Hi this Question follows after the answer of [Douglas Zare](https://mathoverflow.net/questions/12327/extension-of-some-feature-of-sde-ornstein-uhlenbeck-type) to this post :


So let's be given a positive function $V(x)$ which is smooth enough to have Lispchitzian derivative at all point. Moreover this function admits multiple zeros denoted $(\theta\_1,...,\theta\_n)$ and let $X\_t$ the diffusion process following the stochastic equation below :


$dX\_t=V'(X\_t)dt + \sigma.dW\_t$ where now $\sigma$ is a constant (but will later be replaced by some function of $X\_t$)
and $X\_0=x$


Here are my questions :


1-What are the conditions on $V$ for $X\_t$ :


a- to admit a stationary distribution ? and does this distribution can be explicited in terms of $V,x,\theta\_1,...,\theta\_n$, and $\sigma$ ?


b- to be an ergodic process ?


2-What are the smoothness conditions on $\sigma$ (now a function of $X\_t$) to keep 1/a- or 1/b- true ?


Of course any reference is welcome if the answer is too long for a single post.


Best Regards



Follow up question, Ornstein-Uhlenbeck Extension with n mean-reversion values
Let us assume you are interested in the solution of $dX\_t=\sigma(X\_t)dW\_t+b(X\_t)dt$. If the so-called *speed measure* $m$ has finite mass, the diffusion converges in distribution to $m/|m|$. Here, $m(x)=2/(\sigma^2(x)s'(x))$ where $s$ is the so-called *scale* of the diffusion. Recall that $s$ is uniquely defined, up to affine transformations, as the solution of the differential equation $\frac12\sigma(x)^2s''(x)+b(x)s'(x)=0$.


In your case, $\sigma$ is constant and $b(\cdot)=V'(\cdot)$, hence $s'(x)=\exp(-2V(x)/\sigma^2)$ up to a multiplicative constant. The condition for ergodicity is that the function $\exp(2V(\cdot)/\sigma^2)$ is integrable. In that case, the stationary distribution is proportional to this function.


