int _0^t \mathbb {E}\left( |X_{s}1_{\{|X_{s}|<m\}}(\omega )|\right) ds \\ \le&\sup _{t\ge 0} \mathbb {E}|X_t| \\ <&|x_0| +C. \end{aligned}$$

Note that \(x_0+C\) is fixed, and it is independent with m. Let \(m\rightarrow \infty \), we obtain \(\mathbb {E}|X_{ss}|<\infty \).

\(\square \)

Now we prove that the invariant measure \(\mu \) has a density \(p_{ss}\).

Lemma 3.3

Suppose that \((\mathbf {A})\)-\((\mathbf {C})\) holds. Then the invariant measure \(\mu \) has a density \(p_{ss}\in L^{p'}(\mathbb {R}^n)\) with \(p'<\frac{n}{n-\alpha }\) for \(n\ge 2\) and \( p'<\infty \) for \(n=1\).

Proof

From Lemma 2.2, for each \(f\in C^{\infty }_0(\mathbb {R}^n)\) and \(T>0\), there is a unique solution \(u\in L^{\infty }(0,T;H^{\vartheta }_{\infty }(\mathbb {R}^n))\) solving the following equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t u(t,x) = -Au(t,x)- f(x), \quad (t,x)\in [0,T]\times \mathbb {R}^n,\\&u(T,x)=0, \quad x\in \mathbb {R}^n \end{aligned} \right. \end{aligned}$$

By Itô’s formula, we have

$$\begin{aligned} \mathbb {E}u\left( T,X_T\right)&=\int _{0}^{T}\mathbb {E}\left( \partial _t u(t,x)+Au(t,x)\right) dt \\&= \mathbb {E} \int _{0}^{T} f(X_t)dt \end{aligned}$$

Then the a priori estimate implies that for all \((\frac{n}{\alpha }\vee 1)<p< \infty \),

$$\begin{aligned} \mathbb {E}\int _{0}^{T}|f(X_t)|dt \le \Vert u\Vert _{L^{\infty }\left( 0,T;L^{\infty }\left( \mathbb {R}^n\right) \right) }\le C \Vert f\Vert _{L^p}. \end{aligned}$$

Since \(C^{\infty }_0(\mathbb {R}^n)\) is dense in \(L^p(\mathbb {R}^n)\), by the ergodic property of \(X_t\), we get for all \((\frac{n}{\alpha }\vee 1)<p< \infty \),

$$\begin{aligned} |\int _{\mathbb {R}^n}f(x)\mu (dx)| \le C \Vert f\Vert _{L^p}, \quad \forall f\in L^p\left( \mathbb {R}^n\right) . \end{aligned}$$

Thus \(f \longmapsto \int _{\mathbb {R}^n}f(x)\mu (dx)\) is linear bounded functional on \(L^p(\mathbb {R}^n)\). Then by Riesz’s representation theorem, the unique invariant measure \(\mu \) has a density \(p_{ss}\in p'\) with \(1\le p'<\frac{n}{n-\alpha }\) for \(n\ge 2\) and \( 1\le p'<\infty \) for \(n=1\). \(\square \)

The Adjoint Operator of Generator

In order to obtain a form of the stationary Fokker-Planck equation corresponding to the SDE (1.1), we need to derive the adjoint operator \(A^{*}\), for the generator A.

By assumption \((\mathbf {B})\), the diffusion coefficient \(\sigma (x)\) is a invertible matrix, and \(\sigma (x)\) and \(\sigma ^{-1}(x)\) are uniform bounded for all \(x\in \mathbb {R}^n\). After changing variables \(\sigma (x)y\rightarrow y\), we rewrite the generator A as

$$\begin{aligned} Au(x) =&b(x)\cdot \nabla u(x) +\int _{\mathbb {R}^{n}\setminus \{0\}}\left( u(x+y)-u(x)\right) \frac{\det \left( \sigma ^{-1}(x)\right) }{|\sigma ^{-1}(x)y|^{n+\alpha }}dy \nonumber \\ =&b(x)\cdot \nabla u(x)+\int _{\mathbb {R}^{n}\setminus \{0\}}(u(x+y)-u(x))k(x,y) \nu (dy), \end{aligned}$$
(3.4)

where the kernel function

$$\begin{aligned} k(x,y)=\frac{1}{\det (\sigma (x))}\left( \frac{|y|}{|\sigma ^{-1}(x)y|}\right) ^{n+\alpha }. \end{aligned}$$

By assumption \((\mathbf {B})\), k(xy) is a positive and bounded function on \(\mathbb {R}^n \times \mathbb {R}^n\) satisfying

$$\begin{aligned} 0<r_0 \le k(x,y) \le r_1, \quad k(x,y)=k(x,-y), \end{aligned}$$

and for all \(y,z\in \mathbb {R}^n\),

$$\begin{aligned} |k(x,y)-k(z,y)| \le r_2 |x-z|^{\beta }, \end{aligned}$$

where \(\beta \in (0,1)\) is same in assumption \((\mathbf {B})\).

Consider the generator A in the following form

$$\begin{aligned} Au(x) =&b(x)\cdot \nabla u(x) +\int _{\mathbb {R}^{n}\setminus \{0\}}\left( u(x+y)-u(x)\right) \frac{\det (\sigma ^{-1}(x))}{|\sigma ^{-1}(x)y|^{n+\alpha }}dy \nonumber \\ =&b(x)\cdot \nabla u(x)+\int _{\mathbb {R}^{n}\setminus \{0\}}(u(x+y)-u(x))k(x,y) \nu (dy), \end{aligned}$$
(3.5)

where

$$\begin{aligned} k(x,y)=\frac{1}{\det (\sigma (x))}\left( \frac{|y|}{|\sigma ^{-1}(x)y|}\right) ^{n+\alpha }. \end{aligned}$$

For every \(\varphi \in C_0^{\infty }(\mathbb {R}^n)\), by Fubini Theorem, we have

$$\begin{aligned} \langle \varphi ,Au \rangle _{L^2} =&\int _{\mathbb {R}^n} \varphi (x)(b(x))\cdot \nabla u(x) dx \nonumber \\&+ \int _{\mathbb {R}^{n}\setminus \{0\}}\int _{\mathbb {R}^n}\varphi (x-y) u(x)k(x-y,y)-\varphi (x)u(x)k(x,y) dx \nu (dy) \nonumber \\ =&-\int _{\mathbb {R}^n} u(x) \text {div}\left[ b(x)\varphi (x)\right] dx \nonumber \\&+ \int _{\mathbb {R}^n} u(x) \int _{\mathbb {R}^{n}\setminus \{0\}} k(x-y,y)\varphi (x-y)-k(x,y)\varphi (x) \nu (dy)dx. \end{aligned}$$
(3.6)

Since \(k(x,y)=k(x,-y)\), the adjoint operator \(A^{*}\) is given by

$$\begin{aligned} A^{*}\varphi (x) =&-\text {div} \left[ b(x)\varphi (x)\right] +\int _{\mathbb {R}^{n}\setminus \{0\}}\varphi (x-y)k(x-y,y)-\varphi (x)k(x,y) \nu (dy) \nonumber \\ =&- \text {div} \left[ b(x)\varphi (x)\right] +\int _{\mathbb {R}^{n}\setminus \{0\}}\left( k(x-y,y)-k(x,y)\right) \varphi (x-y)\nu (dy)\nonumber \\&+\int _{\mathbb {R}^{n}\setminus \{0\}}\left( \varphi (x-y)-\varphi (x)\right) k(x,y) \nu (dy) \nonumber \\ =&-\text {div} \left[ b(x)\varphi (x)\right] + \lim _{\epsilon \searrow 0}\int _{\mathbb {R}^n\setminus \{|x-y|>\epsilon \}}\frac{k(y,x-y)-k(x,x-y)}{|x-y|^{n+\alpha }}\varphi (y)dy \nonumber \\&+\int _{\mathbb {R}^{n}\setminus \{0\}}(\varphi (x+y)-\varphi (x))k(x,y) \nu (dy) \nonumber \\ =&A\varphi (x)-2 b(x)\cdot \nabla \varphi (x) -\text {div}(b(x))\varphi (x) + \mathcal {S}\varphi (x), \end{aligned}$$
(3.7)

where \(\mathcal {S}\) is a singular intergal operator defined by

$$\begin{aligned} \mathcal {S}\varphi (x)=\lim _{\epsilon \searrow 0}\int _{ \{|x-y|>\epsilon \}} \mathcal {K}(x,x-y)\varphi (y)dy, \end{aligned}$$
(3.8)

with the kernel

$$\begin{aligned} \mathcal {K}(x,y)=\frac{k(x-y,y)-k(x,y)}{|y|^{n+\alpha }}=\frac{\det (\sigma ^{-1}(x-y))}{|\sigma ^{-1}(x-y)y|^{n+\alpha }}-\frac{\det (\sigma ^{-1}(x))}{|\sigma ^{-1}(x)y|^{n+\alpha }}. \end{aligned}$$

