On fractal minimizers and potentials of occupation measures
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918249831071744 |
|---|---|
| author | Hinz, Michael Tölle, Jonas M. Viitasaari, Lauri |
| author_facet | Hinz, Michael Tölle, Jonas M. Viitasaari, Lauri |
| contents | We consider four prototypes of variational problems and prove the existence of fractal minimizers through the direct method in the calculus of variations. By design these minimizers are Hölder curves or Hölder parametrizations of hypersurfaces whose images generally have a non-integer Hausdorff dimension. Although their origin is deterministic, their regularity properties are roughly similar to those of typical realizations of stochastic processes. As a key tool, we prove novel continuity and boundedness results for potentials of occupation measures of Gaussian random fields. These results complement well-known results for local times, but hold under much less restrictive assumptions. In an auxiliary section, we generalize earlier results on non-linear compositions of fractional Sobolev functions with $BV$-functions to higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On fractal minimizers and potentials of occupation measures Hinz, Michael Tölle, Jonas M. Viitasaari, Lauri Probability Functional Analysis Optimization and Control Primary: 28A78, 31B15, 46E35, 49J05, 49J10, 60G17, Secondary: 26B30, 28A80, 60G15, 60G22 We consider four prototypes of variational problems and prove the existence of fractal minimizers through the direct method in the calculus of variations. By design these minimizers are Hölder curves or Hölder parametrizations of hypersurfaces whose images generally have a non-integer Hausdorff dimension. Although their origin is deterministic, their regularity properties are roughly similar to those of typical realizations of stochastic processes. As a key tool, we prove novel continuity and boundedness results for potentials of occupation measures of Gaussian random fields. These results complement well-known results for local times, but hold under much less restrictive assumptions. In an auxiliary section, we generalize earlier results on non-linear compositions of fractional Sobolev functions with $BV$-functions to higher dimensions. |
| title | On fractal minimizers and potentials of occupation measures |
| topic | Probability Functional Analysis Optimization and Control Primary: 28A78, 31B15, 46E35, 49J05, 49J10, 60G17, Secondary: 26B30, 28A80, 60G15, 60G22 |
| url | https://arxiv.org/abs/2512.14248 |