Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds
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| Format: | Preprint |
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2024
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| _version_ | 1866909470653677568 |
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| author | Shalova, Anna Schlichting, André |
| author_facet | Shalova, Anna Schlichting, André |
| contents | We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_14813 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds Shalova, Anna Schlichting, André Analysis of PDEs Mathematical Physics Probability We study stationary solutions of McKean-Vlasov equation on a high-dimensional sphere and other compact Riemannian manifolds. We extend the equivalence of the energetic problem formulation to the manifold setting and characterize critical points of the corresponding free energy functional. On a sphere, we employ the properties of spherical convolution to study the bifurcation branches around the uniform state. We also give a sufficient condition for an existence of a discontinuous transition point in terms of the interaction kernel and compare it to the Euclidean setting. We illustrate our results on a range of system, including the particle system arising from the transformer models and the Onsager model of liquid crystals. |
| title | Solutions of stationary McKean-Vlasov equation on a high-dimensional sphere and other Riemannian manifolds |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2412.14813 |