  The Hawkes self-excited point process provides an efficient representation of
the bursty intermittent dynamics of many physical, biological, geological and
economic systems. By expressing the probability for the next event per unit
time (called "intensity"), say of an earthquake, as a sum over all past events
of (possibly) long-memory kernels, the Hawkes model is non-Markovian. By
mapping the Hawkes model onto stochastic partial differential equations that
are Markovian, we develop a field theoretical approach in terms of probability
density functionals. Solving the steady-state equations, we predict a power law
scaling of the probability density function (PDF) of the intensities close to
the critical point $n=1$ of the Hawkes process, with a non-universal exponent,
function of the background intensity $\nu_0$ of the Hawkes intensity, the
average time scale of the memory kernel and the branching ratio $n$. Our
theoretical predictions are confirmed by numerical simulations.
