Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Exponential asymptotics for time–space Hamiltonians

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

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

\[\int_{0}^{t}\int_{0}^{t}{\frac{1}{\vert r-s\vert^{\alpha_{0}}}}\gamma (B_{r}-B_{s})\,\mathrm{d}s\,\mathrm{d}r,\quad t\ge0,\] where $(B_{s},s\ge0)$ is a $d$-dimensional Brownian motion, the kernel $\gamma(\cdot):\mathbb{R}^{d}\rightarrow [0,\infty)$ 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.


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

\[\int_{0}^{t}\int_{0}^{t}{\frac{1}{\vert r-s\vert^{\alpha_{0}}}}\gamma(B_{r}-B_{s})\,\mathrm{d}s\,\mathrm{d}r,\quad t\ge0,\] où $(B_{s},s\ge0)$ est un mouvement brownien de dimension $d$, le noyau $\gamma(\cdot):\mathbb{R}^{d}\rightarrow [0,\infty)$ 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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1529-1561.

Received: 17 November 2012
Revised: 19 September 2013
Accepted: 22 September 2013
First available in Project Euclid: 21 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60K37: Processes in random environments 60K40: Other physical applications of random processes 60G55: Point processes 60F10: Large deviations

Time–space Hamiltonian Brownian motion Feynman–Kac large deviations


Chen, Xia; Hu, Yaozhong; Song, Jian; Xing, Fei. Exponential asymptotics for time–space Hamiltonians. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1529--1561. doi:10.1214/13-AIHP588.

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