Local regularity for time-dependent tug-of-war games with varying probabilities

TL;DR

This paper investigates the local regularity of value functions in time-dependent tug-of-war games, establishing Lipschitz continuity for constant probabilities and H"older and Harnack estimates for variable probabilities, linking to a nonlinear parabolic PDE.

Contribution

It provides new regularity results for value functions in time-dependent tug-of-war games with variable probabilities, connecting game theory to nonlinear PDE analysis.

Findings

Lipschitz continuity for constant probability games

H"older and Harnack estimates for variable probability games

Connection to normalized p(x,t)-parabolic equations

Abstract

We study local regularity properties of value functions of time-dependent tug-of-war games. For games with constant probabilities we get local Lipschitz continuity. For more general games with probabilities depending on space and time we obtain H\"older and Harnack estimates. The games have a connection to the normalized $p(x,t)$-parabolic equation $(n+p(x,t))u_t=\Delta u+(p(x,t)-2) \Delta_{\infty}^N u$.