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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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