I have given a diffusion equation $$ dX_t = -\nabla V(X_t) \, dt + \sigma dB_t.$$
I found here(1) a characterization when $X_t$ is reversible, aslong as $\sigma=1$. Is this also true for $\sigma \neq 1$?
It is said here(2) that the general result for arbitrary $\sigma$ can be found here(3). Unfortunately I do not have access to this document, and I also assume that the notation will probably be unreadable for me and not as plain and clean as here(1).
Therefore, I would like to know if it is easy to get from here(1) to the general case where $\sigma \neq 1$.
Also I would like to know if the stationary measure $\mu$ is unique if $X_t$ is reversible.
I would recommend one of the classic books
[1] Stochastic Differential Equations. An Introduction with Applications By B. Øksendal
http://www.springer.com/us/book/9783540047582
[2] Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach by Holden, Øksendal, Ubøe and Zhang
http://link.springer.com/book/10.1007/978-1-4684-9215-6
[3] Numerical Solution of Stochastic Differential Equations by Kloeden, and Platen
http://www.springer.com/us/book/9783540540625