arXiv research

Abstract: Nonlinear random walk with distributionally robust transition probabilities.

The paper characterizes stochastic incompleteness in Riemannian manifolds.

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…

The paper proves wellposedness of flows on manifolds with bounded geometry.

New methods improve stability of Sinkhorn algorithm in machine learning.

Flat semigroups can represent normal weighted homogeneous surface singularities.

New infinite family of hyperbolic L-space knots with specific semigroups.

The paper establishes new inequalities for Finsler measure spaces.

Introduces intrinsic Hopf-Lax semigroup linking to intrinsic slope.

Proves representability of complex semigroup systems.

The problem behind this paper is the proper measurement of the degree of quality/acceptability/distance to arbitrage of trades. We are narrowing the class of coherent acceptability indices introduced by Cherny and Madan (2007) by imposing an additional mathematical property. For this, we introduce the notion of a conca…

This monograph develops the theory of covariant Schrödinger semigroups acting on sections of vector bundles over noncompact Riemannian manifolds from scratch. Contents: I. Sobolev spaces on vector bundles II. Smooth heat kernels on vector bundles III. Basis differential operators in Riemannian manifolds IV. Some specif…

The paper studies a semigroup generated by finite intervals and characterizes its properties.

The paper studies dynamical properties in semigroups modulo ideals.

Intertwining curvature bounds for graphs and quantum Markov semigroups verified.

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 paper associates knots to numerical semigroups and shows their Alexander polynomials coincide with semigroups' Poincaré series.

We consider the dynamics of rational semigroups (semigroups of rational maps) on the Riemann sphere. We provide proof that a random backward iteration algorithm to draw the pictures of the Julia sets, previously proven to work in the context of iteration of a rational map of degree two or more, extends to finitely gene…

The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.

We investigate the dynamics of semigroups generated by a family of polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. The Julia set of such a semigroup may not be connected in general. We show that for such a polynomial semigroup, if $A$ and $B$ are two connected compo…

We extend a result regarding the Random Backward Iteration algorithm for drawing Julia sets (known to work for certain rational semigroups containing a non-Möbius element) to a class of Möbius semigroups which includes certain settings not yet been dealt with in the literature, namely, when the Julia set is not a thick…

Graphs approximate semigroups for diffusion on Riemannian manifolds.

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the…

DM uses semigroup property to tune diffusion time for better data analysis.

Constructs free semigroups with critical exponents close to but less than ambient groups.

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δφ(u)$, $φ$ being an ar…

Paper proposes deep learning for operators in semigroups, improving dynamical system modeling.

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.

Paper analyzes a new Hopf-Lax semigroup in metric spaces.

Establishes geometric properties of elements in the positive semigroup of a general real semisimple Lie group.

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

Various semigroups of noninvertible supermatrices of the special (antitriangle) shape having nilpotent Berezinian which appear in supersymmetric theories are defined and investigated. A subset of them continuously represents left and right zero semigroups and rectangular bands. The ideal properties of higher order rect…

Study estimates gaps in semigroup products, proving embedding properties.

We investigate the dynamics of $2$-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space ${\cal B}$ of such semigroups. We show that for a parameter $h$ in the intersection of ${\cal B}$, the hyperbolicity locus ${\c…

We consider isotropic Lévy processes on a compact Riemannian manifold, obtained from an $\mathbb{R}^d$-valued Lévy process through rolling without slipping. We prove that the Feller semigroups associated with these processes extend to strongly continuous contraction semigroups on $L^p$, for $1\leq p<\infty$, and that t…

We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a …

We investigate the Banach Lie groupoids and inverse semigroups naturally associated to W*-algebras. We also present statements describing relationship between these groupoids and the Banach Poisson geometry which follows in the canonical way from the W*-algebra structure.

Generates semigroups for differential expressions on Riemannian manifolds.

In those lecture notes, we review some applications of heat semigroups methods in Riemannian and sub-Riemannian geometry. The notes contain parts of courses taught at Purdue University, Institut Henri Poincaré, Levico Summer School and Tata Institute.

Study quantum diffusion on spectral triples and spinor bundles.

New stability theory for Sinkhorn semigroups with explicit decay rates.

Through the main example of the Ornstein-Uhlenbeck semigroup, the Bakry-Emery criterion is presented as a main tool to get functional inequalities as Poincaré or logarithmic Sobolev inequalities. Moreover an alternative method using the optimal mass transportation, is also given to obtain the logarithmic Sobolev inequa…

Construction of a semigroup with 15 generators and 84 relations is given. The center of this semigroup is in one-to-one correspondence with the set of all isotopy classes of non-oriented singular knots (links with finitely many double intersections in general position) in three-dimensional space.

We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish se…

Let $M$ be a compact Riemannian manifold with boundary $\pp M$ and $L= \DD+Z$ for a $C^1$-vector field $Z$ on $M$. Several equivalent statements, including the gradient and Poincaré/log-Sobolev type inequalities of the Neumann semigroup generated by $L$, are presented for lower bound conditions on the curvature of $L$ …

We prove Bismut-type formulae for the first and second derivatives of a Feynman-Kac semigroup on a complete Riemannian manifold. We derive local estimates and give bounds on the logarithmic derivatives of the integral kernel. Stationary solutions are also considered. The arguments are based on local martingales, althou…

The paper constructs and analyzes self-similar blowup solutions for a wave map equation.