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 {K(x,t)(Du)ut(x,t)=12⟨D2uJ(x,t)(Du),J(x,t)(Du)(x,t)⟩u(x,t)=F(x)in ΩT,on Γ, where ΩT=Ω×(0,T]ΩT=Ω×(0,T] and ΓΓ 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, ut(x,t)=⟨D2u(x,t)Du|Du|(x,t),Du|Du|(x,t)⟩.