Remark 3.1

Since \(k(x,-y)=k(x,y)\), we have

$$\begin{aligned} A^{*}\varphi (x) =&-\text {div} \left[ b(x)\varphi (x)\right] +\int _{\mathbb {R}^{n}\setminus \{0\}}\varphi (x+y)k(x+y,y)-\varphi (x)k(x,y) \nu (dy) . \end{aligned}$$

When \(n=1\), function \(k(x,y)=|\sigma (x)|^{\alpha }\). So

$$\begin{aligned} A^{*}\varphi (x)=-\text {div} \left[ b(x)\varphi (x)\right] - (-\Delta )^{\frac{\alpha }{2}}\left( |\sigma (x)|^{\alpha }\varphi (x)\right) . \end{aligned}$$

It is consistent with the form of Fokker-Planck equation which was given in [39]. The existence and uniqueness of weak solutions to Fokker-Planck equations when b and \(\sigma \) are Lipschitz in [39].

Regularity of Invariant Measure

The probability density of invariant measure \(p_{ss}(x)\) satisfies the nonlocal elliptic equation \(A^{*}p_{ss}(x)=0\) in weak sense:

$$\begin{aligned} \int _{\mathbb {R}^n} A\varphi (x)p_{ss}(x)dx=0, \quad \forall \varphi \in C^{\infty }_0\left( \mathbb {R}^n\right) . \end{aligned}$$

But in order to make sure that the conjugate variable which is given in next section is well-defined, we need higher regularity of \(p_{ss}\).

Lemma 3.4

Assume that \((\mathbf{A})\)-\((\mathbf{C})\) hold. Then

  1. (i)

    Singular integral operator \(\mathcal {S}\) is a bounded linear operator from \(L^p(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\), for all p with \(1<p<\infty \).

  2. (ii)

    \(\mathcal {S}^{*}=-\mathcal {S}\) in \(L^p(\mathbb {R}^n)\) with \(1<p<\infty \).

Proof

(i) By Calderón-Zygmund theory, it suffices to show that the kernel \(\mathcal {K}\) satisfies

$$\begin{aligned} |\nabla _y \mathcal {K}(x,y)| \le C |y|^{-n-1} \quad \text {for almost every } x\in \mathbb {R}^n, y\ne 0. \end{aligned}$$

From \((\mathbf {B})\) and \((\mathbf {C})\), for almost every \(x\in \mathbb {R}^n, y\in \mathbb {R}^{n}\setminus \{0\}\), we have

$$\begin{aligned} |\nabla _y \mathcal {K}(x,y)| \le&\frac{|\nabla _y\det \left( \sigma (x-y)\right) |}{|\det (\sigma (x))|^2|\sigma ^{-1}(x-y)y|^{n+\alpha }}\\&+(n+\alpha )\frac{|\det \left( \sigma ^{-1}(x-y)\right) |}{|\sigma ^{-1}(x-y)y|^{n+\alpha +1}}|\nabla _y \left( \sigma ^{-1}(x-y)y\right) | \\&+ \left( n+\alpha \right) \frac{|\det \left( \sigma ^{-1}(x)\right) |}{|\sigma ^{-1}(x)y|^{n+\alpha -1}}|\nabla _y (\sigma ^{-1}(x)y)|\\ \le&C |y|^{-n-1}. \end{aligned}$$

(ii) From (3.7), for each \(\varphi (x)\in C^{\infty }_0(\mathbb {R}^n)\), we get

$$\begin{aligned}&A^{**}\varphi (x)= A^{*}\varphi (x)- (2 b(x)\cdot \nabla +\text {div}(b(x))^{*}\varphi (x) + \mathcal {S}^{*}\varphi (x)\\&=A\varphi (x)+\mathcal {S}^{*}\varphi (x)+\mathcal {S}\varphi (x)=A\varphi (x) \end{aligned}$$

Thus \(\mathcal {S}^{*}\varphi (x)=-\mathcal {S}\varphi (x)\) for each \(\varphi (x)\in C^{\infty }_0(\mathbb {R}^n)\). Since \(L^p(\mathbb {R}^n)\) with \(1<p<\infty \) is a reflexive Banach space and \(C^{\infty }_0(\mathbb {R}^n)\) is dense in \(L^p(\mathbb {R}^n)\), \(\mathcal {S}^{*}=-\mathcal {S}\) in \(L^p(\mathbb {R}^n)\) with \(1<p<\infty \). The proof is complete. \(\square \)

Lemma 3.5

Assume that \((\mathbf{A})\)-\((\mathbf{C})\) hold. Then

  1. (i)

    The semigroup \(P^{*}_t\) associated with Fokker-Planck equation (2.5) can be extended to a strongly continuous semigroup on \(L^p(\mathbb {R}^n)\) for all \(1\le p \le \infty \) with \(\Vert P^{*}_t\Vert _{L(L^p)} \le C\) for some constant \(C>0\);

  2. (ii)

    The resolvent set \(\rho (A^{*})\supset (0,\infty )\), and \(\Vert (\lambda I-A^{*})^{-1}\Vert _{L(L^p)}\le \frac{C}{\lambda }\) for all \(\lambda >0\).

Proof

By theorem 2.1, for every \(f \in L^1(\mathbb {R}^n)\), let \(p_0=|f|/\Vert f\Vert _{L^1}\). Then \(p_0\) is a probability density, and the probability density at t for all \(t>0\) is given by

$$\begin{aligned} p_t(y)=\int _{\mathbb {R}^n}p_0(x)p(0,x;t,y)dx. \end{aligned}$$

Note that have \(P^{*}_t f = \Vert f\Vert _{L^1}p_t\). From the definition of transition probability, \(p_t \in L^1(\mathbb {R}^n)\) and

$$\begin{aligned} \Vert p_t\Vert _{L^1}=\Vert p_0\Vert _{L^1}, \quad t\ge 0. \end{aligned}$$

So for \(\forall t>0\), \(\Vert P^{*}_t f\Vert _{L^1}=\Vert f\Vert _{L^1}\), and \(P^{*}_t\) is a strongly continuous contraction semigroup on \(L^1(\mathbb {R}^n)\).

The two-side estimate in theorem 2.2 shows that for each \(f \in L^{\infty }(\mathbb {R}^n)\),

$$\begin{aligned} \Vert P^{*}_t f\Vert _{L^{\infty }}\le C t^{-\frac{n}{\alpha }}\Vert f\Vert _{L^{\infty }}, \quad \forall t>0. \end{aligned}$$

for some constant \(C>0\). Moreover, we have

$$\begin{aligned} \lim _{t\rightarrow 0}P^{*}_t f =f \quad \text {in} \ L^{\infty }\left( \mathbb {R}^n\right) . \end{aligned}$$

So \(P^{*}_t\) is a strongly continuous semigroup on \(L^{\infty }(\mathbb {R}^n)\) with \(\Vert P^{*}_t\Vert _{L(L^{\infty })}\le C\).

By interpolation inequality, for all \(f\in L^{1}\cap L^{\infty }\), \(1<p<\infty \)

$$\begin{aligned} \Vert P^{*}_t f\Vert _{L^p} \le Ct^{-\frac{n}{\alpha p'}}\Vert f\Vert _{L^p}, \quad \forall t>0, \end{aligned}$$

where \(\frac{1}{p}+\frac{1}{p'}=1\). So \(P^{*}_t\) is a strong strongly continuous semigroup on \(L^p\) with \(\Vert P^{*}_t\Vert _{L(L^p)}\le C\).

The Hille-Yosida-Phillips Theorem implies that the resolvent set \(\rho (A^{*})\supset (0,\infty )\). Moreover, \(\Vert (\lambda I-A^{*})^{-1}\Vert _{L(L^p)}\le \frac{C}{\lambda }\) for all \(\lambda >0\). \(\square \)

Now we consider the weighted \(L^1\) space

$$\begin{aligned} \mathbf {L}^1 = \left\{ f\in L^1 | \int _{\mathbb {R}^n}|x||f(x)| dx < \infty \right\} , \end{aligned}$$

which is equipped with the following norm

$$\begin{aligned} \Vert f\Vert _{\mathbf {L}^1} =\Vert |x|f\Vert _{L^1}= \int _{\mathbb {R}^n}|f(x)||x| dx. \end{aligned}$$

Let \(\mathcal {L}^p=L^p \cap \mathbf {L}^1\), \(1\le p \le \infty \) be a Banach space equipped with norm

$$\begin{aligned} \Vert f\Vert _{\mathcal {L}^p}=\Vert f\Vert _{\mathbf {L}^1}+\Vert f\Vert _{L^p}. \end{aligned}$$

We study the adjoint semigroup \(P^{*}_t\) on \( \mathcal {L}^p\) in the following Lemma.

