I would like to explain how the construction of a Wilson action in the exact renormalization group (ERG) formalism is related to diffusion equations for the fields. By making the diffusion equations gauge covariant, we have managed to linearize the BRST invariance of U(1) gauge theories. But with YM theories, the BRST invariance remains non-linear, and we have not been able to solve the constraint. All this has been done in collaboration with Hiroshi Suzuki (Kyushu University).