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Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions

Alexander Quaas, Erwin Topp

TL;DR

This work establishes the existence and qualitative structure of ergodic pairs for a fractional Hamilton–Jacobi equation on a bounded domain with censored diffusion. Using the vanishing discount method applied to the discounted problem $\lambda u + (-\Delta)_\Omega^s u + |Du|^m = f$, the authors obtain a minimal large solution and derive precise blow-up rates near the boundary via the distance function through $g_\gamma$. They prove the existence of an ergodic pair $(u,c)$ with $(-\Delta)_\Omega^s u + |Du|^m = f - c$ in $\Omega$ and $u(x)\to\infty$ as $x\to\partial\Omega$, with $u\in C^{2s+\alpha}(\Omega)$ and $u$ in the $\gamma$-class, and they characterize the ergodic constant $c$ as a critical value. The results reveal partial uniqueness in the $\gamma$-class and minimality up to additive constants, while addressing the nonlocal state-dependent operator and boundary blow-up. The findings contribute to understanding large-time behavior and stochastic control with state constraints in nonlocal diffusion settings, and they lay groundwork for further uniqueness and Liouville-type results for censored operators.

Abstract

In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain $Ω\subset \mathbb{R}^N$. We restrict ourselves to the case in which the nonlinear gradient term has a scaling less or equal than the fractional order of the diffusion. In similarity to its second-order counterpart, we provide existence of ergodic pairs involving solutions that blow-up on $\partial Ω$. We use the celebrated vanishing discount method, where the analysis of the approximated solutions have its own interest, leading to qualitative properties for the ergodic problem such as precise blow-up rates for the solution and characterization of the ergodic constant. The main difficulties arise from the state-dependency of the operator, from which the arguments of the local case based on well-known invariance properties of the Laplacian are not longer at disposal.

Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions

TL;DR

This work establishes the existence and qualitative structure of ergodic pairs for a fractional Hamilton–Jacobi equation on a bounded domain with censored diffusion. Using the vanishing discount method applied to the discounted problem , the authors obtain a minimal large solution and derive precise blow-up rates near the boundary via the distance function through . They prove the existence of an ergodic pair with in and as , with and in the -class, and they characterize the ergodic constant as a critical value. The results reveal partial uniqueness in the -class and minimality up to additive constants, while addressing the nonlocal state-dependent operator and boundary blow-up. The findings contribute to understanding large-time behavior and stochastic control with state constraints in nonlocal diffusion settings, and they lay groundwork for further uniqueness and Liouville-type results for censored operators.

Abstract

In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain . We restrict ourselves to the case in which the nonlinear gradient term has a scaling less or equal than the fractional order of the diffusion. In similarity to its second-order counterpart, we provide existence of ergodic pairs involving solutions that blow-up on . We use the celebrated vanishing discount method, where the analysis of the approximated solutions have its own interest, leading to qualitative properties for the ergodic problem such as precise blow-up rates for the solution and characterization of the ergodic constant. The main difficulties arise from the state-dependency of the operator, from which the arguments of the local case based on well-known invariance properties of the Laplacian are not longer at disposal.
Paper Structure (11 sections, 17 theorems, 207 equations)

This paper contains 11 sections, 17 theorems, 207 equations.

Table of Contents

  1. Introduction.
  2. Literature review and further discussion.
  3. Technical lemmas
  4. Barriers.
  5. Regularity estimates.
  6. The discounted problem.
  7. On the regime $s \leq 1/2$.
  8. The ergodic problem
  9. Qualitative properties of the ergodic problem
  10. Characterization of the ergodic constant.
  11. On uniqueness.

Key Result

Theorem 1.1

Let $s \in (1/2, 1)$ and $s + 1/2 < m \leq 2s$. Then, for each $\lambda > 0$ and $f \in C(\Omega)$ bounded from below such that there exists a viscosity solution $u \in C^{1, \alpha}(\Omega)$ to problem eq-blow-up. It is the minimal large solution to eq in the sense that for every $v \in C(\Omega)$ viscosity solution to eq-blow-up, then $u \leq v$ in $\Omega$. It is also the unique solution in th

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 24 more