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arXiv · 2504.11789

The vanishing discount problem for nonlocal Hamilton-Jacobi equations

Andrea Davini·Hitoshi Ishii
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Abstract

We establish a convergence result for the vanishing discount problem in the context of nonlocal HJ equations. We consider a fairly general class of discounted first-order and convex HJ equations which incorporate an integro-differential operator posed on the $d$-dimensional torus, and we show that the solutions converge to a specific critical solution as the discount factor tends to zero. Our approach relies on duality techniques for nonlocal convex HJ equations, building upon Hahn-Banach separation theorems to develop a generalized notion of Mather measure. The results are applied to a specific class of convex and superlinear Hamiltonians.

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Andrea Davini, Hitoshi Ishii. 2025-04-16. The vanishing discount problem for nonlocal Hamilton-Jacobi equations. https://arxiv.org/abs/2504.11789

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