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

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.