First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908583647510528 |
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| author | Pinzi, Alessandro |
| author_facet | Pinzi, Alessandro |
| contents | The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t)_{t\in[0,T]}\subset \mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ associated to a non-local drift $b:[0,T]\times\mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$ and a non-local diffusion term $a:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \operatorname{Sym}_+(\mathbb{R}^{d\times d})$. Then, we show that any solution to that equation can be lifted to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and also to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we use this superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized KFP. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_07542 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation Pinzi, Alessandro Analysis of PDEs Functional Analysis Probability 60G57, 35R15, 35Q84, 60H15 The goal of this paper is to define an evolution equation for a curve of random probability measures $(M_t)_{t\in[0,T]}\subset \mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ associated to a non-local drift $b:[0,T]\times\mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$ and a non-local diffusion term $a:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \operatorname{Sym}_+(\mathbb{R}^{d\times d})$. Then, we show that any solution to that equation can be lifted to a superposition of solutions to a non-linear Kolmogorov-Fokker-Planck equation and also to a superposition of weak solutions to the McKean-Vlasov equations. Finally, we use this superposition result to show how existence and uniqueness can be transferred from the equation on random measures to the associated non-linear Kolmogorov-Fokker-Planck equation and to the McKean-Vlasov equation, assuming uniqueness of the linearized KFP. |
| title | First order equation on random measures as superposition of weak solutions to the McKean-Vlasov equation |
| topic | Analysis of PDEs Functional Analysis Probability 60G57, 35R15, 35Q84, 60H15 |
| url | https://arxiv.org/abs/2510.07542 |