A linear stochastic biharmonic heat equation: hitting probabilities
Consider the linear stochastic biharmonic heat equation on a $d$-dimensional torus ( $d=1,2,3)$, driven by a space-time white noise and with periodic boundary conditions: $$ \left(\frac{\partial}{\partial t}+(-\Delta)^2\right) v(t, x)=\sigma \dot{W}(t, x),(t, x) \in(0, T] \times \mathbb{T}^d, $$ $v(...
| Autores: | , |
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| Formato: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2022 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/192887 |
| Acesso em linha: | https://hdl.handle.net/2445/192887 |
| Access Level: | acceso abierto |
| Palavra-chave: | Probabilitats Processos estocàstics Equacions en derivades parcials Processos gaussians Probabilities Stochastic processes Partial differential equations Gaussian processes |
| Resumo: | Consider the linear stochastic biharmonic heat equation on a $d$-dimensional torus ( $d=1,2,3)$, driven by a space-time white noise and with periodic boundary conditions: $$ \left(\frac{\partial}{\partial t}+(-\Delta)^2\right) v(t, x)=\sigma \dot{W}(t, x),(t, x) \in(0, T] \times \mathbb{T}^d, $$ $v(0, x)=v_0(x)$. We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for $d=2$, they include a $z\left(\log \frac{c}{z}\right)^{1 / 2}$ term. Consider $D$ independent copies of the random field solution to (0.1). Applying the criteria proved in Hinojosa-Calleja and Sanz-Solé (Stoch PDE Anal Comp 2021. https://doi.org/10.1007/s40072-021-001901), we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets.This yields results on the polarity of sets and on the Hausdorff dimension of the path process. |
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