Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models. (English) Zbl 1259.60094

Chen, Xia

Ann. Probab. 40, No. 4, 1436-1482 (2012).

Author’s abstract: Let \(B_s\) be a \(d\)-dimensional Brownian motion and \(\omega(dx)\) be an independent Poisson field on \(\mathbb{R}^d\). The almost sure asymptotics for the logarithmic moment generating function \[ \log\operatorname{E}_0\exp\left\{\pm\theta\int_0^t \overline V(B_s)ds\right\}\qquad (t\to\infty) \] are investigated in connection with the renormalized Poisson potential of the form \[ \overline V(x)=\int_{\mathbb{R}^d}\frac{1}{|y-x|^p}[\omega(dy)-dy],\qquad x\in\mathbb{R}^d. \] The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.

Reviewer: Josef Steinebach (Köln)

MSC:

60J65	Brownian motion
60K37	Processes in random environments
60K40	Other physical applications of random processes
60G55	Point processes (e.g., Poisson, Cox, Hawkes processes)
60F10	Large deviations

Keywords:

renormalization; Poisson field; Brownian motion in Poisson potential; parabolic Anderson model; Feynman-Kac representation; large deviations

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