Massive Particle Systems, Wasserstein Brownian Motions, and the Dean-Kawasaki Equation
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arXiv
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| Format: | Preprint |
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2024
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| author | Schiavo, Lorenzo Dello |
| author_facet | Schiavo, Lorenzo Dello |
| contents | We develop a unifying theory for four different objects: (1) infinite systems of interacting massive particles; (2) solutions to the Dean-Kawasaki equation with singular drift and space-time white noise; (3) Wasserstein diffusions with a.s. purely atomic reversible random measures; (4) metric measure Brownian motions induced by Cheeger energies on $L^2$-Wasserstein spaces.
For the objects in (1)-(3) we prove existence and uniqueness of solutions, and several characterizations, on an arbitrary locally compact Polish ambient space $M$ with exponentially recurrent Feller driving noise. In the case of the Dean-Kawasaki equation, this amounts to replacing the Laplace operator with some arbitrary diffusive Markov generator $\mathsf{L}$ with ultracontractive semigroup. In addition to a complete discussion of the free case, we consider singular interactions, including, e.g., mean-field repulsive isotropic pairwise interactions of Riesz and logarithmic type under the assumption of local integrability.
We further show that each Markov diffusion generator $\mathsf{L}$ on $M$ induces in a natural way a geometry on the space of probability measures over $M$. When $M$ is a manifold and $\mathsf{L}$ is a drifted Laplace-Beltrami operator, this geometry coincides with the geometry of $L^2$-optimal transportation. The corresponding `geometric Brownian motion' coincides with the 'metric measure Brownian motion' in (4). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Massive Particle Systems, Wasserstein Brownian Motions, and the Dean-Kawasaki Equation Schiavo, Lorenzo Dello Probability Functional Analysis 60G57, 60H17, 60J46, 49Q22, 70F45 We develop a unifying theory for four different objects: (1) infinite systems of interacting massive particles; (2) solutions to the Dean-Kawasaki equation with singular drift and space-time white noise; (3) Wasserstein diffusions with a.s. purely atomic reversible random measures; (4) metric measure Brownian motions induced by Cheeger energies on $L^2$-Wasserstein spaces. For the objects in (1)-(3) we prove existence and uniqueness of solutions, and several characterizations, on an arbitrary locally compact Polish ambient space $M$ with exponentially recurrent Feller driving noise. In the case of the Dean-Kawasaki equation, this amounts to replacing the Laplace operator with some arbitrary diffusive Markov generator $\mathsf{L}$ with ultracontractive semigroup. In addition to a complete discussion of the free case, we consider singular interactions, including, e.g., mean-field repulsive isotropic pairwise interactions of Riesz and logarithmic type under the assumption of local integrability. We further show that each Markov diffusion generator $\mathsf{L}$ on $M$ induces in a natural way a geometry on the space of probability measures over $M$. When $M$ is a manifold and $\mathsf{L}$ is a drifted Laplace-Beltrami operator, this geometry coincides with the geometry of $L^2$-optimal transportation. The corresponding `geometric Brownian motion' coincides with the 'metric measure Brownian motion' in (4). |
| title | Massive Particle Systems, Wasserstein Brownian Motions, and the Dean-Kawasaki Equation |
| topic | Probability Functional Analysis 60G57, 60H17, 60J46, 49Q22, 70F45 |
| url | https://arxiv.org/abs/2411.14936 |