Question

Hi this Question follows after the answer of Douglas Zare 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

Answer 1

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.