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

Abstract The Carleman approach is a popular method in the field of deterministic classical dynamics for replacing a finite number d of nonlinear differential equations by an infinite-dimensional linear system. Here, this approach is applied to a system of d stochastic differential equations for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:math> when the forces and the diffusion-matrix elements are polynomials, to write the linear system governing the dynamics of the averaged values <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>⋯</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>d</mml:mi> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> labeled by the d integers <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . The natural decomposition of the Carleman matrix into blocks associated with the global degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:math> is useful to identify the models with 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, which can be either additive or multiplicative or square root or with two types of noise per coordinate, with many examples in dimensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . In d = 1, the Carleman matrix governing the dynamics of the moments <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> is diagonal for the geometric Brownian motion, whereas 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 toward steady states with power-law tails. In dimension d = 2, the Carleman matrix governing the dynamics of the correlations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow=