An explicit Schilder‐type theorem for super‐Brownian motions

Like ordinary Brownian motion, super‐Brownian motion, a central object in the theory of superprocesses, is a universal object arising in a variety of settings. Schilder‐type theorems and Cramér‐type theorems are two of the major topics for large‐deviation theory. A Schilder‐type (which is also a Cramér‐type) sample large deviation for super‐Brownian motions with a good rate function represented by a variation formula was established in 1993 and 1994; since then there have been very valuable contributions for giving an affirmative answer to the question of whether this sample large deviation holds with an explicit good rate function. In this paper, thanks to previous results on this issue and the Brownian snake, we establish such a large deviation for nonzero finite initial measures. © 2010 Wiley Periodicals, Inc.