Properties of Measures and Processes Related to Brownian Motion on Infinite-Dimensional Heisenberg-Like Groups.

摘要

Brownian motion on infinite-dimensional Heisenberg-like groups has the form (Bt, B 0t + 12&smallint; t0 w (Bs, dBs))where B and B0 are Brownian motions on a separable Banach space and finite-dimensional Hilbert space, respectively. For measures on finite dimensional spaces, there are two equivalent definitions of smoothness - one that relies on Lebesgue measure and one that doesn't. In this dissertation, we present an adaptation of the latter definition of smoothness for measures on infinite-dimensional Heisenberg-like groups and show that the law of Brownian motion on these groups satisfies this definition of smoothness. We also note that Zt := &smallint;t 0 w(Bs, dBs) makes sense as an infinite-dimensional analogue of the Levy area process and prove a small deviations estimate for this process. Finally, we present a Chung-like law of the iterated logarithm and functional law of the iterated logarithm for Zt as applications of the small deviations result.