Let $V(x) = x^2 / 2+ W(x)$ where $W(x)$ is a smooth function with compact support. Let $f$ denote the probability density. $f(x) = \frac{e^{-V(x)}}{\int e^{-V(x)}dx}$. Consider the stochastic differential equation : $dx(t) = dB(t) - V'(B(t))dt$ with $X(0) = x_0$
where $x_0$ is a random variable independent of $B(s)$ with density $f(x)$.
Find the density of the random variable $X(t)$.
I am pretty sure this looks like an application of Kolmogorov forward equation. But the right hand terms are not expressed in terms of $X(t)$ so I was wondering, if someone can show me how it is done as I have never seen an example using Kolmogorov forward equation before. (Thanks in advance)