  In this paper we investigate a path dependent optimal control problem on the
process space with both drift and volatility controls, with possibly degenerate
volatility. The dynamic value function is characterized by a fully nonlinear
second order path dependent HJB equation on the process space, which is by
nature infinite dimensional. In particular, our model covers mean field control
problems with common noise as a special case. We shall introduce a new notion
of viscosity solutions and establish both existence and comparison principle,
under merely Lipschitz continuity assumptions. The main feature of our notion
is that, besides the standard smooth part, the test function consists of an
extra singular component which allows us to handle the second order derivatives
of the smooth test functions without invoking the Ishii's lemma. We shall use
the doubling variable arguments, combined with the Ekeland-Borwein-Preiss
Variational Principle in order to overcome the noncompactness of the state
space. A smooth gauge-type function on the path space is crucial for our
estimates.
