Thermal fluctuations in a Josephson-fluxon oscillator are considered with emphasises on the stochastic dynamics of fluxon and the non-trivial continuum-mode excitations. Analytical expressions for the radiation line-width of the oscillator are derived from perturbation theory and standard stochastic process theory. The averaged kinetic energy of a low-velocity fluxon is shown to be kBT/2. The continuum excitations are explored in the framework of the inverse scattering transform approach. The averaged energy per wave mode turns out to be kBT,and this justifies the adiabatic approximation applied in the fluxon stochastic dynamics.