AN INHOMOGENEOUS EVOLUTION EQUATION INVOLVING THE NORMALIZED INFINITY LAPLACIAN WITH A TRANSPORT TERM

In this paper, we prove the uniqueness and stability of viscosity solutions of the following initial-boundary problem related to the random game named tug-of-war with a transport term { u(t) -Delta(N)(infinity) u - <xi, Du > = f (x, t), in QT, u = g, on partial derivative(p)Q(T,) where -Delta(N)(infinity) u = 1/vertical bar Du vertical bar(2) Sigma(n)(i,j=) ux(i) ux(j) ux(i)x(j) denotes the normalized infinity Laplacian, xi : Q(T) -> Rn is a continuous vector field, f and g are continuous. When xi is a fixed field and the inhomogeneous term f is constant, the existence is obtained by the approximate procedure. When f and xi are Lipschitz continuous, we also establish the Lipschitz continuity of the viscosity solutions. Furthermore we establish the comparison principle of the generalized equation with the first order term with initial-boundary condition u(t) (x, t) -Delta(N)(infinity) u (x, t) - H (x, t,Du (x, t)) = f (x, t), where H (x, t, p) : Q(T) x R-n -> R is continuous, H (x, t, 0) = 0 and grows at most linearly at in fi nity with respect to the variable p. And the existence result is also obtained when H (x, t, p) = H (p) and f is constant for the generalized equation.