论文标题
为了寻找必要和充分的条件来解决抛物线模型，以粗糙的噪音
In search of necessary and sufficient conditions to solve parabolic Anderson model with rough noise
论文作者
论文摘要
本文试图获得必要和足够的条件，以用分离的高斯噪声求解抛物线的安德森模型：$ \ frac {\ partial} {\ partial t} u（t，x）= \ frac {1} {1} {2} {2} {2} {2} {2}ΔU（t，x）+U（t，x）+U（t，x）+U（t，x）是[1/2，1）$和空间hurst参数$ h $ $ =（h_1，\ cdots，h_d）$ $ $ \ in（0，1） ^d $，和$ \ dot {w}（w}（w}（t，x）= \ frac { \ partial x_1 \ cdots \ partial x_d} w（t，x）$。当$ d = 1 $，而当$（h_0，h）\ in（\ frac 12,1）\ times（\ frac 1 {20}，\ frac 12）$时，我们表明条件$ 2H_0+H> 5/2 $是必需的，足以确保存在Parabolic Anderson模型的独特解决方案。当$ d \ ge 2 $时，我们会在赫斯特参数上找到必要且充分的条件，以便解决方案候选者的每个混乱都是正方形的。
This paper attempts to obtain necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises: $\frac{\partial}{\partial t}u(t,x)=\frac{1}{2}Δu(t,x)+u(t,x)\dot{W}(t,x)$, 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.
