The link between the Brownian motion and the Laplace operator is well-known.

What stochastic process plays an analogous role with respect to the $p$-Laplace operator?

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    $\begingroup$ It's an interesting question. One piece of the question would be in what sense the role is "analogous". There's a standard notion of the generator of a Markov process, but this always yields a linear operator. $\endgroup$ – Nate Eldredge Feb 25 at 0:36
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    $\begingroup$ The following reference by Peres and Sheffield discusses a counterpart of Brownian motion for the p-Laplacian projecteuclid.org/euclid.dmj/1221656864 $\endgroup$ – Nawaf Bou-Rabee Feb 25 at 1:07

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