Regularity theory for a new class of fractional parabolic stochastic evolution equations
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| Format: | Preprint |
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2022
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| author | Kirchner, Kristin Willems, Joshua |
| author_facet | Kirchner, Kristin Willems, Joshua |
| contents | A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^γX(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $γ\in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert space $H$ and the spatiotemporal driving noise $\dot{W}^Q$ is the formal time derivative of an $H$-valued cylindrical $Q$-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process $X$ are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of $A$. In addition, the covariance of $X$ and its long-time behavior are analyzed.
These abstract results are applied to the cases when $A := L^β$ and $Q:=\tilde{L}^{-α}$ are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Matérn fields to space-time. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_00248 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Regularity theory for a new class of fractional parabolic stochastic evolution equations Kirchner, Kristin Willems, Joshua Probability Analysis of PDEs 60G15, 60H15 (Primary) 47D06, 35R11 (Secondary) A new class of fractional-order stochastic evolution equations of the form $(\partial_t + A)^γX(t) = \dot{W}^Q(t)$, $t\in[0,T]$, $γ\in (0,\infty)$, is introduced, where $-A$ generates a $C_0$-semigroup on a separable Hilbert space $H$ and the spatiotemporal driving noise $\dot{W}^Q$ is the formal time derivative of an $H$-valued cylindrical $Q$-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process $X$ are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of $A$. In addition, the covariance of $X$ and its long-time behavior are analyzed. These abstract results are applied to the cases when $A := L^β$ and $Q:=\tilde{L}^{-α}$ are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Matérn fields to space-time. |
| title | Regularity theory for a new class of fractional parabolic stochastic evolution equations |
| topic | Probability Analysis of PDEs 60G15, 60H15 (Primary) 47D06, 35R11 (Secondary) |
| url | https://arxiv.org/abs/2205.00248 |