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.
Publié le : 2016-07-12
Classification:
Large deviations,
Piecewise Deterministic Markov Processes,
Hamilton-Jacobi equations,
Viscosity solutions,
Scaling limits,
Front acceleration,
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP],
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
@article{hal-01344939,
author = {Bouin, Emeric and Calvez, Vincent and Grenier, Emmanuel and Nadin, Gr\'egoire},
title = {Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations},
journal = {HAL},
volume = {2016},
number = {0},
year = {2016},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-01344939}
}
Bouin, Emeric; Calvez, Vincent; Grenier, Emmanuel; Nadin, Grégoire. Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations. HAL, Tome 2016 (2016) no. 0, . http://gdmltest.u-ga.fr/item/hal-01344939/