The impact of thermal noise on kink motion through the curved region of the long Josephson junction is studied. On the basis of the Fokker–Planck equation the analytical formula that describes the probability of transmission of the kink over the potential barrier is proposed. The analytical results are compared with the simulations based on the field model. It has been numerically shown that above one Kelvin the probabilities of crossing the barrier are correctly described by the formula proposed in the paper. •The formula that describes the probability of transmission of the kink is proposed.•The analytical results are compared with the simulations based on the field model.•Above one Kelvin the probabilities of crossing the barrier are correctly described.