Let X^H = {X^H（8）,8∈ R^N1} and XK = {X^K（t）,t ∈R^2} be two independent anisotropic Gaussian random fields with values in R^d with indices H = （H1,... ,HN1） ∈ （0, 1）^N1, K = （K1,..., KN2）∈ （0, 1）^N2, respectively. Existence of intersections of the sample paths of XH and XK is studied. More generally, let E1 R^N1, E2 R^N2 and F R^d be Borel sets. A necessary condition and a sufficient condition for P{（X^H（E1） ∩ X^K（E2）） ∩ F ≠ Ф} 〉 0 in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 x E2 x F in the metric space （R^N1＋N2＋d, ρ） are proved, whereρ is a metric defined in terms of H and K. These results are applicable to solutions of stochastic heat equations driven by space-time Gaussian noise and fractional Brownian sheets.
The main goal of this paper is to study the sample path properties for the harmonisable-type N-parameter multifractional Brownian motion, whose local regularities change as time evolves. We provide the upper and lower bounds on the hitting probabilities of an （N, d）-multifractional Brownian motion. Moreover, we determine the Hausdorff dimension of its inverse images, and the Hausdorff and packing dimensions of its level sets.
Let {us （x） ： s 〉 0, x ∈ JR} be a random string taking values in ]Rd. The main goal of this paper is to discuss the characteristics of the polar functions of {us （x） ： s ≥ 0, x ∈ JR}. The relationship between a class of continuous functions satisfying the HSlder condition and a class of polar-functions of {us（x） ： s 〉 0, x ∈ R} is presented. The Hausdorff and packing dimensions of the set that the string intersects a given non-polar continuous function are determined. The upper and lower bounds are obtained for the probability that the string intersects a given function in terms of respectively Hausdorff measure and capacity.
Let X（1） = {X（1）（s）, s ∈ R＋} and X（2） = {X（2）（t）, t ∈ R＋} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X （1） and X （2） are studied. More gener-ally, let E1, E2？（0,∞） and F ？Rd be Borel sets. A necessary condition and a suffcient condition for P{X（1）（E1）∩X（2）（E2）∩F 6=？}〉0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space （R＋× R＋× Rd,ρb）, whereρb is an unsymmetric metric defined in R＋× R＋× Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.
This paper studies fractal properties of polar sets for random string processes. We give upper and lower bounds of the hitting probabilities on compact sets and prove some sufficient conditions and necessary conditions for compact sets to be polar for the random string process. Moreover, we also determine the smallest Hausdorff dimensions of non-polar sets by constructing a Cantor-type set to connect its Hausdorff dimension and capacity.
