Before inflation, what sets the initial value of the inflaton field? [This is a version of the question that I've revised based on helpful comments from Dan.]
I haven't studied inflation at a technical level. My picture of the process is that we have an inflaton field $\phi$ which is a scalar and has a potential $V(\phi)$. Before inflation, the field starts at some $\phi_o$ that doesn't equal the value $\phi_m$ that minimizes $V$. It then rolls downhill to the minimum.
Assuming I have this right, what I don't understand is (1) why $\phi$ initially has a single value $\phi_o$ everywhere, and (2) what sets $\phi_o$. Before the onset of inflation, the universe's temperature was much higher than any scale set by $V$. When the age of the universe was on the order of the Planck time, presumably its temperature was on the Planck scale. If the temperature was that high, wouldn't thermal fluctuations cause the field to sample a wide range of values of $\phi$, contradicting #1? And if it could sample all those values, I would expect that thermodynamically, it would consist overwhelmingly of values of $\phi$ close to $\phi_m$, not some other value $\phi_o$.
 A: Let us assume single field inflation
1) $\phi$ does not necessarily have the some value of initial condition everywhere,
 $$\phi(t,\vec{x}) = \bar{\phi}(t)+\delta\phi(t,\vec{x}) $$ 
usually it is the $\bar{\phi}(t)$ having same initial condition, while $\delta\phi(t,\vec{x})$ is treated as perturbations. In addition to that, what is your definition of "initial"? If one chooses the synchronous gauge in which the time foliation is defined by constant $\phi$  everywhere (independent of $\vec{x}$), then $\phi$ is always the same everywhere.
2) The temperature during inflation is the Gibbons-Hawking temperature for de Sitter space, $T\propto H$ , which is of the order of hubble parameter and hence the order of $V$. People have worried about the fluctuation of the inflation field, hawking, Bardeen etc. They found that the fluctuation is small but I could not find the paper I have in mind. I will try to find it. It is safe to discuss inflation with generically small fluctuations.
You asked a good question which puzzles many people. The main concern of your question is the initial condition of inflation. It seems to some people that theorists are just hiding the problems in the initial condition. However, to many physicists, a theory explaining many features with one input is no doubt a successful theory. I guess you already know the utility of inflationary theories so I am not explaining it further. There are theories postulating that our universe is created by quantum creation in which the initial conditions are set by chance, there are other universe having very different initial conditions. String theorists have also calculated how likely universes like ours are created in the context of string theory.
Inflation is a huge research field and there are still many problems to be solved. 
A: I found this paper (http://arxiv.org/abs/1401.2253) that discusses pre-inflationary dynamics, and specifically talks about "Kinetic Dominance", where the $\dot\phi^2$ terms dominate over $V(\phi)$, and this ends up giving some initial conditions that are independent of the specific form of the potential.
This might not answer the question of "why just one initial value?" since it seems to posit that the actual value of the field at the onset of inflation is tunable.
