# Prove the Laplacian kernel is a valid kernel

By a valid kernel, we mean the Gram matrix induced by it is positive semi-definite (and symmetric).

Let $$K(x,y)=\exp(-\lambda \|x-y \|)$$, where $$\lambda >0.$$ We say $$K$$ is a Laplacian kernel.

I know the exponential of a valid kernel is still a valid kernel. But unfortunately, $$K_1(x,y)=\|x-y \|$$ is not a valid kernel and my strategy is not working.

I have googled a lot and found a solution involving Fourier transform and reproducing kernel Hilbert spaces(abbr. RKHS). But I am new to this area and not really familiar with RKHS. I wonder if there is an elementary way to prove this. Many thanks.

• Thanks for help gg. Sorry for my late reply. Reading stochastic theory takes some time. May I ask you two questions? I refer to the Wiki page of OU process and understand how to derive $cov(x_s,x_t)=\frac{\sigma^2}{2\theta}\left( \exp(-\theta |t-s|) - \exp(-\theta(t+s)) \right)$. But then the Wiki page(under the section \textit{Mathematical Properties} says, for the stationary process, the covariance of $x_s$ and $x_t$ is $\frac{\sigma^2}{2\theta}\left( \exp(-\theta |t-s|) \right)$. Can you help me understand this computation? Thanks. Jul 17, 2022 at 16:22