Some Limit Theorems for Super-Brownian Motion and Semilinear Differential Equations

Abstract

The empirical measure, a generalization of occupation times, of a super-Brownian motion is studied. In our case the empirical measure tends almost surely to Lebesgue measure as time $t \rightarrow \infty$. Asymptotic probabilities of deviation from this central behavior by various orders (large, not very large and normal deviations) are estimated. Extension to similar superprocesses, that is, Dawson-Watanabe processes, is discussed. Our analytic approach also produces new results for semilinear PDE's.