What causes fluctuations in the inflaton field value? From this article

During inflation there are quantum fluctuations in the inflaton field.

And this

The field φ experiences quantum fluctuations, as the uncertainty principle tells us it must.

And this

However, the field φ, like every other field, is subject to quantum mechanical fluctuations. As it rolls down the hill, in some regions φ fluctuates downwards and in others it fluctuates upwards.


This answer states that the fluctuation is in the measurement of the field not in the field itself. Quantum fields do not fluctuate. That is, there’s no quantum fluctuations causing the field value to fluctuate. 
What is really meant by “quantum fluctuations” in the inflaton field? Fluctuations from what?
$~$How does it cause the inflaton field to fluctuate to a higher field value in some regions and to a lower field value in some others?
 A: The field itself doesn't have any specific value (at a given point in space and at a given moment of time), and doesn't "fluctuate" like a thermodynamical quantity.  In a way, it is a "blurred" quantity and is represented as a field operator in QM.  If an ideal observer could do some field measurement there, he would find any value from some given set of values, with some probabilities distribution (according to the QM rules).
QM is a representation theory.  It associates an operator to any measurable quantity.  But then, there is no operator really living "out there" (how could there be an operator propagating in real space!?).  The operator is just an human abstract construction, a tool to make predictions.  In some way, QM (and all of physics anyway) is a complicated system of "epicycles" specifically made to do measurement predictions (a bit like the old Ptolemaic geocentric system in astronomy).
So the "blurry field" slide down its potential and could appears to be fluctuating when several measurements are done at the same time, on similar copies of the universe.  But the field itself isn't a definite number that could "fluctuate".   It is the measured value that could be different, according to several measurements.  According to QM, Nature is fondamentally statistical in nature!
A: I think the difference between these two explanations is a differing view of a quantum field. I have found that some people define as a quantum field the field itself with the creation and annihilation operators that generate the particles in quantum field theory, while others consider the space filled with a quantum field and the creation and annihilation operators acting on this unchanging field.
The latter is the way the Feynman diagrams are used to write the interaction integrals, example: the electron field is the plane wave wavefunction of the Dirac solution for the electron, constant on every space time point, on which creation and annihilation operators work, creating the fluctuations.
For gravity which is only effectively quantized it is too much to talk of the "Inflaton particle" so the first "definition" is assumed , carrying the quantum fluctuations in the word "field".
