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 . .}. The relationship between a class of continuous functions satisfying the Hölder condition and a class of polar-functions of {u s (x): s . 0, x . .} 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.