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OF STABLE PROCESSES

The primary motivation for this study is a result of P61ya [8] which states that two points starting simultaneously in the plane and performing, independently, simple random walks will meet infinitely often with probability one. Dvoretzky and ErdSs [3] remark that three points starting simultaneously in the plane and performing independent simple random walks will not meet infinitely often; however, on the line they will meet with probability one. On the other hand, four points will not meet infinitely often even in R The purpose of this paper is to consider some similar questions about stable processes. First we ask for what values of a and N will two independent stable processes of index a in R meet? This question can be answered very easily because if X(t) and Y(t are the two processes and they start simultaneously, they will meet if and only if the process X(t) Y(t) returns to the origin. Since X(t) Y(t) is also a stable process of index a and a stable process of index a in R will return to the origin with positive probability (actually with probability one) if and only if a > 1 and N 1, this solves the problem of which values of a and N give rise to processes which will collide. It is also easy to verify that three independent stable processes can never have a simultaneous collision. The next problem is then to find the Hausdorff dimension of the intersection when a > 1 and N 1, i.e. the dimension of the set