The paper considers instantly coalescing, or instantly annihilating, systems of one-dimensional Brownian particles on the real line. Under maximal entrance laws, the distribution of the particles at a fixed time is shown to be Pfaffian point processes closely related to the Pfaffian point process describing one dimensional distribution of real eigenvalues in the real Ginibre ensemble of random matrices. As an application, an exact large time asymptotic for the n-point density function for coalescing particles is derived.

We consider Funaki's model of a random string taking values in R d . It is specified by the following stochastic PDE,whereẆ =Ẇ (x, t) is two-parameter white noise, also taking values in R d . We find the dimensions in which the string hits points, and in which it has double points of various types. We also study the question of recurrence and transience.

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