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Consider a Brownian motion $W$ reflected at the boundary of a domain $D$ in Euclidean space. I want to look at the process obtained by "restricting" it to the boundary.

I was thinking of taking the local time $L(t)$ at the boundary, and then defining my new process as $W(L^{-1}(t))$.

Is this the appropriate way to do it? Does it define a Markov process on the boundary? If so, what is known about this process for, say, a $d$-dimensional round ball (eg on its regularity/ size and frequency of jumps)?

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    $\begingroup$ You probably mean $W(L^{-1}(t))$ which would be a reasonable definition and would give you a Markov process on the boundary that locally looks like a Cauchy process. (In the case of $D$ a half-plane it would precisely be a Cauchy process.) $\endgroup$
    – Martin Hairer
    Commented Jul 29, 2020 at 19:35
  • $\begingroup$ Thank you, that answers it. Fixed the error you mentioned. $\endgroup$
    – alesia
    Commented Jul 29, 2020 at 20:06
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    $\begingroup$ This has been studied since 1960s, by Molchanov, Hsu, Kolsrud and others. If you like exact references, you may have a look at page 4 of my recent preprint at arXiv:1912.00072. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 29, 2020 at 21:26

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