Strong solutions of a stochastic differential equation with irregular random drift

Abstract

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$\mathrm{d}X=u(\omega ,t,X)\mathrm{d}t+\frac{1}{2}\sigma (\omega ,t,X){\sigma }^{\prime }(\omega ,t,X)\mathrm{d}t+\sigma (\omega ,t,X)\mathrm{d}W(t),$$ where the drift coefficient $u$ is random and irregular. The random and regular noise coefficient $\sigma$ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that $u$ belongs to $L^p(\Omega ;L^{\infty }([0,T];{\dot{H}}^1(\mathbb{R})))$ for any finite $p\ge 1$, $\mathbb{E}{\left|u(t)-u(0)\right|}_{{\dot{H}}^1(\mathbb{R})}^2\to 0$ as $t\downarrow 0$, and $u$ satisfies the one-sided gradient bound ${\partial }_{x}u(\omega ,t,x)\le K(\omega ,t)$, where the process $K(\omega ,t)>0$ exhibits an exponential moment bound of the form $\mathbb{E}\exp (p{\int }_t^{T}K(s)\mathrm{d}s)\lesssim t^{-2p}$ for small times $t$, for some $p\ge 1$. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter--Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation (SPDE).

20 pages

Cited by in corpus (1)

• Global existence of dissipative solutions to the Camassa--Holm equation with transport noise