Abstract
We consider the Wiener sausage among Poissonian obstacles. The obstacle is called hard if Brownian motion entering the obstacle is immediately killed, and is called soft if it is killed at certain rate. It is known that Brownian motion conditioned to survive among obstacles is confined in a ball near its starting point. We show the weak law of large numbers, large deviation principle in special cases and the moment asymptotics for the volume of the corresponding Wiener sausage. One of the consequence of our results is that the trajectory of Brownian motion almost fills the confinement ball.
Similar content being viewed by others
References
Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics (New York), vol. 38. Springer, New York (1998)
Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 28(4), 525–565 (1975)
Povel, T.: Confinement of Brownian motion among Poissonian obstacles in R d, d≥3. Probab. Theory Relat. Fields 114(2), 177–205 (1999)
Sznitman, A.-S.: On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Commun. Pure Appl. Math. 44(8–9), 1137–1170 (1991)
Sznitman, A.-S.: Annealed Lyapounov exponents and large deviations in a Poissonian potential. I, II. Ann. Sci. Éc. Norm. Super. (4) 28(3), 345–370, 371–390 (1995)
Sznitman, A.-S.: Brownian Motion, Obstacles and Random Media. Springer Monographs in Mathematics. Springer, Berlin (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fukushima, R. Asymptotics for the Wiener Sausage among Poissonian Obstacles. J Stat Phys 133, 639–657 (2008). https://doi.org/10.1007/s10955-008-9629-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9629-5