Heat Kernel Measure Research Articles

Matrix-valued modified logarithmic Sobolev inequality for sub-Laplacian on [formula omitted]

We prove that the canonical sub-Laplacian on SU(2) admits a modified log-Sobolev inequality on matrix-valued functions, independent of the matrix sizes. This establishes the first example of a matrix-valued modified log-Sobolev inequality for a sub-Laplacian. We also show that on Lie groups the heat kernel measure pt at time t satisfies matrix-valued modified log-Sobolev inequality with constants in order O(t−1).

Quasi-Invariance for Infinite-Dimensional Kolmogorov Diffusions

We prove Cameron-Martin type quasi-invariance results for the heat kernel measure of infinite-dimensional Kolmogorov and related diffusions. We first study quantitative functional inequalities for appropriate finite-dimensional approximations of these diffusions, and we prove these inequalities hold with dimension-independent coefficients. Applying an approach developed by Baudoin, Driver, Gordina, and Melcher previously, these uniform bounds may then be used to prove that the heat kernel measure for certain of these infinite-dimensional diffusions is quasi-invariant under changes of the initial state.

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modeled on an abstract Wiener space.

Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron--Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon--Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.

Transportation inequalities for Markov kernels and their applications

We study the relationship between functional inequalities for a Markov kernel on a metric space X and inequalities of transportation distances on the space of probability measures P(X). Extending results of Luise and Savaré on Hellinger–Kantorovich contraction inequalities for the particular case of the heat semigroup on an RCD(K,∞) metric space, we show that more generally, such contraction inequalities are equivalent to reverse Poincaré inequalities. We also adapt the “dynamic dual” formulation of the Hellinger–Kantorovich distance to define a new family of divergences on P(X) which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities. We discuss applications including results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.

Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups

On a stratified Lie group G equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on Lp spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group G. Moreover, if G satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.

Smoothing and non-smoothing via a flow tangent to the Ricci flow

We study a transformation of metric measure spaces introduced by Gigli and Mantegazza consisting in replacing the original distance with the length distance induced by the transport distance between heat kernel measures. We study the smoothing effect of this procedure in two important examples. Firstly, we show that in the case of particular Euclidean cones, a singularity persists at the apex. Secondly, we generalize the construction to a sub-Riemannian manifold, namely the Heisenberg group, and show that it regularizes the space instantaneously to a smooth Riemannian manifold.

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

This paper studies the empirical laws of eigenvalues and singular values for random matrices drawn from the heat kernel measures on the unitary groups \({\mathbb {U}}_N\) and the general linear groups \({\mathbb {GL}}_N\), for \(N\in {\mathbb {N}}\). It establishes the strongest known convergence results for the empirical eigenvalues in the \({\mathbb {U}}_N\) case, and the first known almost sure convergence results for the eigenvalues and singular values in the \({\mathbb {GL}}_N\) case. The limit noncommutative distribution associated with the heat kernel measure on \({\mathbb {GL}}_N\) is identified as the projection of a flow on an infinite-dimensional polynomial space. These results are then strengthened from variance estimates to \(L^p\) estimates for even integers p.

Holomorphic functions and the Itô chaos

This paper is concerned with the characterization of spaces of square integrable holomorphic functions on a complex manifold, $G$, in terms of the derivatives of the function at a fixed point $o\in G$. The reproducing kernel properties of square integrable holomorphic functions are reviewed and a number of examples are given. These examples include square integrable holomorphic functions relative to Gaussian measures on complex Euclidean spaces along with their generalizations to heat kernel measures on complex Lie groups. These results are intimately related to the Itô's chaos expansion in stochastic analysis and to the Fock space description of free quantum fields in physics.

Hypoelliptic heat kernels on infinite-dimensional Heisenberg groups

We study the law of a hypoelliptic Brownian motion on an infinite-dimensional Heisenberg group based on an abstract Wiener space. We show that the endpoint distribution, which can be seen as a heat kernel measure, is absolutely continuous with respect to a certain product of Gaussian and Lebesgue measures, that the heat kernel is quasi-invariant under translation by the Cameron–Martin subgroup, and that the Radon–Nikodym derivative is Malliavin smooth.

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

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

Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups

We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in \cite{BaudoinGarofalo2011}.

A subelliptic Taylor isomorphism on infinite-dimensional Heisenberg groups

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

From the heat measure to the pinned Wiener measure on loop groups

For each L p -Wasserstein distance ( p > 1 ) with the cost function induced by the L 2 -distance on loop groups, we show that there exists a unique optimal transport map solving the Monge–Kantorovich problem when the initial probability measure is absolutely continuous with respect to the heat kernel measure. In particular, this provides us a family of measurable maps on loop groups which push the heat kernel measure forward to the pinned Wiener measure.

Central limit theorem for the heat kernel measure on the unitary group

We prove that for a finite collection of real-valued functions f 1 , … , f n on the group of complex numbers of modulus 1 which are derivable with Lipschitz continuous derivative, the distribution of ( tr f 1 , … , tr f n ) under the properly scaled heat kernel measure at a given time on the unitary group U ( N ) has Gaussian fluctuations as N tends to infinity, with a covariance for which we give a formula and which is of order N − 1 . In the limit where the time tends to infinity, we prove that this covariance converges to that obtained by P. Diaconis and S.N. Evans in a previous work on uniformly distributed unitary matrices. Finally, we discuss some combinatorial aspects of our results.

Integrated Harnack inequalities on Lie groups

We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an “integrated” Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang’s Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional “Lie” groups.

Heat kernel analysis on semi-infinite Lie groups

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 L p norms of the Radon–Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure μ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the “Lie algebra” of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the L2(ν)-closure of holomorphic polynomials by their values on the Cameron–Martin subgroup.

Surjectivity of the Taylor map for complex nilpotent Lie groups

AbstractA Hermitian formqon the dual space,*, of the Lie algebra,, of a simply connected complex Lie group,G, determines a sub-Laplacian, Δ, onG. Assuming Hörmander's condition for hypoellipticity, there is a smooth heat kernel measure, ρt, onGassociated toetΔ/4. In a companion paper [6], we proved the existence of a unitary “Taylor” map from the space of holomorphic functions inL2(G, ρt) ontoJt0(a subspace of) the dual of the universal enveloping algebra of. Here we give a very different proof of the surjectivity of the Taylor map under the assumption thatGis nilpotent. This proof provides further insight into the structure of the Taylor map. In particular we show that the finite rank tensors are dense inJt0when the Lie algebra is graded and the Laplacian is adapted to the gradation. We also show how the Fourier–Wigner transform produces a natural family of holomorphic functions inL2(G, ρt), for appropriatet, whenGis the complex Heisenberg group.

Heat kernel analysis on infinite-dimensional Heisenberg groups

We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, { ν t } t > 0 , are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron–Martin type quasi-invariance results, (3) good L p -bounds on the corresponding Radon–Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.