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The Wiener sausage

Let $ \beta(t)$, $ t \geqslant 0$, be the standard Brownian motion in $ {\mathbb{R}}^d$ (the Markov process with generator $ \Delta/2$) starting at 0. Let $ P,E$ denote its probability law and expectation on path space. The Wiener sausage with radius $ a>0$ is the process defined by

$ W^a(t) = \bigcup_{0 \leqslant s \leqslant t} B_a(\beta(s)), ~~~t \geqslant 0, $ (1)

where $ B_a(x)$ is the open ball with radius $ a$ around $ x \in {\mathbb{R}}^d$, i.e., $ W^a(t)$ is the $ a$-environment of the Brownian path up to time $ t$. The Wiener sausage is an important mathematical object, because it is one of the simplest examples of a non-Markovian functional of Brownian motion. It plays a key role in the study of various stochastic phenomena, such as heat conduction and trapping in random media, as well as in the analysis of spectral properties of random Schrödinger operators.