Which sets contain multiple points of Brownian motion?

Abstract

In 1950, Dvoretzky, Erdös and Kakutani [2] showed that in ℝ3 almost all paths of Brownian motion X have double points, or self-intersections of order 2 (there are no triple points [4]); later the same authors proved that almost all sample paths of Brownian motion in the plane have points of arbitrarily high multiplicity (a point x in ℝ2 is a k-tuple point for the path ω, or a self-intersection of order k, if there are times tl < t2 < … < tk such that x = X(t1, ω) = X(t2, ω) = … X(tk, ω)).

The Multiple Points of Fractional Brownian Motion

“…To avoid negative values of the logarithm, we define log + (x) = max{log(x), 0}. Our results extend Theorem 1 and 3 of Tongring [14].…”

“…It is also well known that a Brownian path in R 2 will hit a compact set if and only if the set has positive logarithmic capacity [9]. Nils Tongring [14] combined these results in 1987 to prove that if B is a Brownian motion in R 2 , and E is a fixed compact set in the plane with positive capacity with respect to the function φ(s) = (log(1/s)) k , k a positive integer, then E contains k-tuple points for almost all paths. We say that x is a k-tuple point (or k-multiple point ) for the path ω if there are times…”

Random fractals and Markov processes

“…, this question was considered by Evans (1987b) and Tongring (1988), who proved some sufficient conditions and different necessary conditions for P{Λ ∩ M k = ∅} > 0. Fitzsimmons and Salisbury (1989) proved that the sufficient condition of Evans (1987b) and Tongring (1988) for planar Brownian motion is also necessary. By using the approach of intersection equivalence, Peres (1999, Corollary 15.4) proves the following much more general result.…”

Intersection-equivalence of Brownian paths and certain branching processes

“…The proof of Theorem 1.1 applies, using Proposition 3.2 (ii) and Corollary 4.3 (ii). D One direction of the following corollary was proved by Evans (1987) and Tongring (1988); the full equivalence was proved in a more general setting by Fitzsimmons and Salisbury (1989). Remark.…”