Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients

TL;DR

This paper proves the weak Sobolev differentiability of solutions to SDEs with local Sobolev and super-linear growth coefficients, and investigates their strong Feller property and irreducibility of the diffusion semigroup.

Contribution

It introduces a new characterization for Sobolev differentiability of random fields and applies it to establish differentiability of SDE solutions with complex coefficients.

Findings

Solutions are weakly Sobolev differentiable with respect to initial conditions.

The associated diffusion semigroup exhibits strong Feller property.

The diffusion process is irreducible.

Abstract

By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility of the associated diffusion semigroup.