Many dynamical systems driven by Gaussian white noise (GWN) can be written $$ \dot{x}(t) = F(x,t) + \sqrt{2D}\xi(t),$$ where $F = -dV/dx$ is a potential and $\xi(t)$ is a GWN with mean $0$ and variance $1$. Averaging over realizations of the noise provides the Fokker-Planck equation $$ \frac{\partial}{\partial t} P(x,t) = - \frac{\partial}{\partial x}[F(x,t)P(x,t)] + D\frac{\partial^2}{\partial x^2} P(x,t).$$
I am curious about the case when the potential $V(x)$ contains an infinite barrier. Is this equivalent to the problem with no potential, where the Fokker-Planck equation is solved with a reflecting boundary condition? If so, is there a boundary condition which can represent a dissipative potential barrier, such as a spring-dashpot-style model?