Summary
We consider the generating function\(\mathbb{E}\exp (\lambda |C_\varepsilon (t)|)\) of the voltime of the Wiener sausageC ∈ (t), which is the ε-neighbourhood of the Wiener path in the time interval [0,t]. For γ<0, the limiting behavior fort→∞, up to logarithmic equivalence, had been determined in a celebrated work of Donsker and Varadhan. For γ>0 it had been investigated by van den Berg and Tóth, but in contrast to the case γ<0, there is no simple expression for the exponential rate known. We determine the asymptotic behaviour of this rate for small and large γ.
Article PDF
Similar content being viewed by others
References
van den Berg, M., Tóth, B.: Expontetial estimates for the Wiener sausage. Probab. Theory Relat. Fields88, 249–259 (1991)
Spitzer, F.: Electrostatic capacity, heat flow and Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.3, 110–121 (1964)
Sznitman, A.S.: Some bounds and limiting results for the measure of Wiener sausage of small radius associated with elliptic diffusions. Stochastic Processes Appl.25, 1–25 (1987)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
van den Berg, M., Bolthausen, E. Asymptotics of the generating function for the volume of the Wiener sausage. Probab. Th. Rel. Fields 99, 389–397 (1994). https://doi.org/10.1007/BF01199898
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01199898