Exponential asymptotics for time–space hamiltonians

Chen, Xia ; Hu, Yaozhong ; Song, Jian ; Xing, Fei

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1529-1561.

Abstract (VO)

In this paper, we investigate the long time asymptotics of the exponential moment for the following time–space Hamiltonian

$${\int }_0^t{\int }_0^t\frac{1}{{|r-s|}^{{\alpha }_0}}\gamma (B_r-B_s)\mathrm{d}s\mathrm{d}r,t\ge 0,$$

where $(B_s,s\ge 0)$ is a $d$-dimensional Brownian motion, the kernel $\gamma (·):{ℝ}^d\to [0,\infty )$ is a homogeneous function with singularity at zero; and ${\alpha }_0\in (0,1)$ together with the scaling parameter of $\gamma$ satisfies certain conditions. Our work is partially motivated by the studies of the short-range sample-path intersection, the strong coupling polaron, and the parabolic Anderson models with a time–space fractional white noise potential.

Résumé

Dans ce papier, nous étudions le comportement en temps long du moment exponentiel du Hamiltonien dépendant du temps

$${\int }_0^t{\int }_0^t\frac{1}{{|r-s|}^{{\alpha }_0}}\gamma (B_r-B_s)\mathrm{d}s\mathrm{d}r,t\ge 0,$$

où $(B_s,s\ge 0)$ est un mouvement brownien de dimension $d$, le noyau $\gamma (·):{ℝ}^d\to [0,\infty )$ est une fonction homogène avec une singularité en zéro, ${\alpha }_0\in (0,1)$ et le paramètre de scaling $\gamma$ satisfont certaines conditions. Notre travail est partiellement motivé par l’étude des intersections ą courte portée de trajectoires, le polaron avec couplage fort et le modèle parabolique d’Anderson avec un potentiel donné par un bruit blanc fractionnaire en espace–temps.

Détail

DOI : 10.1214/13-AIHP588

Keywords: Time–space hamiltonian, brownian motion, Feynman–Kac large deviations

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