Existence, uniqueness and ergodic properties for time-homogeneous It\^o-SDEs with locally integrable drifts and Sobolev diffusion coefficients

TL;DR

This paper establishes existence, uniqueness, and ergodic properties of solutions to certain time-homogeneous Itô stochastic differential equations with locally integrable drifts and Sobolev diffusion coefficients, using advanced PDE and Dirichlet form techniques.

Contribution

It introduces new methods to construct weak solutions for SDEs with less regular coefficients and proves their ergodic and regularity properties under minimal assumptions.

Findings

Constructed weak solutions for SDEs with Sobolev diffusion coefficients.

Proved irreducibility and strong Feller properties of the transition function.

Derived conditions for recurrence, ergodicity, and invariant measures.

Abstract

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.