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 existence of viscosity solutions to a parabolic problem of the form $$ {cases} K_{(x,t)}(D u)u_t (x,t)= \frac12 <D^2 u J_{(x,t)}(D u),J_{(x,t)}(D u) (x,t) &{in} Ω_T, u(x,t)=F(x)&{on}Γ, {cases} $$ 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, $u_t(x,t)=<D^{2}u(x,t)\frac{Du}{|Du|}(x,t),\frac{Du}{|Du|}(x,t)>$.

16 pages

References in corpus (2)

• Tug-of-war and the infinity Laplacian
• An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions

Cited by in corpus (2)

• The evolution problem associated with eigenvalues of the Hessian
• Time-dependent tug-of-war games and normalized parabolic $p$-Laplace equations