I have two positive definite kernels $k_1,k_2 : \mathbb R^d \to \mathbb R$ and a probability measure $\rho$ on $\mathbb R^d$. The kernels are of the form $k_i(x,y)=K_i(x-y)$ for some function $K_i:\mathbb R^d \to \mathbb R$. I want to know if there exists a function $\phi : \mathbb R^d \to \mathbb R$ that satisfies
$$\int_{\mathbb R^d} (k_2-k_1)(y,x) \nabla\log(\rho(y)) d\rho(y) = \int_{\mathbb R^d} k_1(y,x) \phi(y) d\rho(y)$$
for (almost) all $x\in\mathbb R^d$.
I've tried applying the Fourier transform convolution theorem. This gives me something resembling a Fredholm integral equation of the first kind - but in $d$ dimensions, with complex functions in the integrand, and integration over all of $\mathbb R^d$. So I'm not sure this is the right approach. Even looking at special cases (mainly Gaussian kernels and measures with simple densities) I haven't been able to get much insight into what $\phi$ should look like.
Any pointers on similar problems or suggestions for useful conditions on $k_1,k_2,\rho$ would be much appreciated.