NO ZERO-CROSSINGS FOR RANDOM POLYNOMIALS AND THE HEAT EQUATION

• 成果类型: Article
• 署名作者: Dembo, Amir; Mukherjee, Sumit
• 署名单位: Stanford University; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP852
• 发表日期: 2015
• 页码: 85-118
• 关键词: real roots

摘要

Consider random polynomial Sigma(n)(i=0) a(i)x(i) of independent mean-zero normal coefficients a(i), whose variance is a regularly varying function (in i) of order alpha. 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(-b alpha+o(1)), and no roots in (1, infinity) with probability n(-b0+o(1)), hence for n even, it has no real roots with probability n(-2b alpha-2b0+o(1)) Here, b(alpha) = 0 when alpha <= -1 and otherwise b(alpha) is an element of (0,infinity ) 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 phi(d) (x, t) to the d-dimensional heat equation initiated by a Gaussian white noise phi(d)(x, o), we confirm that the probability of phi(d)(x, t) not equal 0 for all t is an element of [1,T], is T-b alpha+o(1), for alpha = d/2 - 1.