Asymptotic mean value formulas, nonlocal space-time parabolic operators and anomalous tug-of-war games

TL;DR

This paper develops asymptotic mean value formulas for fractional space-time operators and introduces novel nonlocal, nonlinear parabolic operators inspired by tug-of-war games, modeling complex anomalous systems.

Contribution

It provides new mean value representation formulas for fractional operators and introduces nonlocal nonlinear parabolic operators related to tug-of-war, expanding the mathematical framework for anomalous processes.

Findings

Established asymptotic mean value formulas for fractional heat operators.

Introduced new nonlocal, nonlinear parabolic operators linked to tug-of-war models.

Connected these operators to nonlocal versions of the infinity Laplace operator.

Abstract

The fractional heat operator $(\partial_t-\Delta_x)^s$ and Continuous Time Random Walks (CTRWs) are interesting and sophisticated mathematical models that can describe complex anomalous systems. In this paper, we prove asymptotic mean value representation formulas for functions with respect to $(\partial_t-\Delta_x)^s$ and we introduce new nonlocal, nonlinear parabolic operators related to a tug-of-war which accounts for waiting times and space-time couplings. These nonlocal, nonlinear parabolic operators and equations can be seen as nonlocal versions of the evolutionary infinity Laplace operator.