Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917101196804096 |
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| author | Tramontana, Davide |
| author_facet | Tramontana, Davide |
| contents | The aim of this work is to study the convergence to equilibrium of an $(h,ρ)$-subelliptic random walk on a closed, connected Riemannian manifold $(M,g)$ associated with a subelliptic second-order differential operator $A$ on $M$. In such a random walk, $h$ roughly represents the step size and $ρ$ the speed at which it is carried out. To construct the random walk and prove the convergence result, we employ a technique due to Fefferman and Phong, which reduces the problem to the study of a constant-coefficient operator $\tilde{A}$ that is locally equivalent to our second-order subelliptic operator $A$, in the sense that the diffusion generated by $\tilde{A}$ induces a local diffusion for $A$. By using the compactness of $M$ this local diffusion can be lifted to a global diffusion, and the convergence result is then obtained via the spectral theory of the associated Markov operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_22869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium Tramontana, Davide Analysis of PDEs Probability 58J65, 58J51, 60G50, 60J10 The aim of this work is to study the convergence to equilibrium of an $(h,ρ)$-subelliptic random walk on a closed, connected Riemannian manifold $(M,g)$ associated with a subelliptic second-order differential operator $A$ on $M$. In such a random walk, $h$ roughly represents the step size and $ρ$ the speed at which it is carried out. To construct the random walk and prove the convergence result, we employ a technique due to Fefferman and Phong, which reduces the problem to the study of a constant-coefficient operator $\tilde{A}$ that is locally equivalent to our second-order subelliptic operator $A$, in the sense that the diffusion generated by $\tilde{A}$ induces a local diffusion for $A$. By using the compactness of $M$ this local diffusion can be lifted to a global diffusion, and the convergence result is then obtained via the spectral theory of the associated Markov operator. |
| title | Subelliptic Random Walks on Riemannian Manifolds and Their Convergence to Equilibrium |
| topic | Analysis of PDEs Probability 58J65, 58J51, 60G50, 60J10 |
| url | https://arxiv.org/abs/2506.22869 |