Well-posedness of kinetic McKean-Vlasov equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909556745961472 |
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| author | Pascucci, Andrea Rondelli, Alessio |
| author_facet | Pascucci, Andrea Rondelli, Alessio |
| contents | We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + σ(t, X_t, [X_t])dW_t$ where $[X_t]$ is the law of $X_t$. We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion $W$ is strictly smaller than that of the solution $X$, as commonly required in classical models of collisional kinetic theory. Assuming Hölder continuous coefficients and a weak Hörmander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously unexplored case where the diffusion coefficient $σ$ depends on the law $[X_t]$. Notably, our proof employs a simplified and direct argument eliminating the need for PDEs involving derivatives with respect to the measure argument. A critical ingredient is the sub-Riemannian metric structure induced by the corresponding Fokker-Planck operator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_10987 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of kinetic McKean-Vlasov equations Pascucci, Andrea Rondelli, Alessio Probability 60H10, 35Q83 We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + σ(t, X_t, [X_t])dW_t$ where $[X_t]$ is the law of $X_t$. We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion $W$ is strictly smaller than that of the solution $X$, as commonly required in classical models of collisional kinetic theory. Assuming Hölder continuous coefficients and a weak Hörmander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously unexplored case where the diffusion coefficient $σ$ depends on the law $[X_t]$. Notably, our proof employs a simplified and direct argument eliminating the need for PDEs involving derivatives with respect to the measure argument. A critical ingredient is the sub-Riemannian metric structure induced by the corresponding Fokker-Planck operator. |
| title | Well-posedness of kinetic McKean-Vlasov equations |
| topic | Probability 60H10, 35Q83 |
| url | https://arxiv.org/abs/2501.10987 |