Density of $\int_{0}^{t}W'(B_{s})ds$ where $W'$ is smooth and compactly supported.

Question

Only hints please

Density of $\int_{0}^{t}W'(B_{s})ds$, where $B_{s}$ is 1-d Brownian motion.

The density of $Y_{s}:=W'(B_{s})$ is $g_{Y}(y)=p_{B_{s}}((W')^{-1}(y))|\frac{d(W')^{-1}(y)}{dy}|$.

How to find density of $\int_{0}^{t}Y_{s}ds$?

The original problem is to find density of

$dX_{t}=dB_{t}-V'(B_{t})dt$ where $V(x)=\frac{x^{2}}{2}+W(x)$ and $x_{0}$ has density $\frac{e^{-V(x)}}{\int e^{-V(y)}dy}$.

thank you

Answer 1

Hint: Invariant distribution.