Parabolic Anderson model with a fractional Gaussian noise that is rough in time. (English. French summary) Zbl 1434.60083

Chen, Xia

Ann. Inst. Henri Poincaré, Probab. Stat. 56, No. 2, 792-825 (2020).

Summary: This paper concerns the parabolic Anderson equation \[\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u+u\frac{\partial^{d+1}W^{\mathbf{H}}}{\partial t\,\partial x_1\cdots \,\partial x_d}\] generated by a \((d+1)\)-dimensional fractional noise with the Hurst parameter \(\mathbf{H}=(H_0,H_1,\ldots,H_d)\) with special interest in the setting that some of \(H_0,\dots,H_d\) are less than half. In the author’s recent work [ibid. 55, No. 2, 941–976 (2019; Zbl 1475.60113)], the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when \(H_0<1/2\) with the concern on solvability, Feynman-Kac’s moment formula and intermittency of the system.

MSC:

60F10	Large deviations
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H40	White noise theory
60J65	Brownian motion
81U10	\(n\)-body potential quantum scattering theory

Keywords:

parabolic Anderson equation; Dalang’s condition; fractional; rough and critical Gaussian noises; Feynman-Kac’s representation; Brownian motion; moment asymptotics

Citations:

Zbl 1475.60113

References:

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