Strong solutions of a stochastic differential equation with irregular random drift

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form $$mathrm{d} X= u(omega,t,X), mathrm{d} t + frac12 sigma(omega,t,X)sigma(omega,t,X),mathrm{d} t + sigma(omega,t,X) , mathrm{d}W(t), $$ 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 $pge 1$, $mathbb{E}left|u(t)-u(0)right|_{dot{H}^1(mathbb{R})}^2 to 0$ as $tdownarrow 0$, 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 $mathbb{E} expBig(pint_t^T K(s),mathrm{d} sBig) lesssim {t^{-2p}}$ for small times $t$, for some $pge1$. 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).