I'm interested in a laplace transform of the heat/gaussian kernel, precisely for $x,y\in \Bbb R ^p$ and $\mu > 0$: $$\int_0^\infty e^{-\mu t} \frac 1 {(2\pi t)^{\frac p 2}} e^{-\frac{||x-y||^2}{2t}} \text{d} t = ?$$
My approaches were several substitutions right up to solution operators for the $\mu$-eigenvalue problem of the scaled laplacian. I know there is a explicit representation for $p=3$ and I remember finding an expression for it a while ago, but with aid of a bessel funtion.
I would appreciate any help or useful reference to this problem.
Extracted from the book "Brownian Motion: An Introduction to Stochastic Processes" by Schilling/Partzsch:
$$ \int_0^\infty e^{-\mu t} \frac 1 {(2\pi t)^{\frac p 2}} e^{-\frac{||x-y||^2}{2t}} \text{d} t = \begin{cases} \frac 1 { \sqrt{2\mu}} e^{-\sqrt{2\mu}|y-x|} &: p=1 \\ \frac 1 {\pi^{\frac p 2}} \left(\frac \mu {2(y-x)^2}\right)^{\frac p 4 - \frac 1 2} K_{\frac p 2 -1}(\sqrt{2\mu}(y-x)) &: p \geq 2 \end{cases},$$ where $K_\nu(z)$ is a Bessel function of the third kind. In the book "Essentials of Brownian Motion and Diffusions" by Frank B. Knight he gives expressions for this in form of green functions for the PDE $\mu u - \frac 1 2 \Delta u = 0$, then for p=3: $$\frac 1 {2\pi |x-y|} e^{- \sqrt{2\mu}|x-y|}$$