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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.07803 |
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| _version_ | 1866911732590444544 |
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| author | Agresti, Antonio Veraar, Mark |
| author_facet | Agresti, Antonio Veraar, Mark |
| contents | We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} β, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $β$ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $α\in [0,\frac{1}{2}),$ \[ u\in L^2(Ω;W^{α,2}(0,T;\mathsf{D}(A^{1/2}))) \quad \text{ if and only if }\quad x\in \mathsf{D}(A^α). \] In particular, there is a lack of persistence of temporal regularity from the diffusion coefficient $g$ to the solution, and additional spatial regularity is required to improve time regularity. In particular, this provides a counterexample to a conjectured time-regularity property for monotone stochastic evolution equations posed by D. Breit and M. Hofmanová in [C. R. Math. Acad. Sci. Paris 354 (2016), 33-37]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on a threshold for temporal regularity of stochastic PDEs Agresti, Antonio Veraar, Mark Probability We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} β, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $β$ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $α\in [0,\frac{1}{2}),$ \[ u\in L^2(Ω;W^{α,2}(0,T;\mathsf{D}(A^{1/2}))) \quad \text{ if and only if }\quad x\in \mathsf{D}(A^α). \] In particular, there is a lack of persistence of temporal regularity from the diffusion coefficient $g$ to the solution, and additional spatial regularity is required to improve time regularity. In particular, this provides a counterexample to a conjectured time-regularity property for monotone stochastic evolution equations posed by D. Breit and M. Hofmanová in [C. R. Math. Acad. Sci. Paris 354 (2016), 33-37]. |
| title | A Note on a threshold for temporal regularity of stochastic PDEs |
| topic | Probability |
| url | https://arxiv.org/abs/2509.07803 |