Expansion of an hypoelliptic heat-kernel outside the cut-locus in semi-group theory

2013 ◽  
Author(s):  
Remi Leandre
Keyword(s):  
Group Theory ◽  
Heat Kernel ◽  
Cut Locus ◽  
Semi Group ◽  
Hypoelliptic Heat Kernel

scholarly journals SHORT TIME FULL ASYMPTOTIC EXPANSION OF HYPOELLIPTIC HEAT KERNEL AT THE CUT LOCUS

2017 ◽  
Vol 5 ◽  
Author(s):  
YUZURU INAHAMA ◽  
SETSUO TANIGUCHI
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Compact Manifold ◽  
Cut Locus ◽  
Finite Set ◽  
Rough Path ◽  
Hypoelliptic Heat Kernel ◽  
Short Time

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the ‘cut locus’ case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe’s distributional Malliavin calculus and T. Lyons’ rough path theory.

scholarly journals Malliavin Calculus of Bismut Type for Fractional Powers of Laplacians in Semi-Group Theory

2011 ◽  
Vol 2011 ◽  
pp. 1-26 ◽  
Author(s):  
Rémi Léandre
Keyword(s):  
Group Theory ◽  
Diffusion Process ◽  
Heat Kernel ◽  
Malliavin Calculus ◽  
Regularity Result ◽  
Jump Process ◽  
Semi Group ◽  
Fractional Powers

We translate into the language of semi-group theory Bismut's Calculus on boundary processes (Bismut (1983), Lèandre (1989)) which gives regularity result on the heat kernel associated with fractional powers of degenerated Laplacian. We translate into the language of semi-group theory the marriage of Bismut (1983) between the Malliavin Calculus of Bismut type on the underlying diffusion process and the Malliavin Calculus of Bismut type on the subordinator which is a jump process.

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Semi-group Theory and Scale Spaces

2008 ◽  
pp. 185-203
Author(s):  
Otmar Scherzer ◽  
Markus Grasmair ◽  
Harald Grossauer ◽  
Markus Haltmeier ◽  
Frank Lenzen
Keyword(s):  
Group Theory ◽  
Semi Group ◽  
Scale Spaces

scholarly journals Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

2019 ◽  
Vol 2 (2) ◽  
pp. 18 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
Chokkalingam Ravichandran ◽  
Shanmugam Dhanalakshmi ◽  
Rangasamy Murugesu
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Order System ◽  
Semi Group ◽  
Non Local

In this paper, we establish the existence of piece wise (PC)-mild solutions (defined in Section 2) for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we used the distributed characteristic operators to define a mild solution of the system. We also discussed the controversy related to the solution operator for the fractional order system using weak and strong Caputo derivatives. Examples are given to illustrate the theory.

The word problem for one-relator semigroups

1986 ◽  
Vol 99 (1) ◽  
pp. 33-44 ◽  
Author(s):  
James Howie ◽  
Stephen J. Pride
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Semi Group ◽  
Full Generality ◽  
Analogous Question

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.

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