  The behavior of the quintessence field is studied during inflation. In order
to have a satisfactory model of dark energy, the quintessence field value today
should be as insensible to the initial conditions as possible. Usually, only
the dependence on the initial conditions specified at the end of inflation or,
equivalently, at the beginning of the radiation dominated era, is considered.
Provided the quintessence field is initially within a large but, crucially,
finite interval, its present value becomes independent of the initial value it
started from. The question as to whether inflation naturally drives the
quintessence field to the above-mentioned interval is addressed. Since the
quantum effects turn out to be important, the formalism of stochastic inflation
is used in order to calculate the evolution of the quintessence field.
Moreover, the quantum effects originating from the inflaton field are also
taken into account and are proved to be sub-dominant in most cases. Finally,
the requirement that the quintessence field is on tracks today is shown to
imply quite tight constraints on the initial values of the quintessence and
inflaton fields at the beginning of inflation. In particular, the initial value
of the inflaton field cannot be too large which indicates that the
quintessential scenario seems to be compatible with inflation only if the total
number of e-folds is quite small.
