Stochastic Differential Equations with Critical Drifts

Abstract

We establish the well-posedness of SDE with the additive noise when a singular drift belongs to the critical spaces. We prove that if the drift belongs to the Orlicz-critical space $L^{q,1}([0,T],L_x^p)$ for $p,q\in (1,\infty )$ satisfying $\frac{2}{q}+\frac{d}{p}=1$, then the corresponding SDE admits a unique strong solution. We also derive the Sobolev regularity of a solution under the Orlicz-critical condition.

28 pages

References in corpus (2)

• Noise Prevents Singularities in Linear Transport Equations
• Regular Flows for Diffusions with Rough Drifts

Cited by in corpus (3)

• $L^q(L^p)$-theory of stochastic differential equations
• On time inhomogeneous stochastic Itô equations with drift in $L_{d+1}$
• Strong solutions of stochastic differential equations with square integrable drift