In this paper, we investigate ergodicity in total variation of the process $X_t$, related to a Lévy-driven stochastic differential equation with unbounded coefficients, and describe the speed of convergence to the respective invariant measure. Some examples are provided.
In this paper we show irreducibility and the strong Feller property for transition probabilities of stochastic differential equations with jumps and monotone coefficients. Thus, exponential ergodicity and the spectral gap for the corresponding transition semigroups are obtained.
We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to $α$-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the $L^1$-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived f…
We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results th…
We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H > 1/2 have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander's condition. We show that such systems satisfy a suitable version of the strong Feller property and we conclude that they admit a unique stationary solution that is physical in the sense that it does not "look into the future". The main technical result required for the analysis is a bound on the moments of the inverse of…
The mild sufficient conditions for exponential ergodicity of a Markov process, defined as the solution to SDE with a jump noise, are given. These conditions include three principal claims: recurrence condition R, topological irreducibility condition S and non-degeneracy condition N, the latter formulated in the terms of a certain random subspace of \Re^m, associated with the initial equation. The examples are given, showing that, in general, none of three principal claims can be removed without losing ergodicity of the process. The key point in the approach, developed in the paper, is that the…
First, we establish an abstract ergodic result on $\mR^d$. Classical ergodic results on $\mR^d$ require that the process is irreducible, we weaken it to some weak form of irreducibility in this article. The main method used in this article is coupling. Then, we apply our abstract ergodic result to stochastic differential equations driven by a Lévy noise and obtain a new result.
Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this pape…
We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+σ(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $σ: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal…
We show the strong well-posedness of SDEs driven by general multiplicative Lévy noises with Sobolev diffusion and jump coefficients and integrable drift. Moreover, we also study the strong Feller property, irreducibility as well as the exponential ergodicity of the corresponding semigroup when the coefficients are time-independent and singular dissipative. In particular, the large jump is allowed in the equation. To achieve our main results, we present a general approach for treating the SDEs with jumps and singular coefficients so that one just needs to focus on Krylov's {\it apriori} estimat…
In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure μ on the level set of a smooth function $ξ: \mathbb{R}^d\rightarrow \mathbb{R}^k$, $1\le k < d$. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose inva…
Let $α\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $α$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+σ(X_{t-})d L^α_t, \quad X_0=x\in\mathbb{R}^d, $$ where $b:\mathbb{R}^d\to\mathbb{R}^d$ and $σ:\mathbb{R}^d\to\mathbb{R}^d\otimes\mathbb{R}^d$ are locally $γ$-Hölder continuous with $γ\in(0\vee(1-α)^+,1]$, $L^α_t$ is a $d$-dimensional rotationally invariant $α$-stable process. Under some dissipative and non-degenerate assumptions on $b,σ$, we show the $V$-uniformly exponential ergodicity for the semigroup $P_t$ associated with $(X_t…
We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $α$-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin-Whitt regime that have heavy-tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound …
We study the large time behavior of solutions to fully nonlinear parabolic equations of Hamilton-Jacobi-Bellman type arising typically in stochastic control theory with control both on drift and diffusion coefficients. We prove that, as time horizon goes to infinity, the long run average solution is characterized by a nonlinear ergodic equation. Our results hold under dissipativity conditions, and without any nondegeneracy assumption on the diffusion term. Our approach uses mainly probabilistic arguments relying on new backward SDE representation for nonlinear parabolic, elliptic and ergodic e…
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of th…
We study the ergodic properties of finite-dimensional systems of SDEs driven by non-degenerate additive fractional Brownian motion with arbitrary Hurst parameter $H\in(0,1)$. A general framework is constructed to make precise the notions of ``invariant measure'' and ``stationary state'' for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solution towards the stationary one is (at least) algebraic. A lower bound on the exponent is also given.
