Methods of Stochastic Field Theory in Non-Equilibrium Systems -- Spontaneous Symmetry Breaking of Ergodicity

Recently, a couple of investigations related to symmetry breaking phenomena, 'spontaneous stochasticity' and érgodicity breaking' have led to significant impacts in a variety of fields related to the stochastic processes such as economics and finance. We investigate on the origins and effects of those original symmetries in the action from the mathematical and the effective field theory points of view. It is naturally expected that whenever the system respects any symmetry, it would be spontaneously broken once the system falls into a vacuum state which minimizes an effective action of the dynamical system.