  The Josephson junction is a building block of quantum circuits. Its behavior,
well understood when treated as an isolated entity, is strongly affected by
coupling to an electromagnetic environment. In 1983, Schmid predicted that a
Josephson junction shunted by a resistance exceeding the resistance quantum
$\mathbf{\textit{R}}_\mathrm{Q} = h/4e^2 \approx 6.45$ k$\mathbf{\Omega}$ for
Cooper pairs would become insulating since the phase fluctuations would destroy
the coherent Josephson coupling. However, recent microwave measurements have
questioned this interpretation. Here, we insert a small Josephson junction in a
Johnson-Nyquist-type setup where it is driven by weak current noise arising
from thermal fluctuations. Our heat probe minimally perturbs the junction's
equilibrium, shedding light on features not visible in charge transport. We
find that the Josephson critical current completely vanishes in DC charge
transport measurement, and the junction demonstrates Coulomb blockade in
agreement with the theory. Surprisingly, thermal transport measurements show
that the Josephson junction acts as an inductor at high frequencies,
unambiguously demonstrating that a supercurrent survives despite the Coulomb
blockade observed in DC measurements. The discrepancy between these two
measurements highlights the difference between the low and the high frequency
response of a junction and calls for further theoretical and experimental
inputs on the dynamics of Josephson junctions \textcolor{black}{operating at
high frequencies in highly resistive environments.
