54 research outputs found

    Hypoelliptic heat kernel inequalities on Lie groups

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    This paper discusses the existence of gradient estimates for second order hypoelliptic heat kernels on manifolds. It is now standard that such inequalities, in the elliptic case, are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated "Ricci curvature" takes on the value -\infty at points of degeneracy of the semi-Riemannian metric associated to the operator. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are invariant under left translation. In particular, "L^p-type" gradient estimates hold for p\in(1,\infty), and the p=2 gradient estimate implies a Poincar\'e estimate in this context.Comment: 22 pages, 0 figures; final journal versio

    Heat kernel analysis on semi-infinite Lie groups

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    This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the LpL^p norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.Comment: 35 page

    Small Deviations for Time-Changed Brownian Motions and Applications to Second-Order Chaos

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    We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.Comment: 23 page

    Convergence of the empirical spectral measure of unitary Brownian motion

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    Let {UtN}t≥0\{U^N_t\}_{t\ge 0} be a standard Brownian motion on U(N)\mathbb{U}(N). For fixed N∈NN\in\mathbb{N} and t>0t>0, we give explicit bounds on the L1L_1-Wasserstein distance of the empirical spectral measure of UtNU^N_t to both the ensemble-averaged spectral measure and to the large-NN limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.Comment: 17 pages; rate of convergence for fixed tt sharpened and proof simplified; new result on convergence of paths of empirical measures on compact time interval

    A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

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    Let GG denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on GG 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 201
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