Beyond It\^o versus Stratonovich

TL;DR

This paper introduces a new stochastic calculus framework emerging from biology studies, connecting it to existing interpretations, and demonstrating its physical relevance and experimental validation.

Contribution

It presents a novel stochastic calculus framework with a potential-based transformation and symplectic structure, extending beyond traditional Itô and Stratonovich interpretations.

Findings

Demonstrates the zero mass limit of a generalized Klein-Kramers equation.

Establishes a new stochastic calculus differing from {}-type interpretations.

Provides empirical validation through recent experiments.

Abstract

Recently, a novel framework to handle stochastic processes has emerged from a series of studies in biology, showing situations beyond 'It\^o versus Stratonovich'. Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein-Kramers equation. Moreover, the connection to other integrations becomes evident: the obtained Fokker-Planck equation defines a new type of stochastic calculus that in general differs from the {\alpha}-type interpretation. A unique advantage of this new approach is a natural correspondence between stochastic and deterministic dynamics, which is useful or may even be essential in practice. The core of the framework is a transformation from the usual Langevin equation to a form that contains a potential function with two additional dynamical matrices, which reveals an underlying symplectic structure. The framework has a direct physical meaning and a straightforward experimental realization. A recent experiment has offered a first empirical validation of this new stochastic integration.