  Consider p independent Brownian motions in R^d, each running up to its first
exit time from an open domain B, and their intersection local time l as a
measure on B. We give a sharp criterion for the finiteness of exponential
moments,
  E[exp(\sum_{i=1}^n (int_B f_i(x) l(dx))^{1/p})], where f_1, ...,f_n are
nonnegative, bounded functions with compact support in B. We also derive a law
of large numbers for intersection local time conditioned to have large total
mass.
