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Maria Gordina

Publications and source records attributed to Maria Gordina.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2023

Notions of null sets in infinite-dimensional Carnot groups

We study several notions of null sets on infinite-dimensional Carnot groups. We prove that a set is Aronszajn null if and only if it is null with respect to measures that are convolutions of absolutely continuous (CAC) measures on Carnot subgroups. The CAC measures are the non-abelian analogue of cube measures. In the case of infinite-dimensional Heisenberg-like groups we also show that being null in the previous senses is equivalent to being null for all heat kernel measures. Additionally, we show that infinite-dimensional Carnot groups that have locally compact commutator subgroups have the structure of Banach manifolds. There are a number of open questions included as well.

math.MG↗
arXiv2023

Dirichlet sub-Laplacians on homogeneous Carnot groups: spectral properties, asymptotics, and heat content

We consider sub-Laplacians in open bounded sets in a homogeneous Carnot group and study their spectral properties. We prove that these operators have a pure point spectrum, and prove the existence of the spectral gap. In addition, we give applications to the small ball problem for a hypoelliptic Brownian motion and the large time behavior of the heat content in a regular domain.

math.PR↗
arXiv2022

Complete Modified Logarithmic Sobolev inequality for sub-Laplacian on $SU(2)$

We prove that the canonical sub-Laplacian on $SU(2)$ admits a uniform modified log-Sobolev inequality for all its matrix-valued functions, independent of the matrix dimension. This is the first example of sub-Laplacian that a matrix-valued modified log-Sobolev inequality has been obtained. We also show that on Lie groups, the heat kernel measure $p_t$ at time $t$ admits matrix-valued modified log-Sobolev constants of order $O(t^{-1})$.

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↗
arXiv2021

Weighted $L^2$-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential $U$ allowing for singularities. By modifying the direct approach to convergence in $L^2$ pioneered by F. Hérau and developped by Dolbeault, Mouhot and Schmeiser, we show that the dynamics converges exponentially fast to equilibrium in the topologies $L^2(dμ)$ and $L^2(W^* dμ)$, where $μ$ denotes the invariant probability measure and $W^*$ is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter $γ$ in Langevin dynamics, by providing a lower bound scaling as $\min(γ, γ^{-1})$. The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

math.PR↗
arXiv2021

Diameter theorems on Kähler and quaternionic Kähler manifolds under a positive lower curvature bound

We define the orthogonal Bakry-Émery tensor as a generalization of the orthogonal Ricci curvature, and then study diameter theorems on Kähler and quaternionic Kähler manifolds under positivity assumption on the orthogonal Bakry-Émery tensor. Moreover, under such assumptions on the orthogonal Bakry-Émery tensor and the holomorphic or quaternionic sectional curvature on a Kähler manifold or a quaternionic Kähler manifold respectively, the Bonnet-Myers type diameter bounds are sharper than in the Riemannian case.

math.DG↗
arXiv2021

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.

math.PR↗
arXiv2021

Gamma calculus beyond Villani and explicit convergence estimates for Langevin dynamics with singular potentials

This paper studies convergence to equilibrium for second-order Langevin dynamics under general growth conditions on the potential. Although we are principally motivated by the case when the potential is singular, e.g. when the dynamics has repulsive forces and/or interactions, the results presented in this paper hold more generally. In particular, our main result is that, given (very) basic structural and growth conditions on the potential, the dynamics relaxes to equilibrium exponentially fast in an explicitly measurable way. The ``explicitness" of this result comes directly from the constants appearing in the growth conditions, which can all be readily estimated, and a local Poincaré constant for the invariant measure $μ$. This result is applied to the specific situation of a singular interaction and polynomial confining well to provide explicit estimates on the exponential convergence rate $e^{-σ}$ in terms of the number $N \geqslant 1$ of particles in the system. We will see that $σ\geqslant c/(ρ\vee N^p)$, where $ρ>0$ is the local Poincaré constant for $μ$ and $c>0, p\geqslant 1$ are constants that are independent of $N$.

math.PR↗
arXiv2020

Ornstein-Uhlenbeck processes with singular drifts: integral estimates and Girsanov densities

We consider a perturbation of a Hilbert space-valued Ornstein--Uhlenbeck process by a class of singular nonlinear non-autonomous maximal monotone time-dependent drifts. The only further assumption on the drift is that it is bounded on balls in the Hilbert space uniformly in time. First we introduce a new notion of generalized solutions for such equations which we call pseudo-weak solutions and prove that they always exist and obtain pathwise estimates in terms of the data of the equation. Then we prove that their laws are absolutely continuous with respect to the law of the original Ornstein--Uhlenbeck process. In particular, we show that pseudo-weak solutions always have continuous sample paths. In addition, we obtain integrability estimates of the associated Girsanov densities. Some of our results concern non-random equations as well, while probabilistic results are new even in finite-dimensional autonomous settings.

math.PR↗
arXiv2019

Gradient Bounds for Kolmogorov Type Diffusions

We study gradient bounds and other functional inequalities for the diffusion semigroup generated by Kolmogorov type operators. The focus is on two different methods: coupling techniques and generalized $Γ$-calculus techniques. The advantages and drawbacks of each of these methods are discussed.

math.PR↗
arXiv2018

Left-invariant geometries on $\mathrm{SU}(2)$ are uniformly doubling

A classical aspect of Riemannian geometry is the study of estimates that hold uniformly over some class of metrics. The best known examples are eigenvalue bounds under curvature assumptions. In this paper, we study the family of all left-invariant geometries on $\mathrm{SU}(2)$. We show that left-invariant geometries on $\mathrm{SU}(2)$ are uniformly doubling and give a detailed estimate of the volume of balls that is valid for any of these geometries and any radius. We discuss a number of consequences concerning the spectrum of the associated Laplacians and the corresponding heat kernels.

math.DG↗
arXiv2017

Coupling in the Heisenberg group and its applications to gradient estimates

We construct a non-Markovian coupling for hypoelliptic diffusions which are Brownian motions in the three-dimensional Heisenberg group. We then derive properties of this coupling such as estimates on the coupling rate, and upper and lower bounds on the total variation distance between the laws of the Brownian motions. Finally we use these properties to prove gradient estimates for harmonic functions for the hypoelliptic Laplacian which is the generator of Brownian motion in the Heisenberg group.

math.PR↗
arXiv2016

Hypoelliptic heat kernel on nilpotent Lie groups

The starting point of our analysis is an old idea of writing an eigenfunction expansion for a heat kernel considered in the case of a hypoelliptic heat kernel on a nilpotent Lie group $G$. One of the ingredients of this approach is the generalized Fourier transform. The formula one gets using this approach is explicit as long as we can find all unitary irreducible representations of $G$. In the current paper we consider an $n$-step nilpotent Lie group $G_{n}$ as an illustration of this technique. First we apply Kirillov's orbit method to describe these representations for $G_{n}$. This allows us to write the corresponding hypoelliptic heat kernel using an integral formula over a Euclidean space. As an application, we describe a short-time behavior of the hypoelliptic heat kernel in our case.

math.DG↗
arXiv2016

An application of a functional inequality to quasi-invariance in infinite dimensions

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we can not use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron-Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space.

math.PR↗