Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations
Bouin, Emeric; Calvez, Vincent; Grenier, Emmanuel; Nadin, Grégoire (2016), Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations. https://basepub.dauphine.fr/handle/123456789/17208
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-01344939
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
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Abstract (EN)We establish a large deviation theory for a velocity jump process, where new random velocities are picked at a constant rate from a Gaussian distribution. The Kolmogorov forward equation associated with this process is a linear kinetic transport equation in which the BGK operator accounts for the changes in velocity. We analyse its asymptotic limit after a suitable rescaling compatible with the WKB expansion. This yields a new type of Hamilton Jacobi equation which is non local with respect to velocity variable. We introduce a dedicated notion of viscosity solution for the limit problem, and we prove well-posedness in the viscosity sense. The fundamental solution is explicitly computed, yielding quantitative estimates for the large deviations of the underlying velocity-jump process à la Freidlin-Wentzell. As an application of this theory, we conjecture exact rates of acceleration in some nonlinear kinetic reaction-transport equations.
Subjects / KeywordsLarge deviations; Piecewise Deterministic Markov Processes; Hamilton-Jacobi equations; Viscosity solutions; Scaling limits; Front acceleration
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