  In the presence of Joule heating, the electronic temperature in a metallic
resistor placed at sub-Kelvin temperatures can significantly exceed the phonon
temperature. Electron cooling proceeds mainly through two processes: electronic
diffusion to and from the connecting wires and electron-phonon coupling. The
goal of this paper is to present a general solution of the problem, in a form
that can easily be used in practical situations. As an application, we compute
two quantities that depend on the electronic temperature profile: the second
and the third cumulant of the current noise at zero frequency, as a function of
the voltage across the resistor. We also consider time dependent heating, an
issue relevant for experiments in which current pulses are used, for instance
in time-resolved calorimetry experiments.
