Let $f_0,f_1 : \mathbb R^d \rightarrow \mathbb R$ be two probability density functions. I am looking at the following system of integral equations:
$$f_0(x) = \hat \phi(x) \int_{\mathbb R^d} \phi(z) p_W(0,x,T,z) \, dz$$ $$f_1(z) = \phi(z) \int_{\mathbb R^d} \hat \phi(x) p_W(0,x,T,z) \, dx$$
where $$p_W(s,x,t,y) = \frac{1}{(2\pi(t-s))^{\frac{d}{2}}} e^{\frac{-||x-y||^2}{2(t-s)}}$$ is the d-dimensional heat kernel.
How do we find the functions $\phi$, $\hat \phi$ ? How do we even call this type of systems ? What's the theory behind it ?