Stochastic differential equations with coefficients in Sobolev spaces

TL;DR

This paper investigates stochastic differential equations with Sobolev space coefficients, establishing conditions for the existence of densities and quasi-invariance of measures, and develops methods for stochastic flow uniqueness.

Contribution

It extends the theory of SDEs with Sobolev coefficients by proving existence, uniqueness, and measure transformation properties under new integrability and regularity conditions.

Findings

Existence of densities for solutions under Sobolev regularity.

Quasi-invariance of Lebesgue measure for bounded Lipschitz coefficients.

Development of a method for stochastic flow uniqueness in Sobolev coefficient SDEs.

Abstract

We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d<+\infty$ for some $\lambda_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# \gamma_d$ admits a density with respect to $\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure $\Leb_d$ quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.