Approximate Slow Manifolds in the Fokker-Planck Equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916592037658624 |
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| author | Kuehn, Christian Sulzbach, Jan-Eric |
| author_facet | Kuehn, Christian Sulzbach, Jan-Eric |
| contents | In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the existence of a slow manifold on the PDE level, which is a crucial tool from the geometric singular perturbation theory allowing the reduction of the system to a lower dimensional slowly evolving equation. In particular, we use a projection approach based upon a Sturm- Liouville eigenbasis to convert the Fokker-Planck PDE to an infinite system of PDEs that can be truncated/approximated to any order. Based upon this truncation, we can employ the recently developed theory for geometric singular perturbation theory for slow manifolds for infinite-dimensional evolution equations. This strategy presents a new perspective on the dynamics of multiple time-scale SDEs as it combines ideas from several previously disjoint reduction methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18981 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Approximate Slow Manifolds in the Fokker-Planck Equation Kuehn, Christian Sulzbach, Jan-Eric Analysis of PDEs Probability In this paper we study the dynamics of a fast-slow Fokker-Planck partial differential equation (PDE) viewed as the evolution equation for the density of a multiscale planar stochastic differential equation (SDE). Our key focus is on the existence of a slow manifold on the PDE level, which is a crucial tool from the geometric singular perturbation theory allowing the reduction of the system to a lower dimensional slowly evolving equation. In particular, we use a projection approach based upon a Sturm- Liouville eigenbasis to convert the Fokker-Planck PDE to an infinite system of PDEs that can be truncated/approximated to any order. Based upon this truncation, we can employ the recently developed theory for geometric singular perturbation theory for slow manifolds for infinite-dimensional evolution equations. This strategy presents a new perspective on the dynamics of multiple time-scale SDEs as it combines ideas from several previously disjoint reduction methods. |
| title | Approximate Slow Manifolds in the Fokker-Planck Equation |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2501.18981 |