A continuous time tug-of-war game for parabolic $p(x,t)$-Laplace type equations

TL;DR

This paper introduces a stochastic differential game that characterizes the unique viscosity solution of a parabolic PDE involving the normalized p(x,t)-Laplace operator, covering the full range of p-values.

Contribution

It formulates a continuous-time game for the parabolic p(x,t)-Laplace equation and proves the uniqueness of viscosity solutions in the entire space.

Findings

The game represents the unique viscosity solution to the PDE.

The formulation covers the full range 1<p(x,t)<∞.

Uniqueness of solutions is established under suitable assumptions.

Abstract

We formulate a stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized $p(x,t)$-Laplace operator. Our game is formulated in a way that covers the full range $1<p(x,t)<\infty$. Furthermore, we prove the uniqueness of viscosity solutions to our equation in the whole space under suitable assumptions.