From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions
From directed polymers in spatial-correlated environment to stochastic heat equations driven by fractional noise in 1+1 dimensions
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从空间相关环境中的定向聚合物到 1 1 维分数噪声驱动的随机热方程
DOI：
10.1016/j.spa.2019.09.018
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发表时间：
--
影响因子：
1.4
通讯作者：
Guanglin Rang
中科院分区：
文献类型：
--
作者：
Guanglin Rang
我们考虑了随机环境中定向聚合物配分函数的极限行为，它用一个线性模型来表示，而不是1+1维的一族iid变量。在空间相关性按代数衰减的假设下，利用Alberts等人(2014)发展的方法，我们证明了尺度配分函数作为定义在[0，1]×R上的过程，弱收敛于由分数布朗场驱动的随机热方程解。分数赫斯特参数由随机环境的关联指数确定。这里涉及分数高斯场的多重积分和平稳过程的谱表示。
We consider the limit behavior of partition function of directed polymers in random environment, which is represented by a linear model instead of a family of iid variables in 1+ 1 dimensions. Under the assumption on the environment that its spatial correlation decays algebraically, using the method developed in Alberts et al.(2014), we show that the scaled partition function, as a process defined on [0, 1]× R, converges weakly to the solution to some stochastic heat equations driven by fractional Brownian field. The fractional Hurst parameter is determined by the correlation exponent of the random environment. Here multiple Itô integral with respect to fractional Gaussian field and spectral representation of stationary process are heavily involved.