Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.01538 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914483996196864 |
|---|---|
| author | Holbach, Simon Raimond, Olivier |
| author_facet | Holbach, Simon Raimond, Olivier |
| contents | We consider a self-interacting diffusion $X$ on a smooth compact Riemannian manifold $\mathbb M$, described by the stochastic differential equation \[ dX_t = \sqrt{2} dW_t(X_t)- β(t) \nabla V_t(X_t)dt, \] where $β$ is suitably lower-bounded and grows at most logarithmically, and $V_t(x)=\frac{1}{t}\int_0^t V(x,X_s)ds$ for a suitable smooth function $V\colon \mathbb M^2\to\mathbb R$ that makes the term $-\nabla V_t(X_t)$ self-repelling. We prove that almost surely the normalized occupation measure $μ_t$ of $X$ converges weakly to the uniform distribution $\mathcal U$, and we provide a polynomial rate of convergence for smooth test functions. The key to this result is showing that if $f\colon\mathbb M\to\mathbb R$ is smooth, then $μ_{e^t}(f)$ shadows the flow generated by the ordinary differential equation \[
\dot x_t=-x_t+\mathcal U(f). \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01538 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence to the uniform distribution of moderately self-interacting diffusions on compact Riemannian manifolds Holbach, Simon Raimond, Olivier Probability 60K35 We consider a self-interacting diffusion $X$ on a smooth compact Riemannian manifold $\mathbb M$, described by the stochastic differential equation \[ dX_t = \sqrt{2} dW_t(X_t)- β(t) \nabla V_t(X_t)dt, \] where $β$ is suitably lower-bounded and grows at most logarithmically, and $V_t(x)=\frac{1}{t}\int_0^t V(x,X_s)ds$ for a suitable smooth function $V\colon \mathbb M^2\to\mathbb R$ that makes the term $-\nabla V_t(X_t)$ self-repelling. We prove that almost surely the normalized occupation measure $μ_t$ of $X$ converges weakly to the uniform distribution $\mathcal U$, and we provide a polynomial rate of convergence for smooth test functions. The key to this result is showing that if $f\colon\mathbb M\to\mathbb R$ is smooth, then $μ_{e^t}(f)$ shadows the flow generated by the ordinary differential equation \[ \dot x_t=-x_t+\mathcal U(f). \] |
| title | Convergence to the uniform distribution of moderately self-interacting diffusions on compact Riemannian manifolds |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2307.01538 |