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

TL;DR

This paper investigates the long-term behavior of solutions to periodic, fully nonlinear integro-differential equations, establishing convergence to ergodic solutions and addressing challenges posed by mixed operators and superlinear growth.

Contribution

It constructs solutions to the ergodic problem for complex integro-differential equations and proves their solutions approximate the long-term behavior of the Cauchy problem.

Findings

Solutions converge to ergodic solutions as time approaches infinity

Established existence of solutions to the ergodic (cell) problem for mixed operators

Extended analysis to superlinear gradient cases

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.