  We formulate quantum electrodynamics on the basis of gauge (or BRST)
covariant diffusion equations of fields. This is a particular example of the
gradient flow exact renormalization group (GFERG). The resulting Wilson action
fulfills a simple gauge Ward--Takahashi identity. We solve the GFERG equation
around the Gaussian fixed point to the second order in gauge coupling and
obtain the 1-loop beta function and anomalous dimensions. The anomalous
dimension of the electron field coincides with that of the fermion field
diffused by a gauge covariant flow equation of L\"uscher.
