Permanental processes with kernels that are not equivalent to a symmetric matrix

Kernels of $\alpha$-permanental processes of the form $$v(x,y)=u(x,y)+f(y),\qquad x,y\in S,$$ in which $u(x,y)$ is symmetric, and $f$ is an excessive function for the Borel right process with potential densities $u(x,y)$, are considered. Conditions are given that determine whether $\{v(x,y);x,y\in S\}$ is symmetrizable or asymptotically symmetrizable.