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 where the drift coefficient is random and irregular, with a weak derivative satisfying for some , . The random and regular noise coefficient may vanish. The main contribution is a pathwise uniqueness result under the assumptions that as , and satisfies the one-sided gradient bound , where the process exhibits an exponential moment bound of the form for small times , for some . 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.