It\^o versus H\"anggi-Klimontovich

TL;DR

This paper introduces the Hänggi-Klimontovich integral, analyzes its mathematical properties, and compares its applicability to physical systems against Itô and Stratonovich interpretations, finding it less suitable for certain classical models.

Contribution

The paper provides a rigorous mathematical formulation of the Hänggi-Klimontovich integral and evaluates its effectiveness in modeling statistical mechanical systems.

Findings

Hänggi-Klimontovich integral is less suitable than Itô and Stratonovich for classical physical models.

Mathematical properties of the Hänggi-Klimontovich integral are characterized.

Application to Langevin particles and relativistic Brownian motion demonstrates its limitations.

Abstract

Interpreting the noise in a stochastic differential equation, in particular the It\^o versus Stratonovich dilemma, is a problem that has generated a lot of debate in the physical literature. In the last decades, a third interpretation of noise, given by the so-called H\"anggi-Klimontovich integral, has been proposed as better adapted to describe certain physical systems, particularly in statistical mechanics. Herein, we introduce this integral in a precise mathematical manner and analyze its properties, signaling those that have made it appealing within the realm of physics. Subsequently, we employ this integral to model some statistical mechanical systems, such as the random dispersal of Langevin particles and the relativistic Brownian motion. We show that, for these classical examples, the H\"anggi-Klimontovich integral is worse adapted than the It\^o integral and even the Stratonovich one.