Mehler kernel

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In mathematics, the Mehler kernel is the heat kernel of the Hamiltonian of the harmonic oscillator. Mehler (1866) gave an explicit formula for it called Mehler's formula. The Kibble–Slepian formula generalizes Mehler's formula to higher dimensions.

The Mehler kernel φ(x, y, t) is a solution to

$\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}-x^2\varphi$

Mehler's kernel is

$\frac{\exp(-\coth(2t)(x^2+y^2)/2 - \text{cosech}(2t)xy)}{\sqrt{2\pi\sinh(2t)}}.$

By a simple transformation this is, apart from a multiplying factor, the bivariate Gaussian probability density given by

$\frac 1{2\pi \sqrt{1-\rho^2}}\exp\left(\frac{(x^2+y^2)- 2\rho xy}{1-\rho^2}\right)$

It can be written as an infinite series involving the one dimensional probability densities and Hermite polynomials of x and y (see the link to Slepian).

References

Grigor'yan, Alexander (2009), Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics 47, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4935-4, MR 2569498
Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung", Journal für Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, JFM 066.1720cj (See p.174, equation (18))

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