Nous nous intéressons à un processus aléatoire sur qui alterne des phases de mouvements rectilignes uniformes et change de vitesse à des temps exponentiels. Nous étudions plus précisément l'équation de Kolmogorov après rééchelonnement hyperbolique , , puis nous effectuons une transformée de Hopf–Cole qui nous donne une équation cinétique suivie par un potentiel. Nous montrons la convergence pour de ce potentiel vers la solution de viscosités d'une équation de Hamilton–Jacobi où le hamiltonien peut présenter une singularité , ce qui est assez inédit dans ce type d'études.
We study a random process on moving in straight lines and changing randomly its velocity at random exponential times. We focus more precisely on the Kolmogorov equation in the hyperbolic scale , with , before proceeding to a Hopf–Cole transform, which gives a kinetic equation on a potential. We show convergence as of the potential towards the viscosity solution to a Hamilton–Jacobi equation where the Hamiltonian may lack regularity, which is quite unseen in this type of studies.
Accepté le :
Publié le :
@article{CRMATH_2017__355_2_170_0, author = {Caillerie, Nils}, title = {Large deviations of a velocity jump process with a {Hamilton{\textendash}Jacobi} approach}, journal = {Comptes Rendus. Math\'ematique}, pages = {170--175}, publisher = {Elsevier}, volume = {355}, number = {2}, year = {2017}, doi = {10.1016/j.crma.2016.12.011}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2016.12.011/} }
TY - JOUR AU - Caillerie, Nils TI - Large deviations of a velocity jump process with a Hamilton–Jacobi approach JO - Comptes Rendus. Mathématique PY - 2017 SP - 170 EP - 175 VL - 355 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.12.011/ DO - 10.1016/j.crma.2016.12.011 LA - en ID - CRMATH_2017__355_2_170_0 ER -
%0 Journal Article %A Caillerie, Nils %T Large deviations of a velocity jump process with a Hamilton–Jacobi approach %J Comptes Rendus. Mathématique %D 2017 %P 170-175 %V 355 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.12.011/ %R 10.1016/j.crma.2016.12.011 %G en %F CRMATH_2017__355_2_170_0
Caillerie, Nils. Large deviations of a velocity jump process with a Hamilton–Jacobi approach. Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 170-175. doi : 10.1016/j.crma.2016.12.011. http://www.numdam.org/articles/10.1016/j.crma.2016.12.011/
[1] A Hamilton–Jacobi approach for front propagation in kinetic equations, Kinet. Relat. Models, Volume 8 (2015) no. 2, pp. 255-280
[2] A kinetic eikonal equation, C. R. Acad. Sci. Paris, Ser. I, Volume 350 (2012) no. 5–6, pp. 243-248
[3] P. Bressloff, O. Faugeras, On the Hamiltonian structure of large deviations in stochastic hybrid systems, 2015, hal-01072077v2.
[4] Singular measure as principal eigenfunction of some nonlocal operators, Appl. Math. Lett., Volume 26 (2013) no. 8, pp. 831-835
[5] Viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc., Volume 277 (1983), pp. 1-42
[6] Some properties of viscosity solutions of Hamilton–Jacobi equations, Trans. Amer. Math. Soc., Volume 282 (1984), pp. 487-502
[7] The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. R. Soc. Edinb., Sect. A, Volume 111 (1989), pp. 359-375
[8] Averaging and large deviations principles for fully piecewise deterministic Markov process and applications to molecular motors, Markov Process. Relat. Fields, Volume 16 (2010), pp. 497-548
[9] Random Perturbations of Dynamical Systems, Grundlehren der Mathematischen Wissenschaften, vol. 260, Springer-Verlag, New York, 1998
Cité par Sources :