  A field theoretical framework is developed for the Hawkes self-excited point
process with arbitrary memory kernels by embedding the original non-Markovian
one-dimensional dynamics onto a Markovian infinite-dimensional one. The
corresponding Langevin dynamics of the field variables is given by stochastic
partial differential equations that are Markovian. This is in contrast to the
Hawkes process, which is non-Markovian (in general) by construction as a result
of its (long) memory kernel. We derive the exact solutions of the
Lagrange-Charpit equations for the hyperbolic master equations in the Laplace
representation in the steady state, close to the critical point of the Hawkes
process. The critical condition of the original Hawkes process is found to
correspond to a transcritical bifurcation in the Lagrange-Charpit equations. We
predict a power law scaling of the PDF of the intensities in an intermediate
asymptotics regime, which crosses over to an asymptotic exponential function
beyond a characteristic intensity that diverges as the critical condition is
approached. We also discuss the formal relationship between quantum field
theories and our formulation. Our field theoretical framework provides a way to
tackle complex generalisation of the Hawkes process, such as nonlinear Hawkes
processes previously proposed to describe the multifractal properties of
earthquake seismicity and of financial volatility.
