No zero-crossings for random polynomials and the heat equation
No zero-crossings for random polynomials and the heat equation
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随机多项式和热方程没有零交叉
DOI：
10.1214/13-aop852
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发表时间：
2012
影响因子：
2.3
通讯作者：
S. Mukherjee
中科院分区：
文献类型：
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作者：
A. Dembo;S. Mukherjee
考虑随机多项式∑ni=0aixi，其独立均值为零的正态系数ai，其方差是α阶的规则变化函数（在i中）。我们导出了中心高斯过程持续指数连续性的一般判据，并利用这些判据证明了这样的多项式在[0，1]中没有根的概率为n−Bα+o（1），在（1，∞）中没有根的概率为n− b 0 +o（1），因此对于偶数n，它没有真实的根的概率为n− 2 B α− 2b 0 +o（1）.当α≤−1时，bα=0，否则Bα∈（0，∞）与详细的规则变化的方差函数无关，对应于光滑样本路径的显式平稳高斯过程的持久性概率.进一步，通过精确求解高斯白色噪声引起的d维热方程的解d（x，t），我们证实了对于所有t∈[1，T]，d（x，t）的概率d（x，t）0，是T−Bα+o（1），其中α=d/2−1。
Consider random polynomial ∑ni=0aixi of independent mean-zero normal coefficients ai, whose variance is a regularly varying function (in i) of order α. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in [0,1] with probability n−bα+o(1), and no roots in (1,∞) with probability n−b0+o(1), hence for n even, it has no real roots with probability n−2bα−2b0+o(1). Here, bα=0 when α≤−1 and otherwise bα∈(0,∞) is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution ϕd(x,t) to the d-dimensional heat equation initiated by a Gaussian white noise ϕd(x,0), we confirm that the probability of ϕd(x,t)≠0 for all t∈[1,T], is T−bα+o(1), for α=d/2−1.