arXiv research

Heat semigroups used to solve geometric inequalities on manifolds.

New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.

Theory of covariant Schrödinger semigroups on Riemannian manifolds developed.

The aim of this paper is to show that the dynamics of $L^p$ heat semigroups ($p>2$) on a symmetric space of non-compact type is very different from the dynamics of the $L^p$ heat semigroups if $p\leq 2$. To see this, it is shown that certain shifts of the $L^p$ heat semigroups have a chaotic behavior if $p>2$ and that …

We look at the semigroup generated by a system of heat equations. Applications to testing normality and option pricing are addressed.

Study quantum diffusion on spectral triples and spinor bundles.

The paper establishes new inequalities for Finsler measure spaces.

Graphs with non-negative curvature have improved heat semigroup estimates.

The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t…

We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.

We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner-Weitzenbock type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat…

Study inequalities on hyperbolic spaces and Riemannian manifolds using symmetrization and heat semigroup.

Establishes exponential contraction in Wasserstein distance on manifolds and flows.

We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gr…

The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.

New criterion for wave operators on Kato-Ricci manifolds.

On a doubling metric measure space endowed with a "carré du champ", we consider $L^p$ estimates $(G_p)$ of the gradient of the heat semigroup and scale-invariant $L^p$ Poincaré inequalities $(P_p)$. We show that the combination of $(G_p)$ and $(P_p)$ for $p\ge 2$ always implies two-sided Gaussian heat kernel bounds. Th…

Unified treatment of two extension problems using heat equation in Heisenberg group.

In this paper we first derive several results concerning the $L^p$ spectrum of arithmetic locally symmetric spaces whose $\Q$-rank equals one. In particular, we show that there is an open subset of $\C$ consisting of eigenvalues of the $L^p$ Laplacian if $p <2$ and that corresponding eigenfunctions are given by certain…

For negatively curved manifolds, a condition is found for intrinsic ultracontractivity of heat semigroups.

Graph diffusion processes approximate manifold heat semigroups using graph transition matrices.

The paper connects fractional Laplacians on spheres to number theory and differential geometry.

Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.

In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature $CDE'(n,0)$ the Sobolev inequality, Nash inequa…

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type: $|\nabla P_t f| \le K P_t(|\nabla f|)$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-…

The paper characterizes stochastic completeness on Riemannian manifolds using nonlocal conditions.

We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.

We study a "div-grad type" sub-Laplacian with respect to a smooth measure and its associated heat semigroup on a compact equiregular sub-Riemannian manifold. We prove a short time asymptotic expansion of the heat trace up to any order. Our main result holds true for any smooth measure on the manifold, but it has a spec…

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving…

We study curvature dimension inequalities for the sub-Laplacian on contact Riemannian manifolds. This new curvature dimension condition is then used to obtain: 1) Geometric conditions ensuring the compactness of the underlying manifold (Bonnet-Myers type results); 2) Volume estimates of metric balls; 3) Gradient bounds…

The paper studies heat behavior on curved spaces without radiality assumption.

The study characterizes heat flow and concentration on directed graphs with a lower Ricci curvature bound.

Conservation of heat in manifolds with boundary under mixed conditions.

We develop the celebrated semigroup approach à la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenböck formula established by Sturm and the author. We show the $L^1$-gradient estimate on Finsler manifolds (under some additional assumptions in the n…

We study the horizontal Laplacian $Δ^H$ associated to the Hopf fibration $S^3\to S^2$ with arbitrary Chern number $k$. We use representation theory to calculate the spectrum, describe the heat kernel and obtain the complete heat trace asymptotics of $Δ^H$. We express the Green functions for associated Poisson semigroup…

New criterion for wave operators without injectivity radius assumptions, leading to spectral stability for Ricci flow.

Let $M$ be a complete connected Riemannian manifold with boundary $\pp M$, $Q$ a bounded continuous function on $\pp M$, and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. By using the reflecting diffusion process generated by $L$ and its local time on the boundary, a probabilistic formula is presented for the semigro…

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on …

The study identifies types of manifolds using variational formulas and integral-differential formulas.

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times.…

Study asymptotically almost periodic solutions on real hyperbolic manifolds.

The paper extends spectral estimates to hyperbolic surfaces with hyperbolic ends.

The paper derives Harnack inequalities for evolving Riemannian manifolds without dimensionality restrictions.

The paper characterizes stochastic incompleteness in Riemannian manifolds.

The paper proves boundedness of a Riesz transform on weighted manifolds.

By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE'(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative cur…

Let $(X,d,μ)$ be a complete metric measure space, with $μ$ a locally doubling measure, that supports a local weak $L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on $(X,d,μ)$. Gradient estimates for Cheeger-harmonic func…