Lemma 3.6

Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold. Then

  1. (i)

    The semigroup \(P^{*}_t\) is a strongly continuous semigroup on \(\mathcal {L}^p\) for all \(1\le p \le \infty \) with \(\Vert P^{*}_t\Vert _{L(\mathcal {L}^p)} \le C\), for some constant \(C\ge 1\).

  2. (ii)

    The resolvent set \(\rho (A^{*})\supset (0,\infty )\), and \(\Vert (\lambda I-A^{*})^{-1}\Vert _{L(\mathcal {L}^p)}\le \frac{C}{\lambda }\), for all \(\lambda >0\).

Proof

For each \(f\in \mathcal {L}^p\), we have \(|x|f\in L^1\). Let \(p_0 =|f|/\Vert f\Vert _{L^1}\), and \(X_0\) is a random variable on \((\Omega , \mathcal {F}, \mathbb {P})\) with density \(p_0\). Consider the SDE (1.1) with \(F(t)=0\). Then from Lemma 3.1, \(\mathbb {E}|X_t|<\infty \) for all \(t>0\). Thus

$$\begin{aligned} \Vert P^{*}_t f\Vert _{\mathbf {L}^1}=\Vert |x|\left( P^{*}_t f\right) \Vert _{L^1}\le C\Vert |x|f\Vert _{L^1}=C\Vert f\Vert _{\mathbf {L}^1}, \quad \forall t>0. \end{aligned}$$

Moreover, from Lemma 3.5, we have

$$\begin{aligned} \Vert P^{*}_t f\Vert _{L^p}\le C\Vert f\Vert _{L^p}, \quad \forall t>0. \end{aligned}$$

Thus the semigroup \(P^{*}_t\) is a strongly continuous semigroup on \(\mathcal {L}^p\) for all \(1\le p \le \infty \) and

$$\begin{aligned} \Vert P^{*}_t f\Vert _{\mathcal {L}^p} \le C\Vert f\Vert _{\mathcal {L}^p}, \quad \forall t>0. \end{aligned}$$

where \(C\ge 1\) is a constant. So (i) is proved. The Hille-Yosida-Phillips Theorem implies the result in (ii). \(\square \)

Lemma 3.7

Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold. Then

  1. (i)

    The density \(p_{ss}\) is positive, i.e. \(p_{ss}(x)>0\) for all \(x\in \mathbb {R}^n\);

  2. (ii)

    For each \(1\le p \le \infty \), the density of the invariant measure \(p_{ss}\in L^p(\mathbb {R}^n)\) and \(p_{ss}\in \mathcal {L}^p\).

Proof

Lemma 3.3 shows that the density of invariant measure \(p_{ss} \in L^{p'}(\mathbb {R}^n)\), with \(p'<\frac{n}{n-\alpha }\) for \(n\ge 2\) and \( p'<\infty \) for \(n=1\). In addition, from definition of invariant measure, the density of invariant measure \(p_{ss}\) satisfies

$$\begin{aligned} p_{ss}(y)=\int _{\mathbb {R}^n}p_{ss}(x)p(0,x;t,y)dx \end{aligned}$$

for all \(t>0\). The two-sides estimates implies that \(p(0,x;t,y)>0\). Thus we obtain that \(p_{ss}(x)>0\) for each \(x\in \mathbb {R}^n\), and \(p_{ss}(x)\in L^{\infty }(\mathbb {R}^n)\). Then the interpolation inequality implies that \(p_{ss} \in L^p(\mathbb {R}^n)\) for all \(1\le p\le \infty \). From Lemma 3.2, \(\int _{\mathbb {R}^n}|x|p_{ss}(x)dx < \infty \). Thus we have \(p_{ss} \in \mathcal {L}^p\) for all \(1\le p \le \infty \). \(\square \)

Now we consider the following nonlocal elliptic equation

$$\begin{aligned} A^{*}u(x)-\lambda u(x)=f(x), \quad x\in \mathbb {R}^n, \end{aligned}$$
(3.9)

where \(A^{*}\) is defined as (3.7), \(f\in L^p(\mathbb {R}^n)\) for some \(1< p< \infty \), and \(\lambda \ge 0\). The solvability and a prior estimate for nonlocal elliptic equations (3.9) are given as following.

Lemma 3.8

Assume that \((\mathbf{A})\)-\((\mathbf{C})\) hold. For some \(\lambda _1 \ge 1\) large enough and for all \(\mu \ge \lambda _1\), \(\vartheta \in [0,\alpha )\), and for \(f\in L^p(\mathbb {R}^n)\), there exists a unique solution \(u\in H^{\vartheta }_p(\mathbb {R}^n)\) to the following nonlocal elliptic equation:

$$\begin{aligned} Au-2b(x)\cdot \nabla u(x)-\mu u(x)=f(x), \quad x\in \mathbb {R}^n. \end{aligned}$$
(3.10)

Moreover, there is a positive constant N, independent of u, such that

$$\begin{aligned} \Vert u\Vert _{H^{\vartheta }_p} \le N\Vert f\Vert _{L^p}. \end{aligned}$$
(3.11)

Proof

If \(f\in C^{\infty }_c(\mathbb {R}^n)\), then we can obtain a unique smooth solution u for (3.10) by

$$\begin{aligned} u(x)=-\mathbb {E}_x \int ^{\infty }_{0}e^{-\mu t} f(Y_t)dt, \end{aligned}$$

where \((Y_t)_{t\ge 0}\) is the Markov process associated to the operator \(\hat{A}u=(A-2b(x)\cdot \nabla ) u\). Now we show the a priori estimate (3.11). Suppose \(u\in H^{\vartheta }_p(\mathbb {R}^n)\) satisfies (3.10). Let \(T>0\) and \(\phi (t)\) be a nonnegative and nonzero smooth function with support (0, T). Let \(\hat{u}(t,x)=\phi (t)u(x)\). Then

$$\begin{aligned} \partial _t \hat{u}= A\hat{u}-2b(x)\cdot \nabla \hat{u}-\mu \hat{u} +\left( u\phi '-f\phi \right) . \end{aligned}$$

By Lemma 2.2, we have

$$\begin{aligned} \Vert u\Vert _{H^{\vartheta }_p}\Vert \phi \Vert _{L^{\infty }}\le \frac{c}{\mu } \left( \Vert u\Vert _{L^p}\Vert \phi '\Vert _{L^{\infty }}+\Vert f\Vert _{L^p}\Vert \phi \Vert _{L^{\infty }}\right) . \end{aligned}$$

Letting \(\mu \) be large enough, we get the a priori estimate

$$\begin{aligned} \Vert u\Vert _{H^{\vartheta }_p} \le N\Vert f\Vert _{L^p}. \end{aligned}$$
(3.12)

Then by a density argument, we get the unique solution \(u\in H^{\vartheta }_p(\mathbb {R}^n)\) of (3.10). The result follows. \(\square \)

Now we consider the weak solution of nonlocal elliptic equation (3.9). A function \(u\in L^p\) is called weak solution to the nonlocal elliptic equation (3.9) if

$$\begin{aligned} \int _{\mathbb {R}^n}u(x)\left( A\varphi (x)+\lambda \varphi (x)\right) dx=\int _{\mathbb {R}^n}f(x)\varphi (x)dx, \quad \forall \varphi \in C^{\infty }_0(\mathbb {R}^n). \end{aligned}$$

We now state the following a priori estimate of weak solution.

Theorem 3.1

Assume that conditions \((\mathbf{A})\)-\((\mathbf{D})\) hold. Let \(f\in L^p(\mathbb {R}^n)\) with some \(1<p<\infty \), and let \(u\in L^p(\mathbb {R}^n)\) be a weak solution to the nonlocal elliptic equation (3.9). Then \(u \in H^{\vartheta }_{p}(\mathbb {R}^n)\) for all \(\vartheta \in [0,\alpha )\).

