Regular Flows for Diffusions with Rough Drifts

Abstract

According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $r/d+2/q<1$. As an application we show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $r/d+2/q<1$. As our second application we derive a Constantin-Iyer type circulation formula for certain weak solutions of Navier-Stokes equation.

References in corpus (3)

• Pathwise uniqueness and continuous dependence for SDEs with nonregular drift
• Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation
• Remarks on the stochastic transport equation with Hölder drift

Cited by in corpus (9)

• Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations
• SDEs with critical time dependent drifts: strong solutions
• SDEs with critical time dependent drifts: weak solutions
• Stochastic Differential Equations with Critical Drifts
• Stochastic differential equations with Sobolev diffusion and singular drift
• Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise
• Strong Uniqueness of Singular Stochastic Delay Equations
• Renormalized Solutions to Stochastic Continuity Equations with Rough Coefficients
• Hypoelliptic diffusions with singular drifts