An evolution problem associated with the Infinity Fractional Laplacian

The Infinity fractional Laplacian, denoted ∆⁽ˢ⁾₍∞₎, is a fully non‑linear, degenerate integro‑differential operator derived from the study of a non‑local Tug‑of‑War game; this operator is the main subject of this work. In the frame of viscosity solutions, we study existence, uniqueness and regularity of solutions to a parabolic problem associated with the operator; given the discontinuous behavior of ∆⁽ˢ⁾₍∞₎ϕ(x) at points where ∇ϕ(x)=0, a stability result is proved, however, it is necessary to ensure that our candidate solution, given by Perron’s Method, has non‑zero gradient in order to apply it. This is done on two different domains, namely an infinite strip and an annular domain; these domains share the property that their complement is made up of two disjoint components, so we are able to consider the same data in both settings and establish growth and decay rates that induce strict monotonicity on our solution.