Proof

Since u is a weak solution to the nonlocal elliptic equation \((A^{*}-\lambda )u(x)=f(x)\), it satisfies

$$\begin{aligned} \int _{\mathbb {R}^n}u(x) A\varphi (x)-\lambda u(x) \varphi (x) dx = \int _{\mathbb {R}^n}f(x)\varphi (x) dx, \quad \forall \varphi \in C^{\infty }_0\left( \mathbb {R}^n\right) . \end{aligned}$$
(3.13)

From (3.7), we have

$$\begin{aligned} \left( A-2b(x)\nabla \right) ^{*}\varphi (x)=A\varphi (x)+\text {div}(b(x))\varphi (x) + \mathcal {S}\varphi (x) \end{aligned}$$

From Lemma 3.4, \(\mathcal {S}^{*}=-\mathcal {S}\) in \(L^p(\mathbb {R}^n)\) with \(1<p<\infty \). So (3.13) is equivalent to

$$\begin{aligned} \int _{\mathbb {R}^n}u(x) \left( A-2b(x)\nabla \right) ^{*}\varphi (x)-\mu u(x) \varphi (x) dx = \int _{\mathbb {R}^n}g(x)\varphi (x) dx, \quad \forall \varphi \in C^{\infty }_0\left( \mathbb {R}^n\right) , \end{aligned}$$

where \(\mu \ge \lambda _1\), \(g(x)=\text {div}(b(x))u(x)-\mathcal {S}u(x)-(\lambda +\mu ) u(x)+f(x)\).

The nonlocal elliptic equation \((A^{*}-\lambda )u=f\) can be rewritten as

$$\begin{aligned} Au(x)-2b(x)\cdot \nabla u(x)-\mu u(Rx)= g(x), \quad x \in \mathbb {R}^n, \end{aligned}$$

where \(\mu \ge \lambda _1\). From assumption (\(\mathbf{C}\)) and Lemma 3.4, \(F:u\rightarrow \text {div}(b(x))u(x)-\mathcal {S}u(x)\) is a linear bounded mapping from \(L^p(\mathbb {R}^n)\) to \(L^p(\mathbb {R}^n)\) for all \(1<p<\infty \). Thus \(g(x)\in L^p(\mathbb {R}^n)\) for all \(1<p<\infty \). Then by Lemma 3.8, there exists a unique solution \(\hat{u}\in H^{\vartheta }_p\) for each \(\vartheta \in [0,\alpha )\) to the following equation

$$\begin{aligned} \left( A -2b(x)\cdot \nabla \right) \hat{u}(x)-\mu \hat{u}(x)=g(x), \quad x\in \mathbb {R}^n, \end{aligned}$$
(3.14)

where \(g(x)=F(u)(x)-(\lambda +\mu ) u(x)+f(x) \in L^p(\mathbb {R}^n)\). Moreover, \(\hat{u}\) satisfies following identity

$$\begin{aligned} \int _{\mathbb {R}^n}\hat{u}(x) \left( A-2b(x)\nabla \right) ^{*}\varphi (x)-\mu \hat{u}(x) \varphi (x) dx = \int _{\mathbb {R}^n}g(x)\varphi (x) dx, \quad \forall \varphi \in C^{\infty }_0\left( \mathbb {R}^n\right) . \end{aligned}$$

Now we show that \(u=\hat{u}\). It is sufficient to show that the weak solution of equation (3.14) is unique. Since the operator \(\hat{A}:=A-2b(x)\nabla \) is also a generater of Markov semigroup \(\hat{P}_t\) which satisfies \(\mathbf {(A)}\)-\(\mathbf {(C)}\). Then by Lemma 3.5, for \(\forall \mu >\lambda _1\), the resolvent operator \(\mu I-\hat{A}^{*}\) is a bijective operator on \(L^p(\mathbb {R}^n)\) with \(1\le p \le \infty \). So the set

$$\begin{aligned} \left\{ \mu I\varphi -\left( A-2b(x)\nabla \right) ^{*}\varphi : \forall \varphi \in C^{\infty }_0\left( \mathbb {R}^n\right) \right\} \end{aligned}$$

is dense in \(L^p(\mathbb {R}^n)\) for all \(1\le p \le \infty \) and \(\mu >\lambda _1 \). It implies that the weak solution of (3.14) is unique, and \(\hat{u} =u\). Thus \(u \in H^{\vartheta }_p(\mathbb {R}^n)\) for all \(\vartheta \in [0,\alpha )\). \(\square \)

Now we have the following regularity result for the density of the invariant measure.

Theorem 3.2

Assume that conditions \((\mathbf{A})\)-\((\mathbf{D})\) hold. Then the unique invariant measure has a density \(p_{ss} \in H^{\vartheta }_{p}(\mathbb {R}^n)\) for all \(\vartheta \in [0,\alpha )\) and \(1< p < \infty \).

Proof

By the definition of the invariant measure, \(p_{ss}\) is the weak solution of nonlocal elliptic equation \(A^{*}p_{ss}=0\), i.e.

$$\begin{aligned} \int _{\mathbb {R}^n} p_{ss}(x) A \varphi (x) dx, \quad \forall \varphi \in C^{\infty }_0(\mathbb {R}^n). \end{aligned}$$

From Lemma 3.7, \(p_{ss} \in L^p(\mathbb {R}^n)\) for all \(1\le p \le \infty \), Then the regularity result in Theorem 3.1 implies that \(p_{ss} \in H^{\vartheta }_p(\mathbb {R}^n)\) for all \(1< p <\infty \) and \(\vartheta \in [0,\alpha )\). \(\square \)

Linear Response Theory

In this section, we derive the response function, and further establish the linear response theory and the Agarwal-type fluctuation-dissipation theorem for SDE (1.1).

The Response Function

In this subsection, we derive the response function. We assume that \((X_{t})_{t\ge 0}\) is a stationary Markov process, which satisfies the SDE (1.1), and its initial distribution is the unique invariant measure \(\mu \) of the corresponding Markov semigroup \(P_t\). Let \((X^{F}_t)_{t\ge 0}\) be the unique strong solution of the perturbed SDE (1.5), which is the perturbed process under perturbation F. We denote \((P_t^{ F})_{t\ge 0}\) the corresponding Markov semigroup of the perturbed process \((X_t^{F})_{t\ge 0}\). The generator of \((X^{ F})_{t\ge 0}\) is denoted by

$$\begin{aligned} A^{F}u(x)=Au(x)+F(t) K(x)\cdot \nabla u(x):=\left( A+F(t) L \right) u(x), \end{aligned}$$

where the external perturbation operator \(Lu=K(x)\cdot \nabla u\). Then the associated Fokker-Planck equation of the perturbed process \((X^{F}_t)_{t\ge 0}\) is

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t p_t^{ F}(x) = A^{*} p_t^{ F}(x)+ F(t) L^{*} p_t^{ F}(x),\\&p_0^{F}(x)=p_{ss}(x), \end{aligned} \right. \end{aligned}$$
(4.1)

where \( L^{*} u(x)=- \text {div}(K(x)u(x))\), and \(p^F_t\) is the probability density of \(X^F_t\).

Now we redefine the response function of an observable in (1.3) mathematically. This definition means that when the system close to the steady state, the change in the expectation value of every observable is linear with the small perturbing source. Therefore, same as in [8], we can understand the response function via the functional derivative of response \(\mathbb {E}O(X_t^{F})-\mathbb {E}O(X_{0})\) with respect to perturbation F.

Definition 4.1

Let \(O(x)\in L^p(\mathbb {R}^n)\) be an observable for some p with \(1< p \le \infty \). Let \((X_{t})_{t\ge 0}\) be a stationary Markov process. For every \(\phi \in L^{\infty }(\mathbb {R}^{+})\), let \((X^{\epsilon \phi }_t)_{t\ge 0}\) be the perturbed process under perturbation \(\epsilon \phi (t)K(x)\) with initial value \(X_0 \). Then a locally integrable function \(\mathcal {R}_{O}\) is called the response function of the observable O if it satisfies

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\frac{1}{\epsilon } \left( \mathbb {E}O\left( X_t^{\epsilon \phi }\right) -\mathbb {E}O\left( X_{0}\right) \right) = \int _0^t \mathcal {R}_{O}(t-s)\phi (s) ds. \end{aligned}$$

The following lemma shows the perturbation property of Markov semigroup \(P_t\).

