Functional integral approach for multiplicative stochastic processes

TL;DR

This paper develops a functional integral formalism for multiplicative stochastic processes, enabling the derivation of correlation functions without discretization and revealing the role of supersymmetry in uniquely defining the process.

Contribution

It introduces a path integral approach incorporating fermionic and bosonic variables, and demonstrates how supersymmetry constrains the stochastic process uniquely.

Findings

Path integral formalism for multiplicative stochastic processes.

Supersymmetry cancels tadpole diagrams, ensuring unique process definition.

Non-perturbative constraints from BRS symmetry on correlation functions.

Abstract

We present a functional formalism to derive a generating functional for correlation functions of a multiplicative stochastic process represented by a Langevin equation. We deduce a path integral over a set of fermionic and bosonic variables without performing any time discretization. The usual prescriptions to define the Wiener integral appear in our formalism in the definition of Green functions in the Grassman sector of the theory. We also study non-perturbative constraints imposed by BRS symmetry and supersymmetry on correlation functions. We show that the specific prescription to define the stochastic process is wholly contained in tadpole diagrams. Therefore, in a supersymmetric theory the stochastic process is uniquely defined since tadpole contributions cancels at all order of perturbation theory.