Large Time Behavior of Periodic Viscosity Solutions for Uniformly Elliptic Integro-Differential Equations

Abstract

In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t+v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of "mixed operators" for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.

References in corpus (4)

• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited
• On the Boundary Ergodic Problem for Fully Nonlinear Equations in Bounded Domains with General Nonlinear Neumann Boundary Conditions
• On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations Associated with Nonlinear Boundary Conditions
• A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise

Cited by in corpus (6)

• Lipschitz regularity for integro-differential equations with coercive hamiltonians and application to large time behavior
• Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion
• On stationary fractional Mean Field Games
• Periodic Homogenization of a Lévy-Type Process with Small Jumps
• Qualitative properties of positive solutions for mixed integro-differential equations
• On Neumann problems for nonlocal Hamilton-Jacobi equations with dominating gradient terms