Strong Feller Property Research Articles

Quasi-stationary distribution for continuous-state branching processes with competition

We study quasi-stationary distribution of the continuous-state branching process with competition introduced by Berestycki et al. (2018). This process is defined as the unique strong solution to a stochastic integral equation with jumps. An important example is the logistic branching process proposed by Lambert (2005). We establish the strong Feller property, trajectory Feller property, Lyapunov condition, weak Feller property and irreducibility, respectively. These properties together allow us to prove that if the competition is strong enough near +∞, then there is a unique quasi-stationary distribution, which attracts all initial distributions with exponential rates.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

One-dimensional SDEs with LPS-type singular drift coefficients and Hölder continuous diffusion coefficients

In this paper, under the assumptions of LPS-type singular drift coefficients and Hölder continuous diffusion coefficients, the existence, uniqueness, stability, non-confluent property and strong Feller property for the solution to a one-dimensional stochastic differential equations are obtained. Our proof are based on the tools of local time and Zvonkin's transform.

Mixing times and hitting times for general Markov processes

The hitting and mixing times are two fundamental quantities associated with Markov chains. In Peres and Sousi [PS15] and Oliveira [Oli12], the authors show that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on general state spaces that satisfy the strong Feller property. Finally, we show that this asymptotic equivalence can be used to find bounds on the mixing times of a large class of Markov chains used in MCMC, such as typical Gibbs samplers and Metropolis–Hastings chains, even though they usually do not satisfy the strong Feller property.

A bidirectional hitting probability for the symmetric Hunt processes

We write σA the first hitting time of set A for the Hunt processes. Let B and BR be compact sets, where BR states far away from B. We assume that the Hunt process is irreducible and conservative and satisfies the Feller property. We consider a relation of the hitting probability of B from BR with the hitting probability of BR from B, without the spatial homogeneity. Our claim is that if the Hunt process satisfies the strong Feller property, then limx→∞Px(σB≤t)=0 implies that limR→∞Py(σBR≤t)=0, for y∈B. Additionally, if the Hunt process is m-symmetric, then both statements are equivalent.

Ergodicity of supercritical SDEs driven by α-stable processes and heavy-tailed sampling

Let α∈(0,2) and d∈N. We consider the stochastic differential equation (SDE) driven by an α-stable process dXt=b(Xt)dt+σ(Xt−)dLtα,X0=x∈Rd, where b: Rd→Rd and σ: Rd→Rd⊗Rd are locally γ-Hölder continuous with γ∈(0∨(1−α)+,1], and Ltα is a d-dimensional symmetric rotationally invariant α-stable process. Under certain dissipative and non-degenerate assumptions on b and σ, we show the V-uniformly exponential ergodicity for the semigroup Pt associated with {Xt(x),t≥0}. Our proofs are mainly based on the heat kernel estimates recently established in (J. Éc. Polytech. Math. 9 (2022) 537–579) to demonstrate the strong Feller property and irreducibility of Pt. Interestingly, when α tends to zero, the diffusion coefficient σ can increase faster than the drift b. As an application, we put forward a new heavy-tailed sampling scheme.

Stochastic differential equations with critically irregular drift coefficients

This paper is concerned with stochastic differential equations (SDEs for short) with irregular coefficients. By utilising a functional analytic approximation approach, we establish the existence and uniqueness of strong solutions to a class of SDEs with critically irregular drift coefficients in a new critical Lebesgue space, where the element may be only weakly integrable in time. To be more precise, let b:[0,T]×Rd→Rd be Borel measurable, where T>0 is arbitrarily fixed and d⩾1. We consider the following SDEXt=x+∫0tb(s,Xs)ds+Wt,t∈[0,T],x∈Rd, where {Wt}t∈[0,T] is a d-dimensional standard Wiener process. For p,q∈[1,+∞), we denote by C[q]([0,T];Lp(Rd)) the space of all Borel measurable functions f such that t1qf(t)∈C([0,T];Lp(Rd)). If b=b1+b2 such that |b1(T−⋅)|∈C[q]([0,T];Lp(Rd)) with 2/q+d/p=1 and ‖b1(T−⋅)‖C[q]([0,T];Lp(Rd)) is sufficiently small, and that b2 is bounded and Borel measurable, then we show that there exists a weak solution to the above equation, and if in addition limt↓0⁡‖t1qb(T−t)‖Lp(Rd)=0, the pathwise uniqueness holds. Furthermore, we obtain the strong Feller property of the semi-group and the existence of density associated with the above SDE. Besides, we extend the classical results concerning partial differential equations (PDEs) of parabolic type with Lq(0,T;Lp(Rd)) coefficients to the case of parabolic PDEs with L[q]∞(0,T;Lp(Rd)) coefficients, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Theorem 3.1). Our results extend Krylov-Röckner and Krylov's profound results of SDEs with singular time dependent drift coefficients [20,23] to the critical case of SDEs with critically irregular drift coefficients in a new critical Lebesgue space.

