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Nonlocal ergodic control problem in \({\mathbb {R}}^d\)

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Abstract

We study the existence–uniqueness of solution \((u, \lambda )\) to the ergodic Hamilton–Jacobi equation

$$\begin{aligned} (-\Delta )^s u + H(x, \triangledown u) = f-\lambda \quad \text {in}\; {\mathbb {R}}^d, \end{aligned}$$

and \(u\ge 0\), where \(s\in (\frac{1}{2}, 1)\). We show that the critical \(\lambda =\lambda ^*\), defined as the infimum of all \(\lambda \) attaining a non-negative supersolution, attains a nonnegative solution u. Under suitable conditions, it is also shown that \(\lambda ^*\) is the supremum of all \(\lambda \) for which a non-positive subsolution is possible. Moreover, uniqueness of the solution u, corresponding to \(\lambda ^*\), is also established. Furthermore, we provide a probabilistic characterization that determines the uniqueness of the pair \((u, \lambda ^*)\) in the class of all solution pair \((u, \lambda )\) with \(u\ge 0\). Our proof technique involves both analytic and probabilistic methods in combination with a new local Lipschitz estimate obtained in this article.

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Acknowledgements

This research of Anup Biswas was supported in part by a SwarnaJayanti fellowship SB/SJF/2020-21/03. Erwin Topp was partially supported by Fondecyt Grant no. 1201897.

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Authors and Affiliations

  1. Indian Institute of Science Education and Research-Pune, Dr. Homi Bhabha Road, Pashan, Pune, 411008, India

    Anup Biswas

  2. Instituto de Matemáticas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21941-909, Brazil

    Erwin Topp

  3. Departmento de Matemáticas y Ciencias de la Computación, Universidad de Santiago de Chile, Casilla 307, Santiago, Chile

    Erwin Topp

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  1. Anup Biswas
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Biswas, A., Topp, E. Nonlocal ergodic control problem in \({\mathbb {R}}^d\). Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02760-1

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