Path-Integral Methods for Multiplicative Noise

Develop path-integral methods for stochastic differential equations with multiplicative noise, where the noise amplitude depends on the system state, that provide consistent transition probability functionals and are applicable beyond special discretization schemes or white-noise assumptions.

Background

The paper distinguishes additive versus multiplicative noise in stochastic differential equations and notes that multiplicative noise is prevalent in physics, finance, and biology. Despite existing path-integral frameworks (e.g., Onsager–Machlup and MSRJD) handling white noise, extending path-integral methods to arbitrary multiplicative noise remains difficult and sensitive to stochastic calculus interpretations.

The authors introduce a generalized formalism using the Parisi–Sourlas method and a cumulant generating function to address this gap, but explicitly acknowledge that the broader development of path-integral methods for multiplicative noise is still an open challenge in the literature.