  For systems in equilibrium at a temperature $T$, thermal noise and energy
damping are related to $T$ through the fluctuation-dissipation theorem (FDT).
We study here an extension of the FDT to an out of equilibrium steady state: a
microcantilever subject to a constant heat flux. The resulting thermal profile
in this spatially extended system interplays with the local energy dissipation
field to prescribe the amplitude of mechanical fluctuations. Using three
samples with different damping profiles (localized or distributed), we probe
this approach and experimentally demonstrate the link between fluctuations and
dissipation. The thermal noise can therefore be predicted a priori from the
measurement of the dissipation as a function of the maximum temperature of the
micro-oscillator.
