ON THE COLLISION LOCAL TIME OF BIFRACTIONAL BROWNIAN MOTIONS

Let [Formula: see text], i = 1,2 be two independent bifractional Brownian motions of dimension 1, with indices Hi ∈ (0, 1) and Ki ∈ (0, 1]. We investigate the collision local time of bifractional Brownian motions [Formula: see text] where δ denotes the Dirac delta function at zero. We show that ℓT exists in L2, and it is Hölder continuous of order 1 - min {H1K1,H2K2}, and furthermore, it is also smooth in the sense of Meyer–Watanabe.

Collision Local Times and Measure-Valued Processes

We consider two independent Dawson-Watanabe super-Brownian motions, Y1 and Y2. These processes are diffusions taking values in the space of finite measures on ℝd. We show that if d ≤ 5 then with positive probability there exist times t such that the closed supports of intersect; whereas if d > 5 then no such intersections occur. For the case d ≤ 5, we construct a continuous, non-decreasing measure–valued process L(Y1, Y2), the collision local time, such that the measure defined by , is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition of L(Y1, Y2). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions with point interactions. In the course of our proofs we obtain smoothness results for the random measures that are uniform in t. These theorems use a nonstandard description of Yi and are of independent interest.