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

Haesung Lee; Haesung Lee

doi:10.3934/math.2025610

AIMS Mathematics

2025, Volume 10, Issue 6: 13555-13574. doi: 10.3934/math.2025610

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

Haesung Lee ^,

Department of Mathematics and Big Data Science, Kumoh National Institute of Technology, Gumi, Gyeongsangbuk-do 39177, Republic of Korea

Received: 29 December 2024 Revised: 22 May 2025 Accepted: 28 May 2025 Published: 12 June 2025
MSC : 15A23, 35A02, 35K15, 35Q84, 47D07, 93D05

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 Itô-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.
- Fokker-Planck equations,
- diffusion equations,
- inequalities,
- semigroups,
- martingale problems,
- Cholesky decomposition
Citation: Haesung Lee. Cholesky decomposition and well-posedness of Cauchy problem for Fokker-Planck equations with unbounded coefficients[J]. AIMS Mathematics, 2025, 10(6): 13555-13574. doi: 10.3934/math.2025610

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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 Itô-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.

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