Consider the kernel methods in machine learning that are used in Support Vector Machine, Gaussian Process, etc. We need to define $k(x;y)$ that measures the similarity between two data points $x;y$. The common choice is the radial basis function RBF kernel: $k(x;y)=exp(- || x-y ||^2)$ .
In most of the cases, $k(x;x)$ is constant and does not depend on $x$. I am wondering if there is any kernel function such that $k(x;x)$ is dependent on $x$. If there is, how is it called? Any reference is helpful.