Lemma 4.1

Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold. Then for each \(f\in L^{p}(\mathbb {R}^n)\) with \(1<p\le \infty \), and \(0\le s\le t \), we have

$$\begin{aligned} P^{\epsilon \phi }_{s,t} f(x)-P_{s,t} f(x)=\epsilon \int _s^t \phi (r) P_{r-s}\left( K(x)\cdot \nabla \left( P^{\epsilon \phi }_{r,t}f( x)\right) \right) dr. \end{aligned}$$

Moreover, for \(t>0\),

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\frac{1}{\epsilon }\left( P^{\epsilon \phi }_{0,t} f(x)-P_{t} f(x)\right) = \int _0^t \phi (s) P_{s}(K(x)\cdot \nabla \left( P_{t-s}f(x)\right) )ds, \quad \text {in} \ L^p\left( \mathbb {R}^n\right) . \end{aligned}$$

Proof

Denote \(P^{\epsilon \phi }_{s,t} f(x)=w_1(s,x)\), \(P_{s,t}f(x)=w_2(s,x)\), and \(w(s,x)=w_1(s,x)-w_2(s,x)\). Then \(w_1\) is the solution of the Kolmogorov backward equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s w_1(s,x) = -A^{\epsilon \phi }(s)w_1(s,x),\quad (s,x)\in [0,t]\times \mathbb {R}^n,\\&w_1(t,x)=f(x),\quad x\in \mathbb {R}^n, \end{aligned} \right. \end{aligned}$$
(4.2)

and \(w_2\) is the solution of the Kolmogorov backward equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s w_2(s,x) = -Aw_2(s,x),\quad (s,x)\in [0,t]\times \mathbb {R}^n,\\&w_2(t,x)=f(x),\quad x\in \mathbb {R}^n. \end{aligned} \right. \end{aligned}$$
(4.3)

Lemma 2.1 implies that these two equations have unique \(H^{\vartheta }_{p}(\mathbb {R}^n)\) solutions for all \(\vartheta \in (1,\alpha )\), such that

$$\begin{aligned} \Vert w_i(s,t)\Vert _{H^{\vartheta }_{p}} \lesssim (t-s)^{-\frac{\vartheta }{\alpha }}\Vert f\Vert _{L^p},\quad i=1,2. \end{aligned}$$

Then for some \(\beta \in (1,\alpha )\), by Sobolev embedding \(H^{\beta }_p \hookrightarrow H^{1}_{p}\),

$$\begin{aligned} \Vert w_i(s,t)\Vert _{H^{1}_{p}} \lesssim (t-s)^{-\frac{\beta }{\alpha }}\Vert f\Vert _{L^p},\quad i=1,2. \end{aligned}$$

Note that w(sx) satisfies following equation

$$\begin{aligned} \left\{ \begin{aligned}&\partial _s w(s,x) = -Aw(s,x)- \epsilon g^{\epsilon }(s,x), \quad (s,x)\in [0,t]\times \mathbb {R}^n,\\&w(t,x)=0, \quad x\in \mathbb {R}^n, \end{aligned} \right. \end{aligned}$$
(4.4)

where \(g^{\epsilon }(s,x):=\phi (s)Lw_1(s,x)=\phi (s)L(P^{\epsilon \phi }_{s,t}f(x))\). From Lemma 2.1, for \(0\le s \le t\), we have

$$\begin{aligned} \Vert g^{\epsilon }(s,x)\Vert _{L^{p}\left( \mathbb {R}^n\right) }\le \Vert \phi \Vert _{L^{\infty }[0,t]}\Vert \nabla w_1(s,x)\Vert _{L^{p}\left( \mathbb {R}^n\right) }\lesssim (t-s)^{-\frac{\beta }{\alpha }}\Vert \phi \Vert _{L^{\infty }[0,t]}\Vert f\Vert _{L^{p}\left( \mathbb {R}^n\right) }. \end{aligned}$$

For some \(\gamma \in [\frac{\alpha }{\beta },\alpha )\),

$$\begin{aligned} \Vert g^{\epsilon }\Vert _{L^{\gamma }\left( s,t;L^{p}\right) }\lesssim (t-s)^{\frac{1}{\gamma }-\frac{\beta }{\alpha }}\Vert \phi \Vert _{L^{\infty }[0,t]}\Vert f\Vert _{L^p\left( \mathbb {R}^n\right) }\le \Vert \phi \Vert _{L^{\infty }[0,t]}\Vert f\Vert _{L^p\left( \mathbb {R}^n\right) }. \end{aligned}$$

Then by Lemma 2.2, there is a unique solution w to (4.4), and it satisfies

$$\begin{aligned} w(s,x)=P^{\epsilon \phi }_{s,t} f(x)-P_{s,t} f(x)=\epsilon \int ^t_s P_{s,r}g^{\epsilon }(r,x)dr=\epsilon \int ^t_s P_{s,r}\phi (r)L\left( P^{\epsilon \phi }_{r,t}f(x)\right) dr, \end{aligned}$$

and

$$\begin{aligned} \Vert P^{\epsilon \phi }_{r,t} f-P_{r,t} f\Vert _{L^{\gamma }\left( s,t;H^1_{p}\right) }&= \Vert w\Vert _{L^{\gamma }\left( s,t;H^1_{p}\right) } \\&\lesssim \epsilon \Vert g^{\epsilon }\Vert _{L^{\gamma }\left( s,t;L^{p}\right) }\\&\lesssim \epsilon c\Vert \phi \Vert _{L^{\infty }[0,t]}\Vert f\Vert _{L^p}. \end{aligned}$$

So

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\Vert P^{\epsilon \phi }_{r,t} f-P_{r,t} f\Vert _{L^{\gamma }(0,t;H^1_{p})} = 0. \end{aligned}$$
(4.5)

This implies that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\Vert g^{\epsilon }(s,x)-\phi (s) L\left( P_{t-s}f(x)\right) \Vert _{L^1\left( 0,t;L^{p}\right) }=0 \end{aligned}$$

Thus for \(t>0\), we have

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\frac{1}{\epsilon }\left( P^{\epsilon \phi }_{0,t} f(x)-P_{t} f(x)\right)&= \lim _{\epsilon \rightarrow 0}\int _0^t g^{\epsilon }(s,x)ds \\&=\int _0^t \lim _{\epsilon \rightarrow 0}g^{\epsilon }(s,x)ds\\&=\int _0^t \phi (s) P_{s}L\left( P_{t-s}f(x)\right) ds \quad \text {in} \ L^p\left( \mathbb {R}^n\right) . \end{aligned}$$

This completes the proof. \(\square \)

Now we state the main results of this subsection.

Theorem 4.1

Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold. Suppose \(F(t)\in L^{\infty }(\mathbb {R}^{+})\), and \(K(x), \text {div}(K(x)) \in L^{\infty }(\mathbb {R}^n)\). Let \(O(x)\in L^p(\mathbb {R}^n)\) be an observable with \(1< p \le \infty \). Then the response function \(\mathcal {R}_{O}\) is given by

$$\begin{aligned} \mathcal {R}_{O}(t)=\int _{\mathbb {R}^n} L\left( P_{t}O(x)\right) p_{ss}(x)dx=\int _{\mathbb {R}^n} O(x) P^{*}_{t}\left( L^{*}p_{ss}(x)\right) dx. \end{aligned}$$

Proof

For every \(t>0\) and \( \phi \in C^{\infty }[0,t]\), we have

$$\begin{aligned} \mathbb {E}O\left( X^{\epsilon \phi }_t\right) = \int _{\mathbb {R}^n}\mathbb {E}_x O(X^{\epsilon \phi }_t)p_{ss}(x)dx=\mathbb {E}P^{\epsilon \phi }_{0,t} O(X_{0}), \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}O\left( X_t\right) = \int _{\mathbb {R}^n}\mathbb {E}_x O(X_t)p_{ss}(x)dx=\mathbb {E}P_{0,t} O\left( X_{0}\right) . \end{aligned}$$

Then it follows from Lemma 4.1 that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\frac{1}{\epsilon } \left( \mathbb {E}O\left( X_t^{\epsilon \phi }\right) -\mathbb {E}O\left( X_{0}\right) \right) =&\lim _{\epsilon \rightarrow 0}\mathbb {E}\frac{1}{\epsilon } \left( P^{\epsilon \phi }_{0,t} O\left( X_{0}\right) -P_{0,t} O\left( X_{0}\right) \right) \\ =&\int _0^t \phi (s) P_{s}L\left( P_{t-s}f\left( X_{0}\right) \right) ds\\ =&\int _0^t\int _{\mathbb {R}^n} \phi (s)P_{s}L\left( P_{t-s}O(x)\right) dsp_{ss}(x)dxds \\ =&\int _0^t \phi (s)\int _{\mathbb {R}^n} L\left( P_{t-s}O(x)\right) p_{ss}(x)dxds \end{aligned}$$

Thus

$$\begin{aligned} \left. \mathcal {R}_{O}(t) =\int _{\mathbb {R}^n} L\left( P_{t}O(x)\right) p_{ss}(x)dx =\int _{\mathbb {R}^n} O(x) P^{*}_{t}\left( - \text {div}\left( K(x)p_{ss}(x)\right) \right) \right) dx. \end{aligned}$$

The proof is complete. \(\square \)

The Conjugate Variable

In nonequilibrium statistical mechanics, the fluctuation-dissipation theorem reveals the response of an observable physical quantity to a small external perturbation, by the correlation function of this observable physical quantity and another observable physical quantity that is a conjugate variable to the perturbation with respect to energy.

