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2020 The continuum parabolic Anderson model with a half-Laplacian and periodic noise
Alexander Dunlap
Electron. Commun. Probab. 25: 1-14 (2020). DOI: 10.1214/20-ECP342

Abstract

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $\partial _{t}u=-(-\Delta )^{\frac {1}{2}}u+\xi u$, where $\xi $ is a periodic spatial white noise. To be precise, we construct limits as $\varepsilon \to 0$ of solutions of $\partial _{t}u_{\varepsilon }=-(-\Delta )^{\frac {1}{2}}u_{\varepsilon }+(\xi _{\varepsilon }-C_{\varepsilon })u_{\varepsilon }$, where $\xi _{\varepsilon }$ is a mollification of $\xi $ at scale $\varepsilon $ and $C_{\varepsilon }$ is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labbé, “A simple construction of the continuum parabolic Anderson model on $\mathbf {R}^{2}$.”

Citation

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Alexander Dunlap. "The continuum parabolic Anderson model with a half-Laplacian and periodic noise." Electron. Commun. Probab. 25 1 - 14, 2020. https://doi.org/10.1214/20-ECP342

Information

Received: 13 March 2020; Accepted: 12 August 2020; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 1448.60135
Digital Object Identifier: 10.1214/20-ECP342

Subjects:
Primary: 60H15

JOURNAL ARTICLE
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