The thermal‐induced escape out of the zero‐voltage state of a Josephson junction is discussed removing the main assumption of the resistively shunted junction model. This model assumes that the junction dissipation is dominated by a linear ohmic shunt resistor. However, the presence of quasiparticles excitations produces dissipation in the junction through nonlinear mechanisms. As a consequence, the dynamical resistance is strongly voltage dependent. In the present work, we study the effect of this voltage‐dependent dissipation on the supercurrent decay in connection with the general problem of a Brownian particle in a metastable state in the presence of a velocity‐dependent friction coefficient. An analytical expression for the lifetime of the metastable state is obtained, providing an answer to the open question of what resistance dominates the junction damping. A comparison of the theory with experimental results is also presented.