First, we have the following Agarwal-type fluctuation dissipation theorem.

Theorem 4.2

(Agarwal-type fluctuation dissipation theorem) Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold, and \(F(t)\in L^{\infty }(\mathbb {R}^{+})\), \( K(x), div(K(x)) \in L^{\infty }(\mathbb {R}^n)\). Then for every observable \(O(x) \in L^p(\mathbb {R}^n)\) for some p with \(1\le p \le \infty \), there exists another observable Y defined as

$$\begin{aligned} Y(x) =\frac{- div\left( K(x)p_{ss}(x)\right) }{p_{ss}(x)}, \end{aligned}$$

such that

$$\begin{aligned} \mathcal {R}_{O}(t)=\mathbb {E}\left( O(X_t)Y\left( X_{0}\right) \right) . \end{aligned}$$

Proof

From Theorem 3.2 and Lemma 3.7, we see that the density \(p_{ss} >0\) and \(p_{ss} \in H^{\vartheta }_p(\mathbb {R}^n)\) for all \(\vartheta \in [0,\alpha )\) and \(1< p < \infty \). So the observable Y is well-defined.

Then the cross correlation function with the invariant measure satisfies

$$\begin{aligned} \mathbb {E}\left( O\left( X_t\right) Y\left( X_{0}\right) \right) =&\mathbb {E}\left( \left( P_t O\left( X_{0}\right) \right) Y\left( X_{ss}\right) \right) \\ =&\int _{\mathbb {R}^n}\left( P_t O(x)\right) Y(x)p_{ss}(x)dx\\ =&\int _{\mathbb {R}^n} O(x)P^{*}_t \left( Y(x)p_{ss}(x)\right) dx \\ =&\int _{\mathbb {R}^n} O(x) P^{*}_{t}\left( L^{*}p_{ss}(x)\right) dx \\ =&\mathcal {R}_{O}(t). \end{aligned}$$

The proof is complete. \(\square \)

Now we define the conjugate variable to the perturbation, and provide the linear response theory of SDE (1.1). Suppose there is a perturbation \(\epsilon K(x)\) applied on the drift term, and the perturbed process has a unique invariant measure with density \(p_{ss}^{\epsilon }\). From [36, 40], the conjugate variable U(x) is given by

$$\begin{aligned} U(x)=-\left. \frac{ \partial \log p_{ss}^{\epsilon }}{\partial \epsilon }\right| _{\epsilon =0} = \left. \frac{\Phi ^{\epsilon }(x)}{\partial \epsilon }\right| _{\epsilon =0}. \end{aligned}$$
(4.6)

In this definition, \(\Phi ^{\epsilon }(x)= -\log p_{ss}^{\epsilon }(x)\) stands for a nonequilibrium potential, or stochastic entropy [21, 36]. For one dimension constant coefficient stochastic dynamic system with \(\alpha \)-stable noises and time independent perturbation, the explicit formula of corresponding invariant measure are known, as in [14]. As specific and simpler examples, for Gaussian noise, Cauchy noise, and Lévy-Smirnoff noise, the explicit expressions for the conjugate variable are derived in [15].

In [8], the authors show that if the conjugate variable U(x) exists, then it has the following form

$$\begin{aligned} U(x)=\frac{v(x)}{p_{ss}(x)}, \end{aligned}$$

where v(x) is a solution of the following elliptic equation

$$\begin{aligned} A^{*}v(x)=L^{*}p_{ss}(x). \end{aligned}$$
(4.7)

Since our perturbation depends on time t, it is difficult to define the conjugate variable as in (4.6) directly. But motivated by above necessary condition, we can define the conjugate variable as \(U(x)=v(x)/p_{ss}(x)\), where v(x) is a solution of (4.7). Before defining the conjugate variable, we need to prove the existence of nonlocal elliptic equation (4.7). Our approach to investigate the solvability of (4.7) is the Fredholm alternative theorem. Before proving it, we first recall the following compactness result, which can be found in [32], Theorem XIII.67.

Definition 4.2

The space \(\mathcal {H}\) is defined by \(\mathcal {H}=\{u\in H^1(\mathbb {R}^n):\int _{\mathbb {R}^n}|x||u|^2dx < \infty \}\) with norm

$$\begin{aligned} \Vert u\Vert _{\mathcal {H}}= \Vert \nabla u\Vert _{L^2}+\Vert |x|u\Vert _{L^2}. \end{aligned}$$

Lemma 4.2

The Sobolev embedding \(\mathcal {H}(\mathbb {R}^n)\hookrightarrow L^p(\mathbb {R}^n)\) is compact for \(2 \le p < 2^{*}\), where

$$\begin{aligned} 2^{*}:=\left\{ \begin{aligned}&\frac{2n}{n-2},\quad 1\le 2 <n\\&\infty ,\quad 2\ge n \end{aligned} \right. \end{aligned}$$

is the Sobolev conjugate of 2.

Applying the above compactness result, we are now ready to establish the following solvability result.

Lemma 4.3

Assume that conditions \((\mathbf{A})\)-\((\mathbf{D})\) hold. Suppose \(f \in \mathcal {L}^p\) for some \(2\vee (\frac{n}{\alpha }+1)<p< \infty \), and it satisfies \(\int _{\mathbb {R}^n}f(x)dx\ne 0\). Then there exists a nonzero solution \(u \in H^{\vartheta }_p\) for all \(\vartheta \in [0,\alpha )\) to the equation

$$\begin{aligned} A^{*}u(x)=f(x), \quad x\in \mathbb {R}^n. \end{aligned}$$
(4.8)

And there exists a nonzero solution \(u_{ss} \in H^{\vartheta }_p\) for all \(\vartheta \in [0,\alpha )\) and all \(2\vee (\frac{n}{\alpha }+1)<p< \infty \) to the equation

$$\begin{aligned} A^{*}u_{ss}(x)=0, \quad x\in \mathbb {R}^n. \end{aligned}$$
(4.9)

Moreover, if \(u_1\), \(u_2\) are both nonzero solutions of the equation (4.8), then \(u_1 -u_2 =cu_{ss}\), where c is a constant.

Proof

From Lemma 3.6, the inverse operator \(T=(I-A^{*})^{-1}\) on \(\mathcal {L}^p\) exists with \(1\le p \le \infty \). Moreover, Theorem 3.1 implies that T is a linear bounded operator from \(\mathcal {L}^p\) to \(H^{\vartheta }_p\cap \mathcal {L}^p \) for all \(\vartheta \in [0,\alpha )\).

Then the nonlocal elliptic equation \(A^{*}u =f\) can be rewritten as

$$\begin{aligned} (I-T)u=h, \end{aligned}$$
(4.10)

where \(h:=-Tf\in \mathcal {L}^p\) with \(2\vee (\frac{n}{\alpha }+1)<p< \infty \). We now claim that \(T:\mathcal {L}^p\rightarrow \mathcal {L}^p\) is a bounded, linear, compact operator. By Sobolev embedding, \(H^{\vartheta }_p(\mathbb {R}^n) \hookrightarrow H^1(\mathbb {R}^n)\) for some \(\vartheta \in (1,\alpha )\). Moreover, since \(p >\frac{n}{\alpha }+1\), we have \(H^{\vartheta }_p(\mathbb {R}^n) \hookrightarrow L^{\infty }(\mathbb {R}^n)\) for some \(\vartheta \in (\frac{n\alpha }{n+\alpha },\alpha )\). Then the Hölder inequality implies that

$$\begin{aligned} \int _{\mathbb {R}^n}|x||u(x)|^2dx \le \Vert u\Vert _{L^{\infty }}\Vert |x|u\Vert _{L^1}\le C\Vert u\Vert _{\mathcal {L}^p}\Vert u\Vert _{H^{\vartheta }_p}. \end{aligned}$$

Thus \((H^{\vartheta }_p \cap \mathcal {L}^p ) \subset \mathcal {H}\) with \(\vartheta \in (1\vee \frac{n\alpha }{n+\alpha },\alpha )\), and

$$\begin{aligned} \Vert u\Vert _{\mathcal {H}} \le C(\Vert u\Vert _{\mathcal {L}^p}+\Vert u\Vert _{H^{\vartheta }_p}). \end{aligned}$$

From Lemma 4.2, \(T:\mathcal {L}^p\rightarrow \mathcal {L}^p\) is a compact operator. So Fredholm alternative holds for the equation \((I-T)u=h\).