Periodic measures for a class of SPDEs with regime-switching

We use the variational approach to investigate periodic measures for a class of stochastic partial differential equations (SPDEs) with regime-switching. The hybrid system is driven by degenerate Lévy noise. We use the Lyapunov function method to study the existence of periodic measures and show the uniqueness of periodic measures by establishing the strong Feller property and irreducibility of the associated time-inhomogeneous semigroup. The main results are applied to stochastic fractional porous medium equations with regime-switching.

A monotone convergence theorem for strong Feller semigroups

For an increasing sequence (T_n) of one-parameter semigroups of sub Markovian kernel operators over a Polish space, we study the limit semigroup and prove sufficient conditions for it to be strongly Feller. In particular, we show that the strong Feller property carries over from the approximating semigroups to the limit semigroup if the resolvent of the latter maps mathbb {1} to a continuous function. This is instrumental in the study of elliptic operators on mathbb {R}^d with unbounded coefficients: our abstract result enables us to assign a semigroup to such an operator and to show that the semigroup is strongly Feller under very mild regularity assumptions on the coefficients. We also provide counterexamples to demonstrate that the assumptions in our main result are close to optimal.

Regularity of transition densities and ergodicity for affine jump‐diffusions

AbstractThis paper studies the transition density and exponential ergodicity for affine processes on the canonical state space . Under a Hörmander‐type condition for diffusion components as well as a boundary nonattainment condition, we derive the existence and regularity of the transition densities and then prove the strong Feller property of the associated semigroup. Moreover, we also show that, under an additional subcriticality condition on the drift, the corresponding affine process is exponentially ergodic in the total variation distance.

Differentiability of the transition semigroup of the stochastic Burgers-Huxley equation and application to optimal control

In this present paper we consider the transition semigroup {Pt}t≥0 related to the stochastic Burgers-Huxley equation which describes a prototype model for describing the interaction between reaction mechanisms, convection effects and diffusion transports. We are proving that it has a regularizing effect in the Banach space of continuous functions C(0,1), that is, the transition semigroup {Pt}t≥0 possesses strong Feller property. Some estimates on derivative of the transition semigroup related to stochastic Burgers-Huxley equation are established based on the exponential moment estimates of the solution of stochastic Burgers-Huxley equation and its first variation equation. Finally, certain application in Hamilton-Jacobi-Bellman equation arising from stochastic optimal control problem is provided to illustrate our results.

Stationary distribution of stochastic population dynamics with infinite delay

This paper examines the stationary distribution of the stochastic Lotka-Volterra model with infinite delay. Since the solutions of stochastic functional or delay differential equations depend on their history, they are non-Markov, which implies that traditional techniques based on the Markov property, cannot be applicable. This paper uses the variable substituting technique to obtain a higher-dimensional stochastic differential equation without delay. In this paper it is shown that the new equation satisfies the strong Feller property and strong Markov property, by which the invariant measure follows from the Krylov-Bogoliubov Theorem. Since distribution of the original model is actually the marginal distribution of the new equation, these results also show that there exists the stationary distribution for the original equation. Moreover, when the noise intensity is strongly dependent on the population size, by Hörmander's theorem, this paper also shows that the stationary distribution of this stochastic Lotka-Volterra system with infinite delay holds a C∞-smooth density.

