The Leibenson process

TL;DR

This paper links the Leibenson equation to a nonlinear Markov process by representing its solutions through a novel McKean--Vlasov SDE with complex coefficients, establishing a probabilistic framework for this generalized PDE.

Contribution

It introduces a probabilistic representation of the Leibenson equation as a nonlinear Markov process via a new McKean--Vlasov SDE with coefficients depending on solution derivatives.

Findings

Probabilistic representation of Leibenson solutions as Markov process

Construction of a novel McKean--Vlasov SDE with derivative-dependent coefficients

Proof of strong solutions despite degeneracy and irregularity

Abstract

Consider the Leibenson equation \begin{equation*} \partial_t u = \Delta_p u^q, \end{equation*} where $\Delta_p f = div(|\nabla f|^{p-2}\nabla f)$ for $p>1$ and $q>0$, which is a simultaneous generalization of the porous media and the $p$-Laplace equation. In this paper we identify the Leibenson equation as a nonlinear Fokker--Planck equation and prove that it has a nonlinear Markov process in the sense of McKean as its probabilistic counterpart. More precisely, we obtain a probabilistic representation of its Barenblatt solutions as the one-dimensional marginal density curve of the unique solutions to the associated McKean--Vlasov SDE. The latter is of novel type, since its coefficients depend pointwise both on its solution's time marginal densities and also on their first and second order derivatives. Moreover, we show that these solutions constitute the aforementioned nonlinear Markov process, which we call the Leibenson process. A further main result of this work is to prove that despite the strong degeneracy of the diffusion and the irregularity of the drift coefficient (which is merely of bounded variation) of the McKean--Vlasov SDE these solutions are probabilistically strong, i.e., measurable functionals of the driving Brownian motion and the initial condition.