Fokker-Planck equations on homogeneous Lie groups and probabilistic counterparts
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929575575945216 |
|---|---|
| author | Caramellino, Lucia Mendico, Cristian |
| author_facet | Caramellino, Lucia Mendico, Cristian |
| contents | We address the well-posedness of subelliptic Fokker-Planck equations arising from stochastic control problems, as well as the properties of the associated diffusion processes. Here, the main difficulty arises from the possible polynomial growth of the coefficients, which is related to the growth of the family of vector fields generating the first layer of the associated Lie algebra. We prove the existence and uniqueness of the energy solution, and its representation as the transition density of the underlying subelliptic diffusion process. Moreover, we show its Holder continuity in time w.r.t.the Fortet-Mourier distance, where the Holder seminorm depends on the degree of homogeneity of the vector fields. Finally, we provide a probabilistic proof of the Feyman-Kac formula, as a consequence of the uniform boundedness in finite time intervals of all moments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11524 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fokker-Planck equations on homogeneous Lie groups and probabilistic counterparts Caramellino, Lucia Mendico, Cristian Analysis of PDEs Probability We address the well-posedness of subelliptic Fokker-Planck equations arising from stochastic control problems, as well as the properties of the associated diffusion processes. Here, the main difficulty arises from the possible polynomial growth of the coefficients, which is related to the growth of the family of vector fields generating the first layer of the associated Lie algebra. We prove the existence and uniqueness of the energy solution, and its representation as the transition density of the underlying subelliptic diffusion process. Moreover, we show its Holder continuity in time w.r.t.the Fortet-Mourier distance, where the Holder seminorm depends on the degree of homogeneity of the vector fields. Finally, we provide a probabilistic proof of the Feyman-Kac formula, as a consequence of the uniform boundedness in finite time intervals of all moments. |
| title | Fokker-Planck equations on homogeneous Lie groups and probabilistic counterparts |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2402.11524 |