Annals of Probability

Annealed Brownian motion in a heavy tailed Poissonian potential

Ryoki Fukushima

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Consider a $d$-dimensional Brownian motion in a random potential defined by attaching a nonnegative and polynomially decaying potential around Poisson points. We introduce a repulsive interaction between the Brownian path and the Poisson points by weighting the measure by the Feynman–Kac functional. We show that under the weighted measure, the Brownian motion tends to localize around the origin. We also determine the scaling limit of the path and also the limit shape of the random potential.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3462-3493.

First available in Project Euclid: 12 September 2013

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Brownian motion random media Poissonian potential localization


Fukushima, Ryoki. Annealed Brownian motion in a heavy tailed Poissonian potential. Ann. Probab. 41 (2013), no. 5, 3462--3493. doi:10.1214/12-AOP754.

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  • [1] Bolthausen, E. (1994). Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22 875–918.
  • [2] Carmona, R. and Lacroix, J. (1990). Spectral Theory of Random Schrödinger Operators. Birkhäuser, Boston, MA.
  • [3] Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations. Pure and Applied Mathematics 137. Academic Press, Boston, MA.
  • [4] Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 525–565.
  • [5] Fukushima, R. (2011). Second order asymptotics for Brownian motion in a heavy tailed Poissonian potential. Markov Process. Related Fields 17 447–482.
  • [6] Gärtner, J. and König, W. (2000). Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10 192–217.
  • [7] Gärtner, J., König, W. and Molchanov, S. A. (2000). Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Related Fields 118 547–573.
  • [8] Gärtner, J. and Molchanov, S. A. (2000). Moment asymptotics and Lifshitz tails for the parabolic Anderson model. In Stochastic Models (Ottawa, ON, 1998). CMS Conference Proceedings 26 141–157. Amer. Math. Soc., Providence, RI.
  • [9] Grüninger, G. and König, W. (2009). Potential confinement property of the parabolic Anderson model. Ann. Inst. H. Poincaré Probab. Stat. 45 840–863.
  • [10] Pastur, L. A. (1977). The behavior of certain Wiener integrals as $t\rightarrow\infty $ and the density of states of Schrödinger equations with random potential. Teoret. Mat. Fiz. 32 88–95.
  • [11] Povel, T. (1999). Confinement of Brownian motion among Poissonian obstacles in ${\mathbf{R}}^{d}$, $d\ge3$. Probab. Theory Related Fields 114 177–205.
  • [12] Sznitman, A.-S. (1991). On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Comm. Pure Appl. Math. 44 1137–1170.
  • [13] Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.