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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.14784 |
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| _version_ | 1866915238673121280 |
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| author | Zhang, Shao-Qin |
| author_facet | Zhang, Shao-Qin |
| contents | We establish an existence result of a solution to a class of probability measure-valued equations, whose solutions can be associated with stationary distributions of many McKean-Vlasov diffusions with gradient-type drifts. Coefficients of the probability measure-valued equation may be discontinuous in the weak topology and the total variation norm. Owing to that the bifurcation point of the probability measure-valued equation is relevant to the phase transition point of the associated McKean-Vlasov diffusion, we establish a local Krasnosel'skii bifurcation theorem. Regularized determinant for the Hilbert-Schmidt operator is used to derive our criteria for the bifurcation point. Concrete examples, including the granular media equation and the Vlasov-Fokker-Planck equation with quadratic interaction, are given to illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Local Bifurcation Theorem for McKean-Vlasov Diffusions Zhang, Shao-Qin Probability Dynamical Systems 60J60, 37G10, 82B26, 46N30 We establish an existence result of a solution to a class of probability measure-valued equations, whose solutions can be associated with stationary distributions of many McKean-Vlasov diffusions with gradient-type drifts. Coefficients of the probability measure-valued equation may be discontinuous in the weak topology and the total variation norm. Owing to that the bifurcation point of the probability measure-valued equation is relevant to the phase transition point of the associated McKean-Vlasov diffusion, we establish a local Krasnosel'skii bifurcation theorem. Regularized determinant for the Hilbert-Schmidt operator is used to derive our criteria for the bifurcation point. Concrete examples, including the granular media equation and the Vlasov-Fokker-Planck equation with quadratic interaction, are given to illustrate our results. |
| title | A Local Bifurcation Theorem for McKean-Vlasov Diffusions |
| topic | Probability Dynamical Systems 60J60, 37G10, 82B26, 46N30 |
| url | https://arxiv.org/abs/2401.14784 |