Given an integral transform $(Tf)(x)=\int_{\Bbb R^d} K(x,y)f(y)dy$, with $K(x,y)=K(y,x)$, how can I find the associated differential operator?
If it helps, I am especially interested in $K(x,y)=e^{-Q(x,y)}$ with $Q(x,y)$ being a polynomial involving only the monomials of the form $x_i^ay_j^b$, with $a+b=2$.