The subelliptic heat kernel on SU(2): Representations, Asymptotics and Gradient bounds

The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities.Comment: Update: Added section + Correction of typo

A gauge model for quantum mechanics on a stratified space

In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified K\"ahler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure furnishes a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is $\mathrm{SU}(2)$, we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.Comment: 38 pages, 9 figures. Changes: comments on the heat kernel and coherent states have been adde

Riesz potentials and nonlinear parabolic equations

The spatial gradient of solutions to nonlinear degenerate parabolic equations can be pointwise estimated by the caloric Riesz potential of the right hand side datum, exactly as in the case of the heat equation. Heat kernels type estimates persist in the nonlinear cas

Free Energies and fluctuations for the unitary Brownian motion

We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang--Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger--Dyson's ones, named here after Makeenko and Migdal

On the Kakutani-Itô-Segal-Gross and Segal-Bargmann-Hall Isomorphisms

AbstractRecently, Gross has shown that the Kakutani-Itô-Segal isomorphism theorem has an extension from the setting of Gaussian measure on a vector space to "heat kernel" measure (pt) on a simply connected Lie group (G) of compact type. The isomorphism relates L2(pt) to a certain completion of the universal enveloping algebra of g=̇ Lie(G). Gross proves this result using the Kakutani-Itô-Segal theorem and an infinite dimensional calculus associated to G-valued Brownian motion. Hijab has greatly simplified and clarified Gross′ proof. Hiiab′s proof avoids most, but not all, of the "infinite dimensional" analysis in the original proof. In this paper, we will build on Hijab′s proof to give a completely "finite dimensional" non-probabilistic proof of Gross′ isomorphism theorem. The proof given here relies heavily on Hall′s beautiful "extension" of the Segal-Bargmann transformation to the setting of compact Lie groups. This theorem relating L2(pt) to a certain L2-space of holomorphic functions (L2(GC) ∩ H) on the complexified Lie group GC will also be generalized to Lie groups of compact type. In the process, it is shown how to characterize, in terms of summability conditions on all the derivatives at the identity in GC, those holomorphic functions which are in L2(GC)