Ju n 20 17 A CENTRAL LIMIT THEOREM FOR THE ANNEALED PATH MEASURES FOR THE STOCHASTIC HEAT EQUATION AND THE CONTINUOUS DIRECTED POLYMER IN d ≥ 3
Ju n 20 17 A CENTRAL LIMIT THEOREM FOR THE ANNEALED PATH MEASURES FOR THE STOCHASTIC HEAT EQUATION AND THE CONTINUOUS DIRECTED POLYMER IN d ≥ 3
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17年6月20日随机热方程和d≥3连续定向聚合物退火路径测度的中心极限定理
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发表时间：
2017
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通讯作者：
Chiranjib Mukherjee
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作者：
Chiranjib Mukherjee
文摘：在d≥3中，我们考虑由dvε给出的软化随机热方程，t=1/2∆vε，t+βε(d−2)/2vε，tdbε，t，初始条件vε，0=1.这里Bε，t是时空平滑的时空高斯白噪声，β>；0是参数.当仅在空间上平滑白噪声势时，在早期工作([MSZ16])中，当平滑参数β0被关闭时，根据平滑的解vε的限制对象中的ε→&>0的值，获得相变。在随机环境中定向聚合物的语言中，解vε表示聚合物路径测量的配分函数，并且上述相变具有弱无序和强无序的特征。在噪声的时空光滑化下，我们考虑了退火场中的聚合物路径度量，证明了对于β>；0和d≥3的任意值，在退火态路径度量下重新标度的布朗路径的分布收敛于具有非简并扩散常数的高斯定律。
Abstract: In d ≥ 3, we consider the mollified stochastic heat equation given by dvε,t = 1/2∆vε,t + βε (d−2)/2 vε,t dBε,t with initial condition vε,0 = 1. Here Bε,t is spatiotemporally smoothened space-time Gaussian white noise and β > 0 is a parameter. When the white noise potential was smoothened only spatially, in an early work ([MSZ16]), a phase transition was obtained, depending on the value of β > 0 in the limiting object of the smoothened solution vε as the smoothing parameter ε → 0 was turned off. In the language of directed polymer in a random environment, the solution vε represent the partition function of the polymer path measure and the aforementioned phase transition was characterized by weak and strong disorder. In the present article, under space-time smoothing of the noise, we consider the polymer path measure in the annealed set up and show that for any value of β > 0 and in d ≥ 3, the distribution of the rescaled Brownian path under the annealed path measure converges to a Gaussian law with a non-degenerate diffusion constant.