Uniqueness Of Invariant Probability Measure Research Articles

Weak solution and invariant probability measure for McKean-Vlasov SDEs with integrable drifts

In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, by Banach's fixed theorem, the existence and uniqueness of invariant probability measure for symmetric McKean-Vlasov SDEs and stochastic Hamiltonian system with integrable drifts are obtained.

Existence, uniqueness and ergodic properties for time-homogeneous Itô-SDEs with locally integrable drifts and Sobolev diffusion coefficients

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 with continuous sample paths on the one-point compactification of $\mathbb{R}^d$ and the corresponding SDE is by a known local well-posedness result pathwise unique up to an explosion time. Just under the given assumptions we show irreducibility and the strong Feller property on $L^{1}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$ of its transition function, and the strong Feller property on $L^{q}(\mathbb{R}^d,m)+L^{\infty}(\mathbb{R}^d,m)$, $q=\frac{dp}{d+p}\in (d/2,p/2)$, of its resolvent, which both include the classical strong Feller property. We present moment inequalities and classical-like non-explosion criteria for the solution which lead to pathwise uniqueness results up to infinity under presumably optimal general non-explosion conditions. We further present explicit conditions for recurrence and ergodicity, including existence as well as uniqueness of invariant probability measures.

Krylov–Safonov estimates for a degenerate diffusion process

This paper proves a Krylov–Safonov estimate for a multidimensional diffusion process whose diffusion coefficients are degenerate on the boundary. As applications the existence and uniqueness of invariant probability measures for the process and Hölder estimates for the associated partial differential equation are obtained.

Fine irreducibility and uniqueness of stationary distribution

The uniqueness of invariant measure is one of the most interesting problems in theory of Markov processes. In this paper, we shall prove that the irreducibility in the sense of fine topology implies the uniqueness of invariant probability measures. It is also proven that this irreducibility is strictly weaker than the strong Feller property plus irreducibility in the sense of original topology, which is the usual uniqueness condition.

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.

On stochastic ordering for diffusion with jumps and applications

We study the stochastic ordering for diffusion processes with jumps. A new comparison theorem for diffusion processes with jumps is obtained. We then apply this comparison theorem to discuss the existence and uniqueness of invariant probability measure for uniformly elliptic diffusion processes with jumps.

Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions

Let be a complete connected Riemannian manifold of dimension and let be a second order elliptic operator on that has a representation in local coordinates, where , for some , and the matrix is non-singular. The aim of the paper is the study of the uniqueness of a solution of the elliptic equation for probability measures , which is understood in the weak sense: for all . In addition, the uniqueness of invariant probability measures for the corresponding semigroups generated by the operator is investigated. It is proved that if a probability measure on satisfies the equation and is dense in , then is a unique solution of this equation in the class of probability measures. Examples are presented (even with and smooth ) in which the equation has more than one solution in the class of probability measures. Finally, it is shown that if , then the semigroup generated by has at most one invariant probability measure.

Information flow in discrete Markov systems

Infinite systems of interacting Markov chains are investigated. Some basic concepts of the ergodic theory of such systems are first presented. In particular, the question of the uniqueness and multiplicity of invariant probability measures is considered. In the case of one-dimensional systems, the question is studied in detail by investigating the flow of information throughout the system and some criteria for the uniqueness of invariant probability measures are obtained.

Information flow in discrete Markov systems

Infinite systems of interacting Markov chains are investigated. Some basic concepts of the ergodic theory of such systems are first presented. In particular, the question of the uniqueness and multiplicity of invariant probability measures is considered. In the case of one-dimensional systems, the question is studied in detail by investigating the flow of information throughout the system and some criteria for the uniqueness of invariant probability measures are obtained.