  We study the time evolution of a state of a relativistic quantum field theory
restricted to a spatial subregion $\Omega$. More precisely, we use the
Feynman-Vernon influence functional formalism to describe the dynamics of the
field theory in the interior of $\Omega$ arising after integrating out the
degrees of freedom in the exterior. We show how the influence of the
environment gets encoded in a boundary term. Furthermore, we derive a
stochastic equation of motion for the field expectation value in the interior.
We find that the boundary conditions obtained in this way are energy
non-conserving and non-local in space and time. Our results find applications
in understanding the emergence of local thermalization in relativistic quantum
field theories and the relationship between quantum field theory and
relativistic fluid dynamics.
