Tohoku Math. J.
73 (2021), 159-198
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Title
EXISTENCE, UNIQUENESS AND ERGODIC PROPERTIES FOR TIME-HOMOGENEOUS ITÔ-SDES WITH LOCALLY INTEGRABLE DRIFTS AND SOBOLEV DIFFUSION COEFFICIENTS
Author
Haesung Lee and Gerald Trutnau
(Received September 24, 2019, revised February 6, 2020) |
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.
Mathematics Subject Classification.
Primary 60H20; Secondary 47D07, 60J35, 31C25, 60J60, 35B65.
Key words and phrases.
Pathwise uniqueness, non-explosion, recurrence, ergodicity, invariant probability measure, strong Feller property, elliptic and parabolic regularity.
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