Alexandroff-Bakelman-Pucci estimate and first exit time in cylindrical domains

Antonio Vitolo; Antonio Vitolo

doi:10.3934/mbe.2026070

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 7: 1908-1925. doi: 10.3934/mbe.2026070

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Alexandroff-Bakelman-Pucci estimate and first exit time in cylindrical domains

Antonio Vitolo ^,

Department of Mathematics, Università di Salerno, via Giovanni Paolo Ⅱ, 132 - 84084 Fisciano (SA), Italy

Received: 09 March 2026 Revised: 09 June 2026 Accepted: 18 June 2026 Published: 01 July 2026

This paper is concerned with elliptic equations satisfied by the mean first exit time of a stochastic process and higher moments. We prove an estimate for all moments of the first exit time in a cylindrical domain, using viscosity solutions. As a further result, we show the existence and the uniqueness of a more general fully nonlinear Dirichlet problem for partial trace equations, which can be associated to nonlinear diffusion.
- fully nonlinear elliptic equations,
- stochastic processes,
- first exit time,
- ABP estimates,
- unbounded domains
Citation: Antonio Vitolo. Alexandroff-Bakelman-Pucci estimate and first exit time in cylindrical domains[J]. Mathematical Biosciences and Engineering, 2026, 23(7): 1908-1925. doi: 10.3934/mbe.2026070

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Abstract

This paper is concerned with elliptic equations satisfied by the mean first exit time of a stochastic process and higher moments. We prove an estimate for all moments of the first exit time in a cylindrical domain, using viscosity solutions. As a further result, we show the existence and the uniqueness of a more general fully nonlinear Dirichlet problem for partial trace equations, which can be associated to nonlinear diffusion.

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