From Lemma 3.7, the density of the unique invariant measure \(p_{ss}\in \mathcal {L}^p\) with \(1\le p \le \infty \).. It implies that the equation \((I-T)u=0\) has a nonzero solution \(p_{ss} \in \mathcal {L}^p\). Then by Fredholm alternative theorem, \(dim N(I-T)=dim N(I-T^{*})>0\). So when \(f=0\), then \(h=-Tf=0\) and there exists a nonzero solution \(u\in N(I-T)\). Furthermore, for every \(f \in \mathcal {L}^p\) with \(f\ne 0\), the equation \(A^{*}u=f\) has a solution \(u_{ss} \in H^{\vartheta }_p\cap \mathcal {L}^p \) if and only if \((f,v)=\int _{\mathbb {R}^n}f(x)v(x) dx=0\) for all \(v\in N(I-T^{*})\), where \(T^{*}\) is a bounded linear operator from \((\mathcal {L}^p)^{*}\) to \((\mathcal {L}^p)^{*}\).

Now we describe the subspace \(N(I-T^{*})\). By definition of \(\mathcal {L}^p\), the smooth bounded function space \(C^{\infty }_b(\mathbb {R}^n)\) is a dense subspace of \((\mathcal {L}^p)^{*}\). If \(w\in C^{\infty }_b(\mathbb {R}^n)\) satisfies equation \((I-T^{*})w=0\), then \(Aw=0\). By maximum principle of nonlocal elliptic operator A, if \(w\in C^{\infty }_b(\mathbb {R}^n)\) is a solution to the equation \(Aw=0\), then the solution w(x) is a constant. Note that for every constant c, \(w(x)=c\) is a solution of \(Aw=0\), and all constant functions constitute a 1-dimension linear subspace of \(C^{\infty }_b(\mathbb {R}^n)\). We obtain that \(N(I-T^{*})=\{f=c | c\in \mathbb {R} \ \text {is a constant}\}\), and the above condition \((f,v)=0\) for all \(v\in N(I-T^{*})\) holds if and only if \(\int _{\mathbb {R}^n}f(x)dx\ne 0 \). Moreover, \(dim N(I-T)=dim N(I-T^{*})=1\), thus \(N(I-T)=\text {span}\{u_{ss}\}\). So if \(u_1\), \(u_2\) are both nonzero solutions of above equation, then \(u_1 -u_2 =cu_{ss}\), where c is a constant. This completes the proof. \(\square \)

Now we define the conjugate variable of perturbation F(t)K(x) as follows.

Definition 4.3

The conjugate variable U(x) of perturbation F(t)K(x) is defined by

$$\begin{aligned} U(x)=\frac{v(x)}{p_{ss}(x)}, \end{aligned}$$

where v(x) is a solution of following nonlocal elliptic equation

$$\begin{aligned} A^{*}v(x)=L^{*}p_{ss}(x). \end{aligned}$$

Here \(A^{*}\) is given by (3.7), and \( L^{*} p_{ss}(x)=- div (K(x)p_{ss}(x))\)

Remark 4.1

Different with the one dimension constant coefficient case in [15], for general SDE (1.1) it is hard to know the explicit formula of the invariant measure and analytical solution of nonlocal elliptic equation (4.7), and we need to consider the numerical solution to obtain the conjugate variable U(x). This question is nontrivial in computational partial differential equations, because it involves the nonlocal operator, as in [16, 41].

Now we state the linear response theory for SDE (1.1).

Theorem 4.3

(Linear response theory) Assume that \((\mathbf{A})\)-\((\mathbf{D})\) hold, \(F(t)\in L^{\infty }(\mathbb {R}^{+})\), \(\text {div}(K(x)) \in L^{\infty }(\mathbb {R}^n)\), \(|x|K(x) \in L^{\infty }(\mathbb {R}^n)\), and \(\int _{\mathbb {R}^n}\text {div}(K(x)p_{ss}(x)) dx \ne 0\). Then for every observable \(O(x) \in L^{p}(\mathbb {R}^n)\) for some p with \(1<p\le \infty \), there exists a conjugate variable U defined as in Definition (4.3), such that

$$\begin{aligned} \mathcal {R}_{O}(t)=\frac{d}{dt}\mathbb {E}\left( O\left( X_t\right) U\left( X_{0}\right) \right) . \end{aligned}$$

Proof

By theorem 3.2, \(p_{ss} \in H^{\vartheta }_p(\mathbb {R}^n)\) for all \(1<p<\infty \) and \(\vartheta \in [0,\alpha )\). By \(K(x)\in H^1_{\infty }(\mathbb {R}^n)\) and Sobolev embedding, \(L^{*}p_{ss}=-\text {div}(K(x)p_{ss}(x))\in L^p(\mathbb {R}^n)\) with \(1\le p \le \infty \). Moreover, by Lemma 3.7, \(p_{ss} \in \mathcal {L}^p\) with \(1\le p \le \infty \). Combing with \(|x|K(x)\in L^{\infty }(\mathbb {R}^n)\), we have \(L^{*}p_{ss} \in \mathcal {L}^p\) for all \(1<p<\infty \). Then Lemma 4.2 implies that there exists a solution v(x) of equation \(A^{*}v(x)=L^{*}p_{ss}(x)\). Since \(p_{ss}(x)>0\) for all \(x\in \mathbb {R}^n\), the conjugate variable \(U(x)=v(x)/p_{ss}(x)\) exists.

From the definition of the conjugate variable U(x), we have

$$\begin{aligned} \mathbb {E}\left( O\left( X_t\right) U\left( X_{0}\right) \right) =\int _{\mathbb {R}^n}O(x)P^{*}_t v(x)dx. \end{aligned}$$

From Lemma 3.5, \(P^{*}_t\) is a strongly continuous semigroup on \(L^p(\mathbb {R}^n)\) for all p with \(1\le p \le \infty \). Then by the dominated convergence theorem and definition of generator A, we have

$$\begin{aligned} \frac{d}{dt}\mathbb {E}\left( O\left( X_t\right) U\left( X_{0}\right) \right)&= \frac{d}{dt}\int _{\mathbb {R}^n}\left( P_t O(x)\right) U(x)p_{ss}(x)dx\\&=\int _{\mathbb {R}^n} O(x)\frac{d}{dt}P^{*}_t (v(x))dx\\&=\int _{\mathbb {R}^n}O(x)P^{*}_t A^{*}v(x)dx \\&=\int _{\mathbb {R}^n}O(x)P^{*}_t \left( L^{*}p_{ss}(x)\right) dx \\&= \mathcal {R}_{O}(t). \end{aligned}$$

This completes the proof. \(\square \)

Remark 4.2

The linear response theory for SDE (1.1) is also called Seifert-Speck type fluctuation-dissipation theorem [40].

Conclusion

We have established a linear response theory for a class of nonlinear stochastic differential equations driven by an \(\alpha \)-stable Lévy process (\(1<\alpha <2\)) and under a perturbation F(t)K(x) on the drift term. In addition, we have developed the Agarwal-type fluctuation-dissipation theorem for these stochastic dynamical systems. Our results show the susceptibility of every observable under an small time-dependent perturbation when the system is close to a steady state. Furthermore, we have also proved a new ergodicity result by the Bogoliubov-Krylov argument, and derived the response function by investigating the perturbation property of the corresponding Markov semigroup \(P_t\). We additionally have shown the existence and regularity for the stationary Fokker-Planck equations by the a priori estimate.

There are still some limitations of our results. We only consider the symmetric \(\alpha \)-stable Lévy noise in this paper. In the non-symmetric case, the generator is more complicated and thus the related partial differential equations in Sect. 3 and their a priori estimates will be technically more involved. However, since the noise considered in this paper is multiplicative, some of our techniques will be helpful for the non-symmetric case, and a version of the linear response theory could be established. Due to the requirement of solvability and regularity of Kolmogorov backward equations for the corresponding SDE, we restrict \(1<\alpha <2\) and require uniform ellipticity assumption in this paper. Moreover, we need the uniformly ellipticity assumption in the current proof of existence of invariant measure. This condition maybe relaxed. For the same reason, we only consider bounded drift term b(x). In order to prove the existence of conjugate variable, it is important to assume that |x|K(x) is bounded. We also ask that the perturbation can be written in the form F(t)K(x), and the perturbation only applied on the drift term. These issues will be the subject of future work.

References

  1. 1.

    Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  2. 2.

    Albeverio, S., Di Persio, L., Mastrogiacomo, E., Smii, B.: A Class of Lévy Driven SDEs and their Explicit Invariant Measures. Potential Anal. bf 45, 229–259 (2016)

    MATH  Google Scholar 

  3. 3.

    Arapostathis, A., Pang, G., Sandrić, N.: Ergodicity of a Lévy-driven SDE arising from multiclass many-server queues. Ann. Appl. Probab. 29, 1070–1126 (2019)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Albeverrio, S., Rüdiger, B., Wu, J.L.: Invariant measures and symmetry property of Lévy type operators. Potential Anal. 13, 147–168 (2000)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Baladi, V.: Linear response despite critical points. Nonlinearity 21, 81–90 (2008)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Bergmann, P.G., Lebowitz, J.L.: New approach to nonequlibrium processes. Phys. Rev. 99, 578 (1955)

    MathSciNet  MATH  ADS  Google Scholar 

  7. 7.

    Chetrite, R., Gawedzki, K.: Fluctuation relations for diffusion processes. Commun. Math. Phys. 282, 469–518 (2008)

    MathSciNet  MATH  ADS  Google Scholar 

  8. 8.

    Chen, X., Jia, C.: Mathematical foundation of nonequilibrium fluctuation-dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients. Stoch. Process Appl. 130, 171–202 (2020)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Chen, Z., Zhang, X.: Heat kernels for time-dependent non-symmetric stable-like operators. J. Math. Anal. Appl. 465, 1–21 (2018)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Duan, J.: An Introduction to Stochastic Dynamics. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  11. 11.

    Da Prato, G.: An Introduction to Infinite-Dimensional Analysis. Springer, Berlin (2006)

    Google Scholar 

  12. 12.

    Dembo, A., Deuschel, J.D.: Markovian perturbation, response and fluctuation dissipation theorem. Ann. I. H. Poincare 46, 822–852 (2010)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Colangeli, M., Lucarini, V.: Elements of a unified framework for response formulae. J. Stat. Mech. P01002, (2014)

  14. 14.

    Dybiec, B., Gudowska-Nowak, E., Sokolov, I.M.: Stationary states in Langevin dynamics under asymmetric Lévy noises. Phys. Rev. E 76, 041122 (2007)

    ADS  Google Scholar 

  15. 15.

    Dybiec, B., Parrondo, J.M.R., Gudowska-Nowak, E.: Fluctuation-dissipation relations under Lévy noises. EPL 98, 50006 (2012)

    ADS  Google Scholar 

  16. 16.

    Gao, T., Duan, J., Li, X., Song, R.: Mean exit time and escape probability for dynamical systems driven by Lévy noise. SIAM J. Sci. Comput. 36, 887–906 (2014)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Gritsun, A., Lucarini, V.: Fluctuations, response, and resonances in a simple atmospheric model. Physica D 349, 62–76 (2017)

    MathSciNet  MATH  ADS  Google Scholar 

  18. 18.

    Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergodic Theory Dyn. Syst. 26, 189–217 (2006)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Gouëzel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differ. Geom. 79, 433–477 (2008)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Hairer, M., Majda, A.J.: A simple framework to justify linear response theory. Nonlinearity 23(4), 909–922 (2010)

    MathSciNet  MATH  ADS  Google Scholar 

  21. 21.

    Hatano, T., Sasa, S.: Steady-State Thermodynamics of Langevin Systems. Phys. Rev. Lett. 86, 3463–3466 (2001)

    ADS  Google Scholar 

  22. 22.

    Lucarini, V.: Revising and extending the linear response theory for statistical mechanical systems: evaluating observables as predictors and predictands. J. Stat. Phys. 173, 1698–1721 (2018)

    MathSciNet  MATH  ADS  Google Scholar 

  23. 23.

    Lucarini, V., Colangeli, M.: Beyond the linear fluctuation-dissipation theorem: the role of causality. J. Stat. Mech. P05013, (2012)

  24. 24.

    Lisowski, B., Valenti, D., Spagnolo, B., Bier, M., Gudowska-Nowak, E.: Stepping molecular motor amid Lévy white noise. Phys. Rev. E 91, 042713 (2015)

    ADS  Google Scholar 

  25. 25.

    Mandelbrot, B.: The variation of certain speculative prices. J. Bus. 36, 394–419 (1963)

    Google Scholar 

  26. 26.

    Mandelbrot, B.: Fractals: Form. Chance and Dimension, Freeman, San Francisco (1977)

    Google Scholar 

  27. 27.

    Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255 (1966)

    MATH  ADS  Google Scholar 

  28. 28.

    Kusmierz, L., Dybiec, B., Gudowska-Nowak, E.: Thermodynamics of Superdiffusion Generated by Lévy-Wiener Fluctuating Forces. Entropy. 20, 658 (2018)

    ADS  Google Scholar 

  29. 29.

    Kusmierz, L., Ebeling, W., Sokolov, I.M., Gudowska-Nowak, E.: Onsagers fluctuation theory and new developments including non-equlibrium Lévy fluctuations. Acta Phys. Pol. B 44, 859–80 (2013)

    MATH  ADS  Google Scholar 

  30. 30.

    Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications. Wiley, New York (2008)

    Google Scholar 

  31. 31.

    Pavliotis, G.A.: Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations. Springer, New York (2016)

    Google Scholar 

  32. 32.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York-London, IV. Analysis of Operators (1978)

  33. 33.

    Ruelle, D.: General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A 245, 220–224 (1998)

    MathSciNet  MATH  ADS  Google Scholar 

  34. 34.

    Ruelle, D.: Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95(1), 393–468 (1999)

    MathSciNet  MATH  ADS  Google Scholar 

  35. 35.

    Ruelle, D.: A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22, 855–870 (2009)

    MathSciNet  MATH  ADS  Google Scholar 

  36. 36.

    Prost, J., Joanny, J.-F., Parrondo, J.M.R.: Generalized Fluctuation-Dissipation Theorem for Steady-State Systems. Phys. Rev. Lett. 103, 090601 (2009)

    ADS  Google Scholar 

  37. 37.

    Sánchez, R., Newman, D.E., Leboeuf, J.-N., Decyk, V.K., Carreras, B.A.: Nature of transport across sheared zonal flows in electrostatic ion-temperature-gradient gyrokinetic plasma turbulence. Phys. Rev. Lett. 101, 205002 (2008)

    ADS  Google Scholar 

  38. 38.

    Schötz, T., Neher, R.A., Gerland, U.: Target search on a dynamic DNA molecule. Phys. Rev. E 84, 051911 (2011)

    ADS  Google Scholar 

  39. 39.

    Schertzer, D., Larcheveque, M., Duan, J., Yanovsky, V.V., Lovejoy, S.: Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises. J. Math. Phys 42, 200–212 (2001)

    MathSciNet  MATH  ADS  Google Scholar 

  40. 40.

    Seifert, U., Speck, T.: Fluctuation-dissipation theorem in nonequilibrium steady states. Eur. Phys. Lett. 89, 10007 (2010)

    ADS  Google Scholar 

  41. 41.

    Wang, X., Duan, J., Li, X., Luan, Y.: Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises. Appl. Math. Comput. 258, 282–295 (2015)

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Wang, J.: Exponential ergodicity and strong ergodicity for SDEs driven by symmetric \(\alpha \)-stable processes. Appl. Math. Lett 26, 654–658 (2013)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Wormell, C.L., Gottwald, G.A.: Linear response for macroscopic observables in high dimensional systems. Chaos 29, 113127 (2019)

    MathSciNet  MATH  ADS  Google Scholar 

  44. 44.

    Xie, L., X, Zhang, X.: Ergodicity of stochastic differential equations with jumps and singular coefficients, Ann. Inst. H. Poincaré Probab. Statist. 56, 175-229(2020)

  45. 45.

    Zwanzig, R.: Nonequilibrium Statistical Mechanics. Oxford University Press, Oxford (2001)

    Google Scholar 

  46. 46.

    Zhang, X.J., Qian, H., Qian, M.: Stochastic theory of nonequilibrium steady states and its applications. Part I. Phys. Rep. 510, 1–86 (2012)

    MathSciNet  ADS  Google Scholar 

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Zhang, Q., Duan, J. Linear Response Theory for Nonlinear Stochastic Differential Equations with \(\alpha \)-Stable Lévy Noises. J Stat Phys 182, 32 (2021). https://doi.org/10.1007/s10955-021-02714-4

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Keywords

  • Linear response theory
  • Invariant measure
  • Fokker-Planck equations
  • \(\alpha \)-stable Lévy process