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 dX=u(ω,t,X)dt+12σ(ω,t,X)∂xσ(ω,t,X)dt+σ(ω,t,X)dW(t),where the drift coefficient u is random and irregular, with a weak derivative satisfying ∂xu=q for some q∈LωpLt∞(Lx2∩Lx1), p∈[1,∞). The random and regular noise coefficient σ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that E‖q(t)−q(0)‖L2(R)2→0 as t↓0, and u satisfies the one-sided gradient bound q(ω,t,x)≤K(ω,t), where the process K(ω,t)>0 exhibits an exponential moment bound of the form Eexp(p∫tTK(s)ds)≲t−2p for small times t, for some p≥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.