Perturbed Brownian motion reflected at a time-dependent boundary

Abstract: Let $B$ be a standard Brownian motion, $x\ge 0,\, ν&lt;1$, and $b:[0,\infty)\to\mathbb R$ is a continuous function locally of finite variation starting from 0. We define the perturbed Brownian motion reflected at the boundary $b$ by establishing strong existence and pathwise uniqueness of a solution to the equation [ W_t=(1-ν)x+B_t+νM_t(W)+\frac12 L_t^0(W-b), \qquad W_t\ge b(t), ] where $M_t(W):=\sup_{0\le s\le t} W_s$ and the process $L<sup>0(W-b)$ is the semimartingale local time at 0 of the process $W-b$. We give a positive result under condition (PB) on the boundary $b$ : for every $T&gt;0$, the upward increment $\sup_{0\le s&lt;t\le T,\,t-s\le h}(b(t)-b(s))⁺ = o(\sqrt{h})$ as $h\downarrow 0$. The proof splits into two regimes: the case $ν&lt;1/2$ is a consequence of the Skorokhod problem in an orthant proved by [Williams 1995], while the case $ν\ge 1/2$ combines a deterministic comparison estimate and a logarithmic upper bound on the number of completed round-trips, following the strategy of [Chaumont and Doney 1999]. For $α\in(0,1/2)$, we also construct an increasing $α$-Hölder boundary for which no continuous adapted solution starting from zero exists for any $ν&lt;1$.