Optimal transition path time of run-and-tumble particles with stochastic resetting.

We present a semianalytical method based on the backward master equation to find the optimal transition path time (TPT) of self-propelled run-and-tumble particles (RTPs) under stochastic resetting. The particle moves in a one-dimensional transition region [x_{A},x_{B}] with two absorbing boundaries. It resets at rate γ to position x_{R}. Upon each resetting event, the particle either reverses its velocity direction with probability η∈[0,1] or retains it with probability 1-η. From the backward master equation we derive exact analytical results for the splitting probability, the mean TPT, and the coefficient of variation (CV) of the TPT distribution. A semianalytical solution for the optimal TPT is also obtained. The mean TPT depends nonmonotonically on γ, α, and η, reaching a minimum at optimal parameter values. For large transition regions, resetting the particle to its initial position can accelerate escape compared with reset-free dynamics. In the strong-resetting limit for η>0 when x_{R}=x_{A}, the probabilistic reversal upon resetting effectively replaces natural tumbling. The asymptotic TPT then becomes independent of the tumbling rate α and reduces to the ballistic traversal time L/v. A speed-reliability trade-off emerges: the parameters that minimize the mean TPT do not generally minimize the CV. Efficient search requires balancing a short mean TPT against high predictability.