arXiv subjects

Liangbing Luo

Publications and source records attributed to Liangbing Luo.

7 recordsLinked to original sources

arXiv2026

Heat content and spectrum for subordinated sub-Laplacians

We study heat content and spectral properties of subordinated sub-Laplacians on arbitrary Carnot groups. We consider restrictions of such operators to bounded open sets with zero Dirichlet boundary condition and study their spectral properties. In particular, for a large class of subordinators we give explicit eigenvalue estimates in terms of the subordinator and the eigenvalues of the sub-Laplacian. We also provide large-time asymptotics of the heat content for the subordinated sub-Laplacian in terms of its spectral gap. For fractional sub-Laplacians we prove short-time asymptotics for the corresponding heat content and relative heat content. Our approach combines semigroup methods, probabilistic techniques, geometric measure theory, and heat kernel estimates.

math.PR↗

arXiv2026

Approximation of harmonic functions on metric measure spaces of controlled geometry via discrete graphs

Given a complete doubling metric measure space $X$ that supports a $2$-Poincar\'e inequality, we approximate harmonic functions on a bounded domain $\Omega$ with a prescribed Newton-Sobolev boundary data. Our approach is based on the approximation of the underlying space $X$ by a family of graphs. This approximated harmonic function is realized as the weak limit of a sequence of functions obtained from the graph minimizers. We prove that such a function is a minimizer with respect to a nonlinear energy form on $N^{1,2}_0(\Omega)$, which is in turn, majorized by the upper gradient energy on $N^{1,2}(X)$. This energy form on $N^{1,2}_0(\Omega)$ is obtained as a $\Gamma$-limit of a sequence of induced energy forms projected from the discrete energy form on the approximating graphs.

math.AP↗

arXiv2025

Logarithmic Sobolev inequalities on infinite-dimensional reduced Heisenberg groups

We construct a family of infinite-dimensional reduced Heisenberg groups which can be viewed as infinite-dimensional homogeneous spaces. Such a space is an analogue of finite-dimensional reduced Heisenberg groups in infinite dimensions. We study properties of the hypoelliptic heat kernel measure on this space, including hypoelliptic logarithmic Sobolev inequalities there.

math.PR↗

arXiv2024

Non-Markovian maximal couplings and a vertical reflection principle on a class of sub-Riemannian manifolds

We develop an approach to constructing non-Markovian, non-co-adapted couplings for sub-Riemannian Brownian motions in sub-Riemannian manifolds with large symmetry groups by treating the specific cases of the three-dimensional Heisenberg group, higher-dimensional non-isotropic Heisenberg groups, SL(2,R) and its universal cover, and SU(2). Our primary focus is on the situation when the processes start from two points on the same vertical fiber, since in general Markovian or co-adapted couplings cannot give the sharp rate for the coupling time in this case. Non-Markovian couplings of this type on sub-Riemannian manifolds were first introduced by Banerjee-Gordina-Mariano, for the three-dimensional Heisenberg group, and were more recently extended by B\'en\'efice to SL(2,R) and SU(2), using a detailed consideration of the Brownian bridge. In contrast, our couplings are based on global isometries of the space, giving couplings that are maximal, as well as making the construction relatively simple and uniform across different manifolds. The coupled processes satisfy a reflection principle with respect to their coupling time, so that the coupling time reduces to the hitting time for one component of the Brownian motion, which is useful in explicitly bounding the tail probability of the coupling time. Further, it's natural to use this coupling as the second stage of a two-stage coupling when considering points on different vertical fibers. We estimate the coupling time in these various situations and give applications to inequalities for the heat semigroup.

math.PR↗

arXiv2023

Construction of a Dirichlet form on metric measure spaces of controlled geometry

Given a compact doubling metric measure space $X$ that supports a $2$-Poincaré inequality, we construct a Dirichlet form on $N^{1,2}(X)$ that is comparable to the upper gradient energy form on $N^{1,2}(X)$. Our approach is based on the approximation of $X$ by a family of graphs that is doubling and supports a $2$-Poincaré inequality. We construct a bilinear form on $N^{1,2}(X)$ using the Dirichlet form on the graph. We show that the $Γ$-limit $\mathcal{E}$ of this family of bilinear forms (by taking a subsequence) exists and that $\mathcal{E}$ is a Dirichlet form on $X$. Properties of $\mathcal{E}$ are established. Moreover, we prove that $\mathcal{E}$ has the property of matching boundary values on a domain $Ω\subseteq X$. This construction makes it possible to approximate harmonic functions (with respect to the Dirichlet form $\mathcal{E}$) on a domain in $X$ with a prescribed Lipschitz boundary data via a numerical scheme dictated by the approximating Dirichlet forms, which are discrete objects.

math.MG↗

arXiv2023

Logarithmic Sobolev inequalities on homogeneous spaces

We consider sub-Riemannian manifolds which are homogeneous spaces equipped with a natural sub-Riemannian structure induced by a transitive action by a Lie group. In such a setting, the corresponding sub-Laplacian is not an elliptic but a hypoelliptic operator. We study logarithmic Sobolev inequalities with respect to the hypoelliptic heat kernel measure on such homogeneous spaces. We show that the logarithmic Sobolev constant can be chosen to depend only on the Lie group acting transitively on such a homogeneous space but the constant is independent of the action of its isotropy group. This approach allows us to track the dependence of the logarithmic Sobolev constant on the geometry of the underlying space, in particular we are able to show that the logarithmic Sobolev constants is independent of the dimension of the underlying spaces in several examples. We illustrate the results by considering the Grushin plane, non-isotropic Heisenberg groups, Heisenberg-like groups, Hopf fibration, $\operatorname{SO}(3)$, $\operatorname{SO}(4)$, and compact Heisenberg manifolds.

math.AP↗

arXiv2021

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 modelled on an abstract Wiener space.

math.AP↗