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Parabolic Anderson model with rough or critical Gaussian noise. (English. French summary) Zbl 07097337
Summary: This paper considers 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 \(\mathbf{H}=(H_{0},H_{1},\ldots,H_{d})\). The existence/uniqueness, Feynman-Kac’s moment formula and the precise intermittency exponents are formulated in the case when some of \(H_{1},\ldots,H_{d}\) are less than one half, and in the case when the Dalang’s condition \[ d-\sum_{k=1}^{n}H_{j}<1\quad\mbox{is replaced by }d-\sum_{k=1}^{n}H_{j}=1. \] Some partial result is also achieved for the case when \(H_{0}<1/2\) which brings insight on what to expect as the Gaussian noise is rough in time.

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