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In mathematics, the '''Mehler kernel''' is the [[heat kernel]] of the Hamiltonian of the [[harmonic oscillator]].
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{{harvs|txt|last=Mehler|authorlink=Gustav Ferdinand Mehler|year=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 ''&phi;''(''x'',&nbsp;''y'',&nbsp;''t'') is a solution to  
 
:<math>\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}-x^2\varphi</math>
 
Mehler's kernel is
 
:<math>
\frac{\exp(-\coth(2t)(x^2+y^2)/2 - \text{cosech}(2t)xy)}{\sqrt{2\pi\sinh(2t)}}.
</math>
 
By a simple transformation this is, apart from a multiplying factor, the bivariate Gaussian probability density given by
 
:<math>
\frac 1{2\pi \sqrt{1-\rho^2}}\exp\left(\frac{(x^2+y^2)- 2\rho xy}{1-\rho^2}\right)
</math>
 
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==
 
*{{Citation | last1=Grigor'yan | first1=Alexander | title=Heat kernel and analysis on manifolds | url=http://books.google.com/books?id=X7QQcVa2EWsC | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=AMS/IP Studies in Advanced Mathematics | isbn=978-0-8218-4935-4 | mr=2569498 | year=2009 | volume=47}}
*{{Citation | last1=Mehler | first1=F. G. | title=Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002152975 | language=German | id={{ERAM|066.1720cj}} | year=1866 | journal=Journal für Reine und Angewandte Mathematik | issn=0075-4102 | issue=66 | pages=161–176}} (See p.174, equation (18))
 
[[Category:Parabolic partial differential equations]]

Latest revision as of 11:45, 4 September 2014

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