Stochastic differential equations with coefficients in Sobolev spaces

Abstract

We consider Itô SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on ${\mathbb{R}}^d$. The diffusion coefficients $A_1,...,A_m$ are supposed to be in the Sobolev space $W_{\text{loc}}^{1,p}({\mathbb{R}}^d)$ 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 $W_{\text{loc}}^{1,p^{\prime }}$ for some $p^{\prime }>1$. Assume $\int_{\R^d} \exp\big[λ_0\bigl(|δ(A_0)| + \sum_{j=1}^m (|δ(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \dγ_d<+\infty$ for some ${\lambda }_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# γ_d$ 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.

31 pages

References in corpus (1)

• Well-posedness of the transport equation by stochastic perturbation