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
The present paper is devoted to the study of the asymptotic behavior of the value functions of both finite and infinite horizon stochastic control problems and to the investigation of their relation with suitable stochastic ergodic control problems. Our methodology is based only on probabilistic techniques, as for instance the so-called randomization of the control method, thus avoiding completely analytical tools from the theory of viscosity solutions. We are then able to treat with the case where the state process takes values in a general (possibly infinite dimensional) real separable Hilbe…
Using elliptic and parabolic regularity results in $L^p$-spaces and generalized Dirichlet form theory, we construct for every starting point weak solutions to SDEs in $\mathbb{R}^d$ up to their explosion times including the following conditions. For arbitrary but fixed $p>d$ the diffusion coefficient $A=(a_{ij})_{1\le i,j\le d}$ is locally uniformly strictly elliptic with functions $a_{ij}\in H^{1,p}_{loc}(\mathbb{R}^d)$ and the drift coefficient $\mathbf{G}=(g_1,\dots, g_d)$ consists of functions $g_i\in L^p_{loc}(\mathbb{R}^d)$. The solution originates by construction from a Hunt process wit…
We consider a sequence $(ξ_n)_{n\ge1}$ of $i.i.d.$ random values living in the domain of attraction of an extreme value distribution. For such sequence, there exists $(a_n)$ and $(b_n)$, with $a_n>0$ and $b_n\in\ER$ for every $n\ge 1$, such that the sequence $(X_n)$ defined by $X_n=(\max(ξ_1,...,ξ_n)-b_n)/a_n$ converges in distribution to a non degenerated distribution. In this paper, we show that $(X_n)$ can be viewed as an Euler scheme with decreasing step of an ergodic Markov process solution to a SDE with jumps and we derive a functional limit theorem for the sequence $(X_n)$ from some met…
We formulate and prove a {\it Local Stable Manifold Theorem\/} for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized usi…
We show the complete proof of the Markov property of the strong solution to a multidimensional SDE whose coefficients involve local time on a hyperplane of the unknown process.
Tamed Euler approximation for fully superlinear growth McKean-Vlasov SDE and their particle systems: sharp rates for strong propagation of chaos, convergence and ergodicity
In this paper we study the problems of invariant and ergodic measures under G-expectation framework. In particular, the stochastic differential equations driven by G-Brownian motion have the unique invariant and ergodic measures. Moreover, the invariant and ergodic measures of G-SDEs are also sublinear expectations. However, the invariant measures may not coincide with ergodic measures, which is different from the classical case.
We present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed including diffusion processes on Riemannian manifolds and Ornstein-Uhlenbeck processes driven by symmetric $α$-stable processes. For SDE driven by $α$-stable process ($α\in (0,2]$) with polynomial drift, the strong ergodicity or not is independent on $α$.
Here we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and non-smooth domains.
We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob--Khas'minskii theorem. The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition o…
We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter $H\in(\frac{1}{3},\frac{1}{2}]$. Our contribution in this work is twofold. First, when the driving vector fields satisfy Hörmander's celebrated "Lie bracket condition," we derive explicit quantitative bounds on the inverse of the Malliavin matrix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential eq…
We study the rate of convergence of some recursive procedures based on some "exact" or "approximate" Euler schemes which converge to the invariant measure of an ergodic SDE driven by a Lévy process. The main interest of this work is to compare the rates induced by exact and approximate Euler schemes. In our main result, we show that replacing the small jumps by a Brownian component in the approximate case preserves the rate induced by the exact Euler scheme for a large class of Lévy processes.
In this note, we study the viability of a bounded open domain in $\mathbb{R}% ^{n}$ for a process driven by a path-dependent stochastic differential equation with Lipschitz data. We extend an invariant result of Cannarsa, Da. Prato and Frankowska [\textit{Indiana Univ. Math. J.} \textbf{59} (2010) 53-78] to a non-Markovian setting.
We study functional stochastic differential equations with a locally unbounded, functional drift focusing on well-posedness, stability and the strong Feller property. Following the non-functional case, we only consider integrability conditions and avoid continuity assumptions as far as possible. Our approach is mainly based on Zvonkin's transformation [18], an extended version of the probabilistic approach of Maslowski and Seidler [12] and the convergence concept for random variables in topological spaces in [2].
In this paper, we derive a characterization theorem for the path-independent property of the density of the Girsanov transformation for {\it degenerated} stochastic differential equations (SDEs), extending the characterization theorem of \cite{twwy} for the non-degenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalizes the corresponding characterization theorem established in \cite{hqwu}.
By Girsanov's thoerem and using the existing log-Harnack inequality for distribution independent SDEs, the log-Harnack inequality is derived for path-distribution dependent stochastic Hamiltonian systems. As an application, the exponential ergodicity in relative entropy is obtained by combining with transportation cost inequality. In addition, the quantitative propagation of chaos in the sense of Wasserstein distance, which together with the coupling by change of measure implies the quantitative propagation of chaos in total variation norm as well as relative entropy are obtained.
In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and Röckner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.
We consider the stochastic differential equation $dX_t = A(X_{t-}) \, dZ_t$, $ X_0 = x$, driven by cylindrical $α$-stable process $Z_t$ in $R^d$, where $α\in (0,1)$ and $d \ge 2$. We assume that the determinant of $A(x) = (a_{ij}(x))$ is bounded away from zero, and $a_{ij}(x)$ are bounded and Lipschitz continuous. We show that for any fixed $γ\in (0,α)$ the semigroup $P_t$ of the process $X_t$ satisfies $|P_t f(x) - P_t f(y)| \le c t^{-γ/α} |x - y|^γ ||f||_\infty$ for arbitrary bounded Borel function $f$. Our approach is based on Levi's method.
The object of the present paper is to find new sufficient conditions for the existence of unique strong solutions to a class of (time-inhomogeneous) stochastic differential equations with random, non-Lipschitzian coefficients. We give an example to show that our conditions are indeed weaker than those relevant conditions existing in the literature. We also derive moment estimations for the maximum process of the solution. Finally, we present a sufficient condition to ensure the non confluence property of the solution of time-homogeneous SDE which, in one dimension, is nothing but stochastic mo…
In this paper, we discuss exponential mixing property for Markovian semigroups generated by segment processes associated with several class of retarded Stochastic Differential Equations (SDEs) which cover SDEs with constant/variable/distributed time-lags. In particular, we investigate the exponential mixing property for (a) non-autonomous retarded SDEs by the Arzelà--Ascoli tightness characterization of the space $\C$ equipped with the uniform topology (b) neutral SDEs with continuous sample paths by a generalized Razumikhin-type argument and a stability-in-distribution approach and (c) jump-d…
This paper is devoted to studying the properties of the exit times of stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs). In particular, we prove that the exit times of $G$-SDEs has the quasi-continuity property. As an application, we give a probabilistic representation for a large class of fully nonlinear elliptic equations with Dirichlet boundary.
In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by $α/2$-subordinated cylindrical Brownian motions with $α\in(1,2)$. To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by $α$-stable noises established in Zhang (arXiv:1204.2630v2).
In this paper, we first show the well-posedness of the SDEs driven by Lévy noises under mild conditions. Then, we consider the existence and uniqueness of periodic solutions of the SDEs. To establish the ergodicity and uniqueness of periodic solutions, we investigate the strong Feller property and the irreducibility of the corresponding time-inhomogeneous semigroups when both small and large jumps are allowed in the equations. Some examples are presented to illustrate our results.
For degenerate stochastic differential equations driven by fractional Brownian motions with Hurst parameter $H>1/2$, the derivative formulas are established by using Malliavin calculus and coupling method, respectively. Furthermore, we find some relation between these two approaches. As applications, the (log) Harnack inequalities and the hyperbounded property are presented.
In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.
In this paper, we investigate stochastic continuity (with respect to the initial value), irreducibility and non confluence property of the solutions of stochastic differential equations with jumps. The conditions we posed are weaker than those relevant conditions existing in the literature. We also provide an example to support our new conditions.
In this paper we prove a derivative formula of Bismut-Elworthy-Li's type as well as gradient estimate for stochastic differential equations driven by $α$-stable noises, where $α\in(0,2)$. As an application, the strong Feller property for stochastic partial differential equations driven by subordinated cylindrical Brownian motions is presented.
In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Lévy process with finite Lévy measure, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.
In this paper, by using Girsanov's transformation and the property of the corresponding reference stochastic differential equations, we investigate weak existence and uniqueness of solutions and weak convergence of Euler-Maruyama scheme to stochastic functional differential equations with Hölder continuous drift driven by fractional Brownian motion with Hurst index $H\in (1/2,1)$.
