Abstract
This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.
Highlights
Let v = {v(x), x ∈ Rk} be a centered continuous Rd-valued Gaussian random field defined on a probability space (Ω, F, P) with i.i.d. components
For a set T ⊂ Rk (e.g., T = (0, ∞)k, or T = [0, 1]k) and an integer m ≥ 2, we say that z ∈ Rd is an m-multiple point of v(x) on T if, with positive probability, there are m distinct points x1, . . . , xm ∈ T such that z = v(x1) = · · · = v(xm)
Several authors have studied the existence of multiple points of Gaussian random fields
Summary
Our main purpose is to continue the work of [5] and extend Talagrand’s approach in [13] to a large class of Gaussian random fields which include fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations with constant coefficients. Sm ∈ T , with high probability, there are small neighbourhoods of si in which the maximum of the increments v(xi) − v(si) (1 ≤ i ≤ m) could be smaller than one would expect from the Hölder regularity This observation allows us to use balls of different radii to construct an efficient random cover for the set of multiple points, which is essential for proving the non-existence of multiple points in the critical dimension. Specific constants will be denoted by K1, K2, c1, etc
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