Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion
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| Format: | Preprint |
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2024
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| author | Labed, Saloua Agram, Nacira Oksendal, Bernt |
| author_facet | Labed, Saloua Agram, Nacira Oksendal, Bernt |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_07029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion Labed, Saloua Agram, Nacira Oksendal, Bernt Probability 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. |
| title | Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2409.07029 |