Winding number of a Brownian particle on a ring under stochastic resetting

We consider a random walker on a ring, subjected to resetting at Poisson-distributed times to the initial position (the walker takes the shortest path along the ring to the initial position at resetting times). In the case of a Brownian random walker the mean first-completion time of a turn is expressed in closed form as a function of the resetting rate. The value is shorter than in the ordinary process if the resetting rate is low enough. Moreover, the mean first-completion time of a turn can be minimised in the resetting rate. At large time the distribution of winding numbers does not reach a steady state, which is in contrast with the non-compact case of a Brownian particle under resetting on the real line. The mean total number of turns (and the variance of the net number of turns) grow linearly with time, with a proportionality constant equal to the inverse of the mean first-completion time of a turn.
Comment: 16 pages, 2 figures; V2: typos corrected; V3: more typos corrected, references added