  Recently, a couple of investigations related to symmetry breaking phenomena,
'spontaneous stochasticity' and 'ergodicity 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.