On a class of Lévy-driven McKean-Vlasov SDEs with Hölder coefficients

In this paper, under the assumptions of Hölder continuous coefficients, the well-posedness, Euler-Maruyama approximation, stability and strong Feller property for the solution to one-dimensional spectrally one sided Lévy processes driven McKean-Vlasov SDEs are considered. Our proof are based on the tools of Yamada-Watanabe approximation technique and Girsanov theorem.

Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation

The present work deals with the global solvability as well as asymptotic analysis of the stochastic generalized Burgers–Huxley (SGBH) equation perturbed by a white-in-time and correlated-in-space noise defined in a bounded interval of $${\mathbb {R}}$$ . We first prove the existence of a unique mild as well as strong solution to the SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of the SGBH equation, that is, irreducibility and the strong Feller property. These two properties guarantee the uniqueness of invariant measures and ergodicity also. Then, under further assumptions on the noise coefficient, we discuss the ergodic behavior of the solution of the SGBH equation by providing a large deviation principle for the occupation measure for large time (Donsker–Varadhan), which describes the exact rate of exponential convergence.

On weak solution of SDE driven by inhomogeneous singular Lévy noise

We study a time-inhomogeneous SDE in R d \mathbb {R}^d driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit ‘principal part’ and a ‘residual part’, subject to certain L ∞ ( d x ) ⊗ L 1 ( d y ) L^\infty (dx)\otimes L^1(dy) and L ∞ ( d x ) ⊗ L ∞ ( d y ) L^\infty (dx)\otimes L^\infty (dy) -estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.

Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results.

Large deviation principle for occupation measures of two dimensional stochastic convective Brinkman-Forchheimer equations

The present work is concerned with two-dimensional stochastic subcritical and critical convective Brinkman-Forchheimer (2 D SCBF) equations perturbed by a white noise (non-degenerate) in smooth bounded domains in We establish two important properties of the Markov semigroup associated with the solutions of 2 D SCBF equations (for the absorption exponent r = 1, 2, 3), that is, irreducibility and strong Feller property. These two properties implies the uniqueness of invariant measures and ergodicity also. Then, we discuss the ergodic behavior of 2 D SCBF equations by providing a Large Deviation Principle (LDP) for the occupation measure for large time (Donsker-Varadhan), which describes the exact rate of exponential convergence.

Mixing and hitting times for Gibbs samplers and other non-Feller processes

The hitting and mixing times are two often-studied quantities associated with Markov chains. Yuval Peres, Perla Sousi and Roberto Oliveira showed that the mixing times and “worst-case” hitting times of reversible Markov chains on finite state spaces are “equivalent”—that is, equal up to some universal multiplicative constant. We have extended this strong connection between mixing and hitting times to Markov chains satisfying the strong Feller property in an earlier work. In the present paper, we further extend the results to include Metropolis–Hastings chains, the popular Gibbs sampler (from statistics), and Glauber dynamics (from statistical physics), which make “one-dimensional” updates and thus do not satisfy the strong Feller property. We also apply this result to obtain decomposition bounds for such Markov chains. Our main tools come from nonstandard analysis.

Strong Feller property of the magnetohydrodynamics system forced by space–time white noise

The magnetohydrodynamics system forced by space–time white noise has been proposed and investigated by physicists and engineers, although without rigorous mathematical proofs of its solution’s fundamental properties. Applying the theory of regularity structures, we prove its well-posedness and strong Feller property. The proof requires a careful treatment of nonlinear terms which are sensitive to specific components of the solution that is a six-dimensional vector field, of which the first three and the last three are velocity and magnetic fields, respectively.

Periodic solutions of hybrid jump diffusion processes

In this paper, we investigate periodic solutions of regime-switching jump diffusions. We first show the well-posedness of solutions to the SDEs corresponding to the hybrid system. Then, we derive the strong Feller property and irreducibility of the associated time-inhomogeneous semigroups. Finally, we establish the existence and uniqueness of periodic solutions. Concrete examples are presented to illustrate the results.