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Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation. I: Lower bound on the volume exponent. (English. French summary) Zbl 1267.82146

Typical trajectories of the Brownian motion are those killed with a homogeneous rate and conditioned to survive till hitting a distant hyperplane. They stay in a tube centered along the direction orthogonal to this hyperplane of diameter \(\sqrt{L}\), whereby \(L\) is the distance between the source of the trajectories and the hyperplane. Adding an inhomogeneous term to the killing rate makes the transversal fluctuation of the trajectories superdiffusive, e.g., of amplitude \(L^{\xi }\) for some \(\xi \in (1/2, 1)\) (\(\xi \) is called the volume exponent). The pertinent inhomogeneity stems from a random (Poissonian) potential \(V\) representing a field of traps whose center locations are given by a Poisson point process, while their radii are distributed according to a common (heavy-tailed) distribution of an unbounded support. The presence of long-range spatial correlations is a crucial difference when compared with Poissonian models studied so far, c.f. a summary of that topic in [A. S. Sznitman, Brownian motion, obstacles and random media. Berlin: Springer (1998; Zbl 0973.60003)].
The present analysis is focused on identifying a lower bound for the volume exponent \(\xi \).

MSC:

82D60 Statistical mechanics of polymers
60K37 Processes in random environments
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics

Citations:

Zbl 0973.60003
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References:

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