The compact support property for solutions to stochastic heat equations with stable noise
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866912531579142144 |
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| author | Hughes, Thomas |
| author_facet | Hughes, Thomas |
| contents | We consider weak non-negative solutions to the stochastic partial differential equation \[ \partial_t Y(t,x) = ΔY(t,x) + Y(t,x)^γ\dot{L}(t,x), \] for $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^d$, where $γ> 0$ and $\dot{L}$ is a one-sided stable noise of index $α\in (1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if $γ\in (2-α, 1)$ for $d=1$, and if $γ\in [1/α,1)$ in dimensions $d \in [2,2/(α-1)) \cap \mathbb{N}$. This complements known results on solutions to the equation with Gaussian noise.
We also establish a stochastic integral formula for the density of a solution and associated moment bounds which hold in all dimensions for which solutions are defined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_04520 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The compact support property for solutions to stochastic heat equations with stable noise Hughes, Thomas Probability 60H15, 60G52, 60G17 We consider weak non-negative solutions to the stochastic partial differential equation \[ \partial_t Y(t,x) = ΔY(t,x) + Y(t,x)^γ\dot{L}(t,x), \] for $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^d$, where $γ> 0$ and $\dot{L}$ is a one-sided stable noise of index $α\in (1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if $γ\in (2-α, 1)$ for $d=1$, and if $γ\in [1/α,1)$ in dimensions $d \in [2,2/(α-1)) \cap \mathbb{N}$. This complements known results on solutions to the equation with Gaussian noise. We also establish a stochastic integral formula for the density of a solution and associated moment bounds which hold in all dimensions for which solutions are defined. |
| title | The compact support property for solutions to stochastic heat equations with stable noise |
| topic | Probability 60H15, 60G52, 60G17 |
| url | https://arxiv.org/abs/2212.04520 |