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 $p\ge 1$, $\mathbb{E}\left|u(t)-u(0)\right|_{\dot{H}^1(\mathbb{R})}^2 \to 0$ 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 $\mathbb{E} \exp\Big(p\int_t^T K(s)\,\mathrm{d} s\Big) \lesssim {t^{-2p}}$ for small times $t$, for some $p\ge1$. 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).