Polar functions and intersections of the random string processes

Let {u _s (x): s ≥ 0, x ∈ ?} be a random string taking values in ? ^d . The main goal of this paper is to discuss the characteristics of the polar functions of {u _s (x): s ≥ 0, x ε ?}. The relationship between a class of continuous functions satisfying the H?lder condition and a class of polar-functions of {u _s (x): s ≥ 0, x ε ?} 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.