0
$\begingroup$

Suppose I have a stochastic process $(X_t)_{t\in \mathbb{R}^d}$ with infinitesimal generator $\mathcal{A}$, for example $\mathcal{A}f(X) = -\mu f'(X) + \frac{1}{2}\sigma^2f''(X)+\lambda \int (f(X')-f(X))d\Phi(X)$, where $\Phi(\cdot)$ is the normal CDF.

I'd like to define a notion of divergence for such stochastic processes, to evaluate how much mass "flows out" from a given point in the state space. Is there such a notion for stochastic processes?

Improve this question
$\endgroup$
3
  • $\begingroup$ I see a Laplacian there, so I am assuming that stochastic process is related to Brownian motion and so it is not differentiable and cannot define divergence quantities per se. Having said that, people have used tools such as "exit probability", "local time" and "occupation measure", "Newtonian capacity" and "polar/non-polar sets" to talk about a stochastic process exiting/entering domains eg. see potential theory chapter in "Brownian motion" by Morters-Peres. $\endgroup$
    – Thomas Kojar
    Feb 12, 2023 at 21:52
  • $\begingroup$ if you have a particular technical question that you want to use "divergence" for, we can try to help out. $\endgroup$
    – Thomas Kojar
    Feb 12, 2023 at 21:53
  • $\begingroup$ I think you can use the theory of stochastic flows here. Nice diffusions define a (random) flow of diffeomorphisms which are as smooth as the coefficients are. Now apply the deterministic result that flows of diffeomorphisms admit a well defined divergence. $\endgroup$
    – Nate River
    Feb 13, 2023 at 2:28

1 Answer 1

Reset to default
0
$\begingroup$

Just turning comment into answer:

I see a Laplacian there, so I am assuming that stochastic process is related to Brownian motion and so it is not differentiable and cannot define divergence quantities per se. Having said that, people have used tools such as "exit probability", "local time" and "occupation measure", "Newtonian capacity" and "polar/non-polar sets" to talk about a stochastic process exiting/entering domains eg. see potential theory chapter in "Brownian motion" by Morters-Peres.

if you have a particular technical question that you want to use "divergence" for, we can try to help out. –

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.