Stochastic differential equations with coefficients in Sobolev spaces

Abstract

We consider the Itô stochastic differential equation d X t = ∑ 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 loc 1 , p ( R d ) with p > d , and to have linear growth. For the drift coefficient A 0 , we distinguish two cases: (i) A 0 is a continuous vector field whose distributional divergence δ ( A 0 ) with respect to the Gaussian measure γ d exists, (ii) A 0 has Sobolev regularity W loc 1 , p ′ for some p ′ > 1 . Assume ∫ R d exp [ λ 0 ( | δ ( A 0 ) | + ∑ j = 1 m ( | δ ( A j ) | 2 + | ∇ A j | 2 ) ) ] d γ d < + ∞ for some λ 0 > 0 . In case (i), if the pathwise uniqueness of solutions holds, then the push-forward ( X t ) # γ d admits a density with respect to γ d . In particular, if the coefficients are bounded Lipschitz continuous, then X t leaves the Lebesgue measure Leb d quasi-invariant. In 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 existence and uniqueness of stochastic flow of maps.