Smoothness of heat kernel measures on infinite-dimensional Heisenberg-like groups

Abstract

We study measures associated to Brownian motions on infinitedimensional Heisenberg-like groups. In particular, we prove that the associated path space measure and heat kernel measure satisfy a strong definition of smoothness.

Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups

“…where ψ X T : G → R, essentially the first logarithmic derivative of ν T , is a "Malliavin smooth" function in the sense that ψ X T is in L ∞− (ν T ) = ∩ p∈[1,∞) L p (ν T ) and all of its derivatives exist and are in also in L ∞− (ν T ); see Lemma 8.7 and Corollaries 8.10 and 8.11. Similar techniques were used in [9] to study elliptic heat kernel measures on G.…”

“…) and all of its derivatives exist and are in also in L ∞− (ν T ); see Lemma 8.7 and Corollaries 8.10 and 8.11. Similar techniques were used in [9] to study elliptic heat kernel measures on G.…”

“…In this section, we expand on the quasi-invariance results in the previous section, and study the integrability and smoothness of the corresponding Radon-Nikodym derivatives. Similar techniques were used to handle the elliptic case in [9]. Again for this section, we fix T > 0.…”