Recent results on multiplicative noise

TL;DR

This paper explores recent advances in understanding Langevin equations with multiplicative noise, including numerical simulations, analytical calculations, and the effects of different stochastic interpretations, revealing complex phase transitions.

Contribution

It provides new numerical and analytical insights into the phase behavior and critical phenomena of multiplicative noise equations, highlighting differences between Stratonovich and Ito interpretations.

Findings

MN equation exhibits weak and strong coupling phases similar to KPZ.

Noise-induced ordering transition occurs only in Stratonovich representation.

A new first-order phase transition at zero spatial coupling is identified.

Abstract

Recent developments in the analysis of Langevin equations with multiplicative noise (MN) are reported. In particular, we: (i) present numerical simulations in three dimensions showing that the MN equation exhibits, like the Kardar-Parisi-Zhang (KPZ) equation both a weak coupling fixed point and a strong coupling phase, supporting the proposed relation between MN and KPZ; (ii) present dimensional, and mean field analysis of the MN equation to compute critical exponents; (iii) show that the phenomenon of the noise induced ordering transition associated with the MN equation appears only in the Stratonovich representation and not in the Ito one, and (iv) report the presence of a new first-order like phase transition at zero spatial coupling, supporting the fact that this is the minimum model for noise induced ordering transitions.