| Abstract: |
| We construct a probabilistic counterpart for the Barenblatt solutions of the doubly nonlinear Leibenson equation
\begin{equation*}
\partial_t u(t,x) = \Delta_p u^q(t,x),\quad (t,x) \in (0,\infty) \times \mathbb{R}^d,
\end{equation*}
where $\Delta_p f = \divv(|\nabla f|^{p-2}\nabla f)$ denotes the $p$-Laplace operator, $p>2 and $q>0$. This counterpart is a family of stochastic processes with one-dimensional time marginal densities given by the Barenblatt solutions. These processes are constructed as weak solutions to a McKean--Vlasov equation whose coefficients depend on the solution law pointwise via the gradient of its density. To this end, we first identify the Leibenson equation as a nonlinear Fokker--Planck equation. Moreover, we prove that these processes form a unique nonlinear Markov process with Barenblatt time marginals, which we call the Leibenson process. Joint work with Viorel Barbu, Sebastian Grube and Michael R\ockner. |
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