Classification of diffusion processes in dimension $d$ via the Carleman approach with applications to models involving additive, multiplicative or square-root noises

TL;DR

This work extends the Carleman embedding—historically a tool for deterministic dynamics—to multidimensional stochastic diffusion processes with polynomial forces and diffusion matrices. By organizing the Carleman matrix into blocks by total degree, the authors derive linear dynamics for all moments and examine how block structure (diagonal, lower- or upper-triangular) yields tractable spectral decompositions. In 1D they recover solvable processes like Geometric Brownian motion and Pearson diffusions with power-law tails, while in 2D they analyze moment correlations, Lyapunov exponents, and tail-behavior via ratio variables and large-deviation formalisms. Across 1D and 2D, the paper connects spectral properties of the Carleman matrix to classical diffusion families (Kesten, Fisher–Snedecor, Student) and provides a unified framework to study additive, square-root, and multiplicative noises, with explicit examples and asymptotics that illuminate tail statistics and regime transitions.

Abstract

The Carleman approach is well-known in the field of deterministic classical dynamics as a method to replace a finite number $d$ of non-linear differential equations by an infinite-dimensional linear system. Here this approach is applied to a system of $d$ stochastic differential equations for $[x_1(t),..,x_d(t)]$ when the forces and the diffusion-matrix elements are polynomials, in order to write the linear system governing the dynamics of the averaged values ${\mathbb E} ( x_1^{n_1}(t) x_2^{n_2}(t) ... x_d^{n_d}(t) )$ labelled by the $d$ integers $(n_1,..,n_d)$. The natural decomposition of the Carleman matrix into blocks associated to the global degree $n=n_1+n_2+..+n_d$ is useful to identify the models that have the simplest spectral decompositions in the bi-orthogonal basis of right and left eigenvectors. This analysis is then applied to models with a single noise per coordinate, that can be either additive or multiplicative or square-root, or with two types of noises per coordinate, with many examples in dimensions $d=1,2$. In $d=1$, the Carleman matrix governing the dynamics of the moments ${\mathbb E} ( x^{n}(t) )$ is diagonal for the Geometric Brownian motion, while it is lower-triangular for the family of Pearson diffusions containing the Ornstein-Uhlenbeck and the Square-Root processes, as well as the Kesten, the Fisher-Snedecor and the Student processes that converge towards steady states with power-law-tails. In dimension $d=2$, the Carleman matrix governing the dynamics of the correlations ${\mathbb E} ( x_1^{n_1}(t) x_2^{n_2}(t) )$ has a natural decomposition into blocks associated to the global degree $n=n_1+n_2$, and we discuss the simplest models where the Carleman matrix is either block-diagonal or block-lower-triangular or block-upper-triangular.