226 research outputs found
Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry
We describe the exponential map from an infinite-dimensional Lie algebra to
an infinite-dimensional group of operators on a Hilbert space. Notions of
differential geometry are introduced for these groups. In particular, the Ricci
curvature, which is understood as the limit of the Ricci curvature of
finite-dimensional groups, is calculated. We show that for some of these groups
the Ricci curvature is
A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups
Let denote an infinite-dimensional Heisenberg-like group, which is a
class of infinite-dimensional step 2 stratified Lie groups. We consider
holomorphic functions on that are square integrable with respect to a heat
kernel measure which is formally subelliptic, in the sense that all appropriate
finite dimensional projections are smooth measures. We prove a unitary
equivalence between a subclass of these square integrable holomorphic functions
and a certain completion of the universal enveloping algebra of the
"Cameron-Martin" Lie subalgebra. The isomorphism defining the equivalence is
given as a composition of restriction and Taylor maps.Comment: Initially posted in June 2011, with minor corrections in November
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Harnack inequalities in infinite dimensions
We consider the Harnack inequality for harmonic functions with respect to
three types of infinite dimensional operators. For the infinite dimensional
Laplacian, we show no Harnack inequality is possible. We also show that the
Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes,
although functions that are harmonic with respect to these processes do satisfy
an a priori modulus of continuity. Many of these processes also have a coupling
property. The third type of operator considered is the infinite dimensional
analog of operators in H\"{o}rmander's form. In this case a Harnack inequality
does hold.Comment: Minor revision of the previous versio
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