A field-theoretic approach to the Wiener Sausage

TL;DR

This paper develops a field-theoretic perturbation approach to analyze the Wiener Sausage, providing exact exponents and amplitudes, and demonstrating the method's versatility for complex boundary conditions and correlation functions.

Contribution

It introduces a single loop renormalised perturbation theory for the Wiener Sausage, offering a new, elegant framework for studying this classical problem in stochastic processes.

Findings

Exact exponents and some amplitudes obtained

Versatile framework for different boundary conditions

Illustrates technical aspects of field-theoretic stochastic analysis

Abstract

The Wiener Sausage, the volume traced out by a sphere attached to a Brownian particle, is a classical problem in statistics and mathematical physics. Initially motivated by a range of field-theoretic, technical questions, we present a single loop renormalised perturbation theory of a stochastic process closely related to the Wiener Sausage, which, however, proves to be exact for the exponents and some amplitudes. The field-theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem. While we recover a number of known, classical results, the field-theoretic techniques deployed provide a particularly versatile framework, which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions. At the same time, we provide a highly instructive, non-trivial example for some of the technical particularities of the field-theoretic description of stochastic processes, such as excluded volume, lack of translational invariance and immobile particles. The aim of the present work is not to improve upon the well-established results for the Wiener Sausage, but to provide a field-theoretic approach to it, in order to gain a better understanding of the field-theoretic obstacles to overcome.