The Ubiquitous Heat Kernel

Jorgenson, Jay; Walling, Lynne H.

doi:10.1007/978-3-642-56478-9_34

Cited by 35 publications

(24 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let I x (t) be the classical I-Bessel function, reviewed in detail in Section 2.1 below. Then, using the general concept of theta functions with inversion formulas constructed from heat kernels (see, e.g., [14]), the authors in [15] prove that for any t > 0 and integer x, the following identity holds for the theta function associated to the discrete torus nZ\Z: where I x (t) denotes the I-Bessel function (see Section 2.2). The generalization of (3) to the d-dimensional discrete torus, defined as the product space…”

Section: Summary Of the Main Resultsmentioning

confidence: 99%

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Chinta¹,

Jorgenson²

2010

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

Abstract. By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz-type zeta functions.

show abstract

Section: Summary Of the Main Resultsmentioning

confidence: 99%

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Chinta¹,

Jorgenson²

2010

Nagoya Mathematical Journal

Self Cite

View full text Add to dashboard Cite

show abstract

“…The result is obtained in the form of the following Theorem 2. Consider the fractional diffusion equation (Metzler and Klafter, 2000;Jorgenson and Lang, 2001)…”

Section: A Fractional Diffusion Equationmentioning

confidence: 99%

Unified Fractional Kinetic Equation and a Fractional Diffusion Equation

Saxena

Mathai

Haubold

2004

Astrophysics and Space Science

View full text Add to dashboard Cite

In earlier papers Saxena et al. (2002, 2003) derived the solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which extended the work of Haubold and Mathai (2000). The object of the present paper is to investigate the solution of a unified form of fractional kinetic equation in which the free term contains any integrable function f(t), which provides the unification and extension of the results given earlier recently by Saxena et al. (2002, 2003). The solution has been developed in terms of the Wright function in a closed form by the method of Laplace transform. Further we derive a closed-form solution of a fractional diffusion equation. The asymptotic expansion of the derived solution with respect to the space variable is also discussed. The results obtained are in a form suitable for numerical computation.Comment: 13 pages, LaTe

show abstract

“…The importance of the heat kernel in mathematics is hard to exaggerate, and we refer to [29,30,42] for numerous examples from various branches of mathematics and theoretical physics. The heat kernel appears typically as the fundamental solution of a partial differential equation, as the integral kernel of an operator semigroup or as the transition density of a stochastic process.…”

Section: Introductionmentioning

confidence: 99%

Heat kernel asymptotics of the subordinator and subordinate Brownian motion

Fahrenwaldt

2018

J. Evol. Equ.

View full text Add to dashboard Cite

For a class of Laplace exponents, we consider the transition density of the subordinator and the heat kernel of the corresponding subordinate Brownian motion. We derive explicit approximate expressions for these objects in the form of asymptotic expansions: via the saddle point method for the subordinator's transition density and via the Mellin transform for the subordinate heat kernel. The latter builds on ideas from index theory using zeta functions. In either case, we highlight the role played by the analyticity of the Laplace exponent for the qualitative properties of the asymptotics.Mathematics Subject Classification: 35K08, 60J55, 41A60

show abstract

The Ubiquitous Heat Kernel

Cited by 35 publications

References 69 publications

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Unified Fractional Kinetic Equation and a Fractional Diffusion Equation

Heat kernel asymptotics of the subordinator and subordinate Brownian motion

Contact Info

Product

Resources

About