2023/07/31 by Hakiki, Youssef, Viens, Frederi
#28A78 #60G15 #60G17 #60J45 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2307.16886
Let X be a d-dimensional Gaussian process in [0,1], where the component are independent copies of a scalar Gaussian process X0 on [0,1] with a given general variance function γ2(r)=Var(X0(r)) and a canonical metric δ(t,s):=(𝔼(X0(t)-X0(s))2)1/2 which is commensurate with γ(t-s). Under a weak regularity condition on γ, referred to below as \mathbf(C0+), which allows γ to be far from Hölder-continuous, we prove that for any Borel set E⊂ [0,1], the Hausdorff dimension of the image X(E) and of the graph GrE(X) are constant almost surely. Furthermore, we show that these constants can be explicitly expressed in terms of dimδ(E) and d. However, when \mathbf(C0+) is not satisfied, the classical methods may yield different upper and lower bounds for the underlying Hausdorff dimensions. This case is illustrated via a class of highly irregular processes known as logBm. Even in such cases, we employ a new method to establish that the Hausdorff dimensions of X(E) and GrE(X) are almost surely constant. The method uses the Karhunen-Loève expansion of X to prove that these Hausdorff dimensions are measurable with respect to the expansion's tail sigma-field. Under similarly mild conditions on γ, we derive upper and lower bounds on the probability that the process X can reach the Borel set F in ℝd from the Borel set E in [0,1]. These bounds are obtained by considering the Hausdorff measure and the Bessel-Riesz capacity of E× F in an appropriate metric ρδ on the product space, relative to appropriate orders. Moreover, we demonstrate that the dimension d plays a critical role in determining whether X|E hits F or not.