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Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion

Saloua Labed, Nacira Agram, Bernt Oksendal

TL;DR

The paper addresses the distribution evolution of solutions to McKean–Vlasov SDEs driven by fractional Brownian motion with $H>1/2$. It derives a distributional Fokker–Planck equation for the law $\mu_t$, and gives a weak form that becomes a concrete PDE with time-dependent diffusion when $\mu_t$ has a density; it also establishes a fractional Feynman–Kac representation linking the forward FP equation to a backward Kolmogorov equation for functionals of the process. The framework uses a white-noise/Fourier approach with a Hilbert space of measures and fractional calculus to handle non-Markovian mean-field dynamics. Two illustrative examples—the law of fractional Brownian motion and a linear MV fractional SDE—demonstrate how fractional noise and mean-field interactions shape the evolution, yielding explicit FP, density, and Feynman–Kac representations.

Abstract

This paper investigates the probability distribution of solutions to McKean--Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst parameter H>1/2. Our main contribution is the derivation of the associated Fokker--Planck equation, which characterizes the time evolution of the law of the solution in a suitable distributional framework. Under mild assumptions, we show that the law-valued process is absolutely continuous in time and provide an explicit weak formulation of the corresponding fractional McKean--Vlasov Fokker--Planck equation. In the case where the law admits a density, we obtain a more explicit partial differential equation with time-dependent diffusion coefficients induced by the fractional noise. We further establish a fractional Feynman--Kac representation, linking the forward Fokker--Planck equation with a backward Kolmogorov equation for functionals of the solution process. This result extends the classical Feynman--Kac framework to mean--field dynamics driven by fractional Brownian motion. To illustrate the theory, we analyze several explicit examples, including the law of fractional Brownian motion itself and linear McKean--Vlasov fractional SDEs. These examples highlight how fractional noise and mean--field interactions jointly affect the probabilistic and analytic structure of the system.

Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion

TL;DR

The paper addresses the distribution evolution of solutions to McKean–Vlasov SDEs driven by fractional Brownian motion with . It derives a distributional Fokker–Planck equation for the law , and gives a weak form that becomes a concrete PDE with time-dependent diffusion when has a density; it also establishes a fractional Feynman–Kac representation linking the forward FP equation to a backward Kolmogorov equation for functionals of the process. The framework uses a white-noise/Fourier approach with a Hilbert space of measures and fractional calculus to handle non-Markovian mean-field dynamics. Two illustrative examples—the law of fractional Brownian motion and a linear MV fractional SDE—demonstrate how fractional noise and mean-field interactions shape the evolution, yielding explicit FP, density, and Feynman–Kac representations.

Abstract

This paper investigates the probability distribution of solutions to McKean--Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst parameter H>1/2. Our main contribution is the derivation of the associated Fokker--Planck equation, which characterizes the time evolution of the law of the solution in a suitable distributional framework. Under mild assumptions, we show that the law-valued process is absolutely continuous in time and provide an explicit weak formulation of the corresponding fractional McKean--Vlasov Fokker--Planck equation. In the case where the law admits a density, we obtain a more explicit partial differential equation with time-dependent diffusion coefficients induced by the fractional noise. We further establish a fractional Feynman--Kac representation, linking the forward Fokker--Planck equation with a backward Kolmogorov equation for functionals of the solution process. This result extends the classical Feynman--Kac framework to mean--field dynamics driven by fractional Brownian motion. To illustrate the theory, we analyze several explicit examples, including the law of fractional Brownian motion itself and linear McKean--Vlasov fractional SDEs. These examples highlight how fractional noise and mean--field interactions jointly affect the probabilistic and analytic structure of the system.
Paper Structure (16 sections, 13 theorems, 123 equations, 2 figures)

This paper contains 16 sections, 13 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Hilbert space of measures
  4. Schwartz space of tempered distributions
  5. fBm and its calculus
  6. The operator $M$
  7. Fractional Brownian motion (fBm)
  8. Fractional white noise calculus
  9. Well-posedness of the McKean-Vlasov fBm equation
  10. A Fokker--Planck Equation for McKean--Vlasov SDEs Driven by fBm
  11. A Feynman--Kac Representation for Fractional McKean--Vlasov Diffusions
  12. Examples
  13. The law of fractional Brownian motion
  14. The law of a linear McKean--Vlasov fractional SDE
  15. Fokker--Planck equation.
  16. ...and 1 more sections

Key Result

Theorem 2.1

(The Bochner--Minlos theorem) There exists a unique probability measure $\mathbb{P}$ on ${\cal B}({\cal S}'(\mathbb{R}))$ with the following property: for all $\phi\in{\cal S}(\mathbb{R})$, where $\Vert\phi\Vert^2=\Vert\phi\Vert^2_{L^2({\mathbb{ R}}^n)},\quad\langle\omega,\phi\rangle= \omega(\phi)$ is the action of $\omega\in{\cal S}'(\mathbb{R})$ on $\phi\in{\cal S}(\mathbb{R})$ and $\mathbb{E}=

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (23)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 13 more