In search of necessary and sufficient conditions to solve parabolic Anderson model with rough noise
| Autor: | Liu, Shuhui, Hu, Yaozhong, Wang, Xiong |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | This paper attempts to obtain necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises: $\frac{\partial}{\partial t}u(t,x)=\frac{1}{2}\Delta u(t,x)+u(t,x)\dot{W}(t,x)$, where $ {W}(t,x)$ is the fractional Brownian field with temporal Hurst parameter $H_0\in [1/2, 1) $ and spatial Hurst parameters $H$ $ =(H_1, \cdots, H_d)$ $ \in (0, 1)^d$, and $\dot{W}(t,x)=\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots \partial x_d}W(t,x)$. When $d=1$ and when $(H_0,H)\in(\frac 12,1)\times(\frac 1{20},\frac 12)$ we show that the condition $2H_0+H>5/2$ is necessary and sufficient to ensure the existence of a unique solution for the parabolic Anderson Model. When $d\ge 2$, we find the necessary and sufficient condition on the Hurst parameters so that each chaos of the solution candidate is square integrable. Comment: 50 pages, 4 figures |
| Databáze: | arXiv |
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