Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\geq 3$
Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\geq 3$
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$dgeq 3$ 中随机热方程和连续定向聚合物的弱无序和强无序
DOI：
10.1214/16-ecp18
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
O. Zeitouni
中科院分区：
文献类型：
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作者：
Chiranjib Mukherjee;A. Shamov;O. Zeitouni
考虑光滑乘性噪声随机热方程u_{\eps，t}= \frac 12 \Delta u_{\eps，t} dt + \beta \eps^{\frac{d-2}{2}}\，\，u_{\eps，t} \，d B_{\eps，t}，\; u_{\eps，0}=1，$$在维度$d\geq 3$，其中$B_{\eps，t}$是空间平滑的（尺度为$\eps$）时空白色噪声，$\beta>0$是参数。我们证明了在（0，\infty）$中存在一个$\bar\beta\，使得当$\beta \bar\beta$时，解表现出弱无序.证明技术使用高斯乘性混沌理论的元素。
We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\eps,t}= \frac 12 \Delta u_{\eps,t} d t+ \beta \eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1,$$ in dimension $d\geq 3$, where $B_{\eps,t}$ is a spatially smoothed (at scale $\eps$) space-time white noise, and $\beta>0$ is a parameter. We show the existence of a $\bar\beta\in (0,\infty)$ so that the solution exhibits weak disorder when $\beta \bar\beta$. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.