I am learning about Stochastic processes. To characterize uniqueness of solutions to a given Stochastic differential equation, I need to find for each continuous function $g :\Bbb{R}^2_+ \to \Bbb{R}$ a function $f:\Bbb{R}^2_+ \to \Bbb{R}$ such that
$$ -\gamma f + (\partial_x f) (y-x) + (\partial_y f) (x-y) + \frac{1}{2} \bigg((\partial_{xx} f) x + (\partial_{yy} f) y\bigg) = g $$
This is I believe an inverse problem which has to do with spectral techniques such as proving that $\gamma \in \rho(L)$ where
$$L f = (\partial_x f) (y-x) + (\partial_y f) (x-y) + \frac{1}{2} \bigg((\partial_{xx} f) x + (\partial_{yy} f) y\bigg)$$
Maybe this can be solved more straightforwardly by finding a Green's function $G(x,y,\tilde{x}, \tilde{y})$ that satisfies $$ -\gamma G - L G = \delta_{x -\tilde{x}} \delta_{y - \tilde{y}}$$
or something similar and then defining
$$f = G*g. $$
How can we find such an $f$? Does it exist? Is there any good reference on the literature for this subject?
Note: Here $(\partial_x f) (y-x) $ means $(\partial_x f(x,y)) \cdot (y-x)$