We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.
"A characterization of the infinitely divisible squared Gaussian processes." Ann. Probab. 34 (2) 728 - 742, March 2006. https://doi.org/10.1214/009117905000000684