Strong Feller Property Research Articles

Perturbation by non-local operators

Suppose that $d\ge 1$ and $0<\beta<\alpha<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to a class of (possibly nonsymmetric) non-local operators $L^b=\Delta^{\alpha/2}+S^b$, where $$ S^bf(x):=A(d, -\beta) \int_{R^d} ( f(x+z)-f(x)- \nabla f(x) \cdot z 1_{\{|z|\leq 1\}} ) \frac{b(x, z)}{|z|^{d+\beta}}dz $$ and $b(x, z)$ is a bounded measurable function on $R^d\times R^d$ with $b(x, z)=b(x, -z)$ for $x, z\in R^d$. Here $A(d, -\beta)$ is a normalizing constant so that $S^b=\Delta^{\beta/2}$ when $b(x, z)\equiv 1$. We show that if $b(x, z) \geq -\frac{{\cal A}(d, -\alpha)}{A(d, -\beta)}\, |z|^{\beta -\alpha}$, then $q^b(t, x, y)$ is a strictly positive continuous function and it uniquely determines a conservative Feller process $X^b$, which has strong Feller property. The Feller process $X^b$ is the unique solution to the martingale problem of $(L^b, {\cal S} (R^d))$, where ${\cal S}(R^d)$ denotes the space of tempered functions on $R^d$. Furthermore, sharp two-sided estimates on $q^b(t, x, y)$ are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of $b(x, z)$. The model considered in this paper contains the following as a special case. Let $Y$ and $Z$ be (rotationally) symmetric $\alpha$-stable process and symmetric $\beta$-stable processes on $R^d$, respectively, that are independent to each other. Solution to stochastic differential equations $dX_t=dY_t + c(X_{t-})dZ_t$ has infinitesimal generator $L^b$ with $b(x, z)=| c(x)|^\beta$.

On the martingale problem and Feller and strong Feller properties for weakly coupled Lévy type operators

This paper considers the martingale problem for a class of weakly coupled Lévy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process (X,Λ). The process (X,Λ), called a regime-switching jump diffusion with Lévy type jumps, is further shown to possess Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.

Large deviations and mixing for dissipative PDEs with unbounded random kicks

We study the problem of exponential mixing and large deviations for discrete-time Markov processes associated with a class of random dynamical systems. Under some dissipativity and regularisation hypotheses for the underlying deterministic dynamics and a non-degeneracy condition for the driving random force, we discuss the existence and uniqueness of a stationary measure and its exponential stability in the Kantorovich–Wasserstein metric. We next turn to the large deviations principle (LDP) and establish its validity for the occupation measures of the Markov processes in question. The proof is based on Kifer’s criterion for non-compact spaces, a result on large-time asymptotics for generalised Markov semigroup, and a coupling argument. These tools combined together constitute a new approach to LDP for infinite-dimensional processes without strong Feller property in a non-compact space. The results obtained can be applied to the two-dimensional Navier–Stokes system in a bounded domain and to the complex Ginzburg–Landau equation.

Harnack inequality and derivative formula for stochastic heat equation with fractional noise

In this note, we establish the Harnack inequality and derivative formula for stochastic heat equation driven by fractional noise with Hurst index $H\in (\frac 14,\frac 12)$. As an application, we introduce a strong Feller property.

Smoothness of continuous state branching with immigration semigroups

In this work we develop an original and thorough analysis of the (non)-smoothness properties of the semigroups, and their heat kernels, associated to a large class of continuous state branching processes with immigration. Our approach is based on an in-depth analysis of the regularity of the absolutely continuous part of the invariant measure combined with a substantial refinement of Ogura's spectral expansion of the transition kernels. In particular, we find new representations for the eigenfunctions and eigenmeasures that allow us to derive delicate uniform bounds that are useful for establishing the uniform convergence of the spectral representation of the semigroup acting on linear spaces that we identify. We detail several examples which illustrate the variety of smoothness properties that CBI transition kernels may enjoy and also reveal that our results are sharp. Finally, our technique enables us to provide the (eventually) strong Feller property as well as the rate of convergence to equilibrium in the total variation norm.

