arXiv subjects

Maria Gordina

Publications and source records attributed to Maria Gordina.

At least 19 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

Free Bakry--\'{E}mery Calculus

We prove a free probabilistic Bakry--\'{E}mery criterion for logarithmic Sobolev inequalities and hypercontractivity. Our approach relies on a noncommutative version of the carr\'{e} du champ operator and its iteration, which we are able to define in the free probability setting. This allows us to formulate a positive curvature condition, which is shown to be sufficient for logarithmic Sobolev inequalities and hypercontractivity in this setting. In the setting of free Gibbs measures, we also show that this curvature condition can be expressed in terms of the positivity of the Hessian of the associated potential.

math.OA↗

arXiv2026

Semiclassical limit of Polyakov-Liouville measure and Q-Curvature Uniformization on even-dimensional manifolds

We study the semiclassical limit of the Polyakov-Liouville measure $\boldsymbol{\nu}_\gamma$, which is a non-Gaussian measure on $H^{-\eps}(M)$ that has recently been extended from Riemann surfaces to general Riemannian manifolds $(M,g)$ of even dimension. We show that under an appropriate rescaling in the semiclassical limit as $\gamma\to0$, the normalized Polyakov-Liouville measure $\Q_\gamma$ concentrates on the unique smooth weight $u$ for which the conformal metric $e^{2u}g$ on $M$ has constant $Q$-curvature.

math.PR↗

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↗

arXiv2025

Spectral theory for L\'evy and L\'evy-Ornstein-Uhlenbeck semigroups on step 2 Carnot groups

We consider non-local perturbations $\Delta^\psi_G$ of sub-Laplacians on a step $2$ Carnot group $G$. The perturbations are by translation-invariant non-local operators acting along the vertical directions in $G$. We use harmonic analysis on $G$ to obtain intertwining relationship between the semigroups generated by $\Delta^\psi_G$ and some strongly continuous contraction semigroups on Euclidean spaces with purely continuous spectrum, and as a result we identify the spectrum of $\Delta^\psi_G$. Further we introduce the L\'evy-Ornstein-Uhlenbeck (OU) semigroup corresponding to $\Delta^\psi_G$. We prove that these Markov semigroups are ergodic, though they are not normal operators on $L^2$ space with respect to the invariant distribution $\mathsf{p}_\psi$. The intertwining relationships allow us to show that all L\'evy-OU generators on $G$ are isospectral, that is, they have the same eigenvalues with the same multiplicities. As a byproduct, we obtain a precise description of the eigenspaces, and also derive explicit formula for the co-eigenfunctions corresponding to some eigenvalues.

math.PR↗

arXiv2025

Geodesics on Grushin spaces

We consider higher-dimensional generalizations of the $\alpha$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using the calculus of generalized trigonometric functions, we obtain explicit conjugate times for geodesics starting at a Riemannian point. We propose a conjectured cut time $\tau=\min\{\tau_j\}$ obtained as the minimum of several candidates, each deriving from the symmetries of components of the Hamiltonian flow. We prove that it provides a lower bound on the first conjugate time, a key step in the extended Hadamard technique. In the three-dimensional case, we combine this method with a density argument to establish the conjecture and obtain the full optimal synthesis.

math.DG↗

arXiv2024

Uniform doubling for abelian products with $\operatorname{SU}(2)$

We prove that the uniform doubling property holds for every Lie group which can be written as a quotient group of $\operatorname{SU}(2) \times \mathbb{R}^n$ for some $n$. In particular, this class includes the four-dimensional unitary group $\operatorname{U}(2)$. As this class contain non-compact as well as compact Lie groups, we discuss a number of analytic and spectral consequences for the corresponding heat kernels.

math.DG↗

arXiv2024

Sharp defective log-Sobolev inequalities on H-type groups

In this paper we prove a sharp defective log-Sobolev inequality on H-type groups. Then we use such an inequality to show exponential integrability of Lipschitz functions with respect to the heat kernel measure. A defective log-Sobolev-type inequality for the Gaussian-like measure with respect to the sub-Riemannian distance is also proved on arbitrary H-type groups.

