Cholesky decomposition and well-posedness of Cauchy problem for Fokker-Planck equations with unbounded coefficients

TL;DR

This paper investigates the well-posedness of the Fokker-Planck equation with unbounded coefficients, utilizing Cholesky decomposition and superposition principles to connect PDE solutions with stochastic processes.

Contribution

It introduces a novel approach combining Cholesky decomposition with superposition principles to analyze the well-posedness of Fokker-Planck equations with low regularity coefficients.

Findings

Established conditions for existence of solutions under general growth constraints.

Connected solution uniqueness to martingale problem and SDE solutions.

Proved ergodicity under additional growth conditions.

Abstract

This paper explores the well-posedness of the Cauchy problem for the Fokker-Planck equation associated with the partial differential operator $L$ with low regularity condition. To address uniqueness, we apply a recently developed superposition principle for unbounded coefficients, which reduces the uniqueness problem for the Fokker-Planck equation to the uniqueness of solutions to the martingale problem. Using the Cholesky decomposition algorithm, a standard tool in numerical linear algebra, we construct a lower triangular matrix of functions $\sigma$ with suitable regularity such that $A = \sigma \sigma^T$. This formulation allows us to connect the uniqueness of solutions to the martingale problem with the uniqueness of weak solutions to It\^{o}-SDEs. For existence, we rely on established results concerning sub-Markovian semigroups, which enable us to confirm the existence of solutions to the Fokker-Planck equation under general growth conditions expressed as inequalities. Additionally, by imposing further growth conditions on the coefficients, also expressed as inequalities, we establish the ergodicity of the solutions. This work demonstrates the interplay between stochastic analysis and numerical linear algebra in addressing problems related to partial differential equations.