Asymptotics for stochastic reaction–diffusion equation driven by subordinate Brownian motion

We study the ergodicity of stochastic reaction–diffusion equation driven by subordinate Brownian motion. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution’s law. These properties imply that this stochastic system admits a unique invariant measure according to Doob’s and Krylov–Bogolyubov’s theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by α-stable type noises do not satisfy Freidlin–Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker–Varadhan type large deviation as time tends to infinity.

A coupled KPZ equation, its two types of approximations and existence of global solutions

This paper concerns the multi-component coupled Kardar–Parisi–Zhang (KPZ) equation and its two types of approximations. One approximation is obtained as a simple replacement of the noise term by a smeared noise with a proper renormalization, while the other one introduced in [6] is suitable for studying the invariant measures. By applying the paracontrolled calculus introduced by Gubinelli et al. [8,9], we show that two approximations have the common limit under the properly adjusted choice of renormalization factors for each of these approximations. In particular, if the coupling constants of the nonlinear term of the coupled KPZ equation satisfy the so-called “trilinear” condition, the renormalization factors can be taken the same in two approximations and the difference of the limits of two approximations are explicitly computed. Moreover, under the trilinear condition, the Wiener measure twisted by the diffusion matrix becomes stationary for the limit and we show that the solution of the limit equation exists globally in time when the initial value is sampled from the stationary measure. This is shown for the associated tilt process. Combined with the strong Feller property shown by Hairer and Mattingly [12], this result can be extended for all initial values.

Continuous-State Branching Processes in Lévy Random Environments

A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a L\'evy process with no jump less than $-1$. We give characterizations of the quenched and annealed transition semigroups of the process in terms of a backward stochastic integral equation driven by another L\'evy process determined by the environment. The process hits zero with strictly positive probability if and only if its branching mechanism satisfies Grey's condition. In that case, a characterization of the extinction probability is given using a random differential equation with singular terminal condition. The strong Feller property of the CBRE-process is established by a coupling method. We also prove a necessary and sufficient condition for the ergodicity of the subcricital CBRE process with immigration.

Bismut formulas and applications for stochastic (functional) differential equations driven by fractional Brownian motions

In this paper by using Malliavin calculus we prove derivative formulas of Bismut type for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, the dimensional-free Harnack type inequalities and the strong Feller property are presented.

Strong Feller Property of Sticky Reflected Distorted Brownian Motion

Using Girsanov transformations we construct from sticky reflected Brownian motion on $$[0,\infty )$$ a conservative diffusion on $$E:=[0,\infty )^n$$ , $$n \in \mathbb {N}$$ , and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies of the mathematical model are motivated by its applications to the dynamical wetting model with $$\delta $$ -pinning and repulsion.

On Feller and Strong Feller Properties and Exponential Ergodicity of Regime-Switching Jump Diffusion Processes with Countable Regimes

This work focuses on a class of regime-switching jump diffusion processes, in which the switching component has countably infinite many states or regimes. The existence and uniqueness of the underlying process are obtained by an interlacing procedure. Then the Feller and strong Feller properties of such processes are derived by the coupling method and an appropriate Radon-Nikodym derivative. Finally the paper studies exponential ergodicity of regime-switching jump-diffusion processes.

Stochastic differential equations with sticky reflection and boundary diffusion

We construct diffusion processes in bounded domains $\Omega $ with sticky reflection at the boundary $\Gamma $ in use of Dirichlet forms. In particular, the occupation time on the boundary is positive. The construction covers a static boundary behavior and an optional diffusion along $\Gamma $. The process is a solution to a given SDE for q.e. starting point. Using regularity results for elliptic PDE with Wentzell boundary conditions we show strong Feller properties and characterize the constructed process even for every starting point in $\overline{\Omega } \backslash \Xi $, where $\Xi $ is given explicitly by the involved densities. By a time change we obtain pointwise solutions to SDEs with immediate reflection under weak assumptions on $\Gamma $ and the drift. A non-trivial extension of the construction yields N-particle systems with the stated boundary behavior and singular drifts. Finally, the setting is applied to a model for particles diffusing in a chromatography tube with repulsive interactions.

Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise

The current paper is devoted to 3D stochastic Ginzburg-Landau equation with degenerate random forcing. We prove that the corresponding Markov semigroup possesses an exponentially attracting invariant measure. To accomplish this, firstly we establish a type of gradient inequality, which is also essential to proving asymptotic strong Feller property. Then we prove that the corresponding dynamical system possesses a strong type of Lyapunov structure and is of a relatively weak form of irreducibility.

Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by &lt;i&gt;α&lt;/i&gt;-stable noise

The current paper is devoted to the ergodicity of stochastic coupled fractional Ginzburg-Landau equations driven by \begin{document}$α$\end{document} -stable noise on the Torus \begin{document}$\mathbb{T}$\end{document} . By the maximal inequality for stochastic \begin{document}$α$\end{document} -stable convolution and commutator estimates, the well-posedness of the mild solution for stochastic coupled fractional Ginzburg-Landau equations is established. Due to the discontinuous trajectories and non-Lipschitz nonlinear term, the existence and uniqueness of the invariant measures are obtained by the strong Feller property and the accessibility to zero.

Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients

By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.

Ergodicity of stochastic Magneto-Hydrodynamic equations driven by α-stable noise

The current paper is devoted to the ergodicity of stochastic Magneto-Hydrodynamic equations driven by α-stable noise with α∈(32,2). By the maximal inequality for the stochastic α-stable convolution and vorticity transformation, the well-posedness of the mild solution for stochastic Magneto-Hydrodynamic equation is established. Due to the discontinuous trajectories, the existence and uniqueness of the invariant measure for stochastic Magneto-Hydrodynamic equation are obtained by the strong Feller property and the accessibility to zero instead of the irreducibility.

Ergodicity of large scale stochastic geophysical flows with degenerate Gaussian noise

This paper considers ergodicity of the stochastic quasigeostrophic equations driven by degenerate Gaussian noise. Uniqueness of invariant measures is shown by proving the asymptotically strong Feller property of the probability transition semigroups.

Large deviation principles for generalized Feynman-Kac functionals and its applications

Large deviation principles of occupation distribution for generalized Feyn-man-Kac functionals are presented in the framework of symmetric Markov processes having doubly Feller or strong Feller property. As a consequence, we obtain the $L^p$-independence of spectral radius of our generalized Feynman-Kac functionals. We also prove Fukushima's decomposition in the strict sense for functions locally in the domain of Dirichlet form having energy measure of Dynkin class without assuming no inside killing.

Self-repelling diffusions on a Riemannian manifold

Let M be a compact connected oriented Riemannian manifold. The purpose of this paper is to investigate the long time behavior of a degenerate stochastic differential equation on the state space $M\times \mathbb{R}^{n}$; which is obtained via a natural change of variable from a self-repelling diffusion taking the form $$dX_{t}= \sigma dB_{t}(X_t) -\int_{0}^{t}\nabla V_{X_s}(X_{t})dsdt,\qquad X_{0}=x$$ where $\{B_t\}$ is a Brownian vector field on $M$, $\sigma >0$ and $V_x(y) = V(x,y)$ is a diagonal Mercer kernel. We prove that the induced semi-group enjoys the strong Feller property and has a unique invariant probability $\mu$ given as the product of the normalized Riemannian measure on M and a Gaussian measure on $\mathbb{R}^{n}$. We then prove an exponential decay to this invariant probability in $L^{2}(\mu)$ and in total variation.

Strong Feller properties for degenerate SDEs with jumps

Under full H\"ormander's conditions, we prove the strong Feller property of the semigroup determined by an SDE driven by additive subordinate Brownian motion, where the drift is allowed to be arbitrarily growth. For this, we extend a criterion due to Malicet-Poly \cite{Ma-Po} and Bally-Caramellino \cite{Ba-Ca} about the convergence of the laws of Wiener functionals in total variations. Moreover, the example of a chain of coupled oscillators is verified.