math.AP↗

arXiv2024

Dirichlet metric measure spaces: spectrum, irreducibility, and small deviations

We show that for ultracontractive irreducible Dirichlet metric measure spaces, the Dirichlet spectrum is discrete for a restriction to any connected open set without any assumption on regularity of the boundary. The main applications include small deviations for the corresponding Hunt process and large time asymptotics for the generalized heat content. Our examples include Riemannian and sub-Riemannian manifolds, as well as non-smooth and fractal spaces.

math.PR↗

arXiv2024

Large deviations principle for sub-Riemannian random walks

We study large deviations for random walks on stratified (Carnot) Lie groups. For such groups, there is a natural collection of vectors which generates their Lie algebra, and we consider random walks with increments in only these directions. Under certain constraints on the distribution of the increments, we prove a large deviation principle for these random walks with a natural rate function adapted to the sub-Riemannian geometry of these spaces.

math.PR↗

arXiv2024

Rigidity of the subelliptic heat kernel on $\operatorname{SU}(2)$

We study heat kernel rigidity for the Lie group $\operatorname{SU}\left( 2 \right)$ kernel equipped with a sub-Riemannian structure. We prove that a metric measure space equipped with a heat kernel of a special form is bundle-isometric to the Hopf fibration $\operatorname{U}\left( 1 \right)\to \operatorname{SU}\left( 2 \right)\to \mathbb{CP}^1$, which coincides with the sub-Riemannian sphere $\operatorname{SU}\left( 2 \right)$.

math.AP↗

arXiv2024

Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra

We use the SLE$_\kappa$ loop measure to construct a natural representation of the Virasoro algebra of central charge $c = c(\kappa) \le 1$. In particular, we introduce a non-degenerate bilinear Hermitian form (and non positive-definite) using the SLE loop measure and show that the representation is indefinite unitary. Our proof relies on the infinitesimal conformal restriction property of the SLE loop measure.

math.PR↗

arXiv2024

Dimension-independent functional inequalities by tensorization and projection arguments

We study stability under tensorization and projection-type operations of gradient-type estimates and other functional inequalities for Markov semigroups on metric spaces. Using transportation-type inequalities obtained by F. Baudoin and N. Eldredge in 2021, we prove that constants in the gradient estimates can be chosen to be independent of the dimension. Our results are applicable to hypoelliptic diffusions on sub-Riemannian manifolds and some hypocoercive diffusions. As a byproduct, we obtain dimension-independent reverse Poincar\'{e}, reverse logarithmic Sobolev, and gradient bounds for Lie groups with a transverse symmetry and for non-isotropic Heisenberg groups.

math.PR↗

arXiv2024

Spectral gap bounds on H-type groups

In this note we provide bounds on the spectral gap for the Dirichlet sub-Laplacians on $H$-type groups. We use probabilistic techniques and in particular small deviations of the corresponding hypoelliptic Brownian motion.

math.PR↗

arXiv2023

The Stochastic Schwarz lemma on Kähler Manifolds by Couplings and Its Applications

We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall-Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete non-compact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau--Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a byproduct, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.

math.DG↗

arXiv2023

Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise

For a family of infinite-dimensional diffusions with degenerate noise, we develop a modified $Γ$ calculus on finite-dimensional projections of the equation in order to produce explicit functional inequalities that can be scaled to infinite dimensions. The choice of our $Γ$ operator appears canonical in our context, as the estimates depend only on the induced control distance. We apply the general analysis to a number of examples, exploring implications for quasi-invariance and uniqueness of stationary distributions.

math.PR↗

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↗

arXiv2023

Non-commutative $L^{p}$ spaces and Grassmann stochastic analysis

We introduce a theory of non-commutative $L^{p}$ spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann Itô processes, Girsanov's formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables including a Grassmann analog of the $Φ^{4}_{2}$ Euclidean QFT in a bounded region and weak solution to singular SPDEs in the spirit of the early work of Jona-Lasinio and Mitter on the stochastic quantisation of $Φ^{4}_{2}$.

math.PR↗