## Differential and Integral Equations

### Tug-of-War games and parabolic problems with spatial and time dependence

#### Abstract

In this paper, we use probabilistic arguments (Tug-of-War games) to obtain the existence of viscosity solutions to a parabolic problem of the form $$\begin{cases} K_{(x,t)}(D u )u_t (x,t)= \tfrac12 \langle D^2 u J_{(x,t)}(D u ),J_{(x,t)}(D u) (x,t)\rangle & \mbox{in } \Omega_T,\\ u(x,t)=F(x) & \mbox{on }\Gamma, \end{cases}$$ where $\Omega_T=\Omega\times(0,T]$ and $\Gamma$ is its parabolic boundary. This problem can be viewed as a version with spatial and time dependence of the evolution problem given by the infinity Laplacian, $$u_t (x,t)= \langle D^2 u (x,t) \frac{D u}{|Du|} (x,t),\, \frac{D u}{|Du|} (x,t)\rangle\, .$$

#### Article information

Source
Differential Integral Equations, Volume 27, Number 3/4 (2014), 269-288.

Dates
First available in Project Euclid: 30 January 2014