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

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.

69 pages

References in corpus (27)

• The large deviation approach to statistical mechanics
• Strong disorder RG approach of random systems
• Structure of Stochastic Dynamics near Fixed Points
• Koopman-von Neumann Approach to Quantum Simulation of Nonlinear Classical Dynamics
• Experimental Realization of a Minimal Microscopic Heat Engine
• Two-temperature Langevin dynamics in a parabolic potential
• Electrical Autonomous Brownian Gyrator
• Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms
• Characterising the nonequilibrium stationary states of Ornstein-Uhlenbeck processes
• Inference in non-equilibrium systems from incomplete information: the case of linear systems and its pitfalls
• Large deviations for Markov processes with stochastic resetting : analysis via the empirical density and flows or via excursions between resets
• Dynamical large deviations of linear diffusions
• Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson Localization models
• Finite-size scaling properties of random transverse-field Ising chains : Comparison between canonical and microcanonical ensembles for the disorder
• Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator
• A novel concept of fractal dimension in deterministic and stochastic Lorenz-63 systems
• Spectral fingerprints of non-equilibrium dynamics: The case of a Brownian gyrator
• Optimal Diffusion Processes
• Inferring entropy production in anharmonic Brownian gyrators
• Ergodicity and mixing bounds for the Fisher-Snedecor diffusion
• Large deviations of the stochastic area for linear diffusions
• Population Extinction on a Random Fitness Seascape
• Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force
• Random Transverse Field Spin-Glass Model on the Cayley tree : phase transition between the two Many-Body-Localized Phases
• Stable cooperation emerges in stochastic multiplicative growth
• An introduction to large deviations with applications in physics
• Star junctions and watermelons of pure or random quantum Ising chains : finite-size properties of the energy gap at criticality