Is there any relation between heat kernel and Laplacian? I know what each of them is but I am not sure about the relation between them.
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The Wikipedia article at http://en.wikipedia.org/wiki/Heat_kernel makes the distinction with its first two equations and clearly shows their relationship (although I admit its notation is abominable). – whuber Feb 06 '14 at 22:04
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@whuber: Thank you but it didn't say more than a few words. – Gigili Feb 07 '14 at 05:55
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Okay, I wasn't clear about that last remark. By following the link to the definition of the Laplace operator we see that in the heat equation $$\frac{\partial K}{\partial t}(t,x,y)=\Delta_x K(t,x,y)$$ the Laplacian "$\Delta_x$" must be with respect to the vector variable $x=(x_1,x_2,\ldots,x_d)$ and is given by $$\Delta_x=\sum_{i=1}^d \frac{\partial^2}{\partial x_i^2}.$$ – whuber Feb 07 '14 at 15:13