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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

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5  
It would probably be a good idea to make the title more descriptive. –  Qiaochu Yuan Jan 31 '11 at 13:48
    
Is it better now ? –  The Bridge Jan 31 '11 at 14:22

1 Answer 1

up vote 5 down vote accepted

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.

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And the generalization to non constant functions $\sigma(\cdot)$ is straightforward. –  Did Jan 31 '11 at 15:00
    
Thanks,great and very clear answer. By the way do you have some references about "speed measures" and "scale functions" that you would recommand ? Regards –  The Bridge Jan 31 '11 at 15:18
1  
I was afraid you would ask this question... A classic reference is the venerable Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus by Rogers and Williams but it might take time to digest. Durrett, as often, is most congenial and you could try his book Stochastic calculus (which I believe has a whole chapter devoted to 1D diffusions, but I do not have the book at hand so, if the chapter exists it will deal with this stuff and otherwise, well, I apologize in advance). –  Did Jan 31 '11 at 15:52
    
Please do not apologize, I know this reference but couldn't remember that it has the subject treated. I should re-read it then. Regards –  The Bridge Jan 31 '11 at 16:07

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