  We investigate the noise current in a thermally biased tunnel junction
between two superconductors with different zero-temperature gaps. When the
Josephson effect is suppressed, this structure can support a nonlinear
thermoelectric effect due to the spontaneous breaking of electron-hole
symmetry, as we recently theoretically predicted. We discuss the possibly
relevant role played by the noise in the junction. While a moderate noise
contribution assists the generation of the thermoelectric signal, further
unveiling the spontaneous nature of the electron-hole symmetry breaking, a
large noise contribution can induce a switching between the two stationary
thermoelectric values, thus hardening the detection of the effect and its
application. We demonstrate that the thermoelectric effect is robust to the
presence of noise for a wide range of parameters and that the spurious
fluctuations of the thermoelectric signal can be lowered by increasing the
capacitance of the junction, for instance by expanding the junction's size. Our
results pave the way to the future experimental observation of the
thermoelectric effect in superconducting junctions, and to improved performance
in quantum circuits designed for thermal management.
