The 'Arcsine' laws of Brownian particles in one dimension describe distributions of three quantities: the time t m to reach maximum position, the time t r spent on the positive side and the time t of the last visit to the origin. Interestingly, the cumulative distribution of all the three quantities are same and given by Arcsine function. In this paper, we study distribution of these three times t m , t r and t in the context of single run-and-tumble particle in one dimension, which is a simple non-Markovian process. We compute exact distributions of these three quantities for arbitrary time and find that all three distributions have delta function part and a non-delta function part. Interestingly, we find that the distributions of t m and t r are identical (reminiscent of the Brownian particle case) when the initial velocities of the particle are chosen with equal probability. On the other hand, for t , only the non-delta function part is same with the other two. In addition, we find explicit expressions of the joint distributions of the maximum displacement and the time at which this maxima occurs. We verify all our analytical results through numerical simulations.

We consider the motion of a randomly accelerated particle in one dimension under stochastic resetting mechanism. Denoting the position and velocity by x and v respectively, we consider two different resetting protocols-(i) complete resetting: here both x and v reset to their initial values x 0 and v 0 at a constant rate r, (ii) partial resetting: here only x resets to x 0 while v evolves without interruption. For complete resetting, we find that the particle attains stationary state in both x and v. We compute the non-equilibrium joint stationary state of x and v from which we obtain the stationary state distribution for position by integrating over v. We also study the late time relaxation of the position distribution function. On the other hand, for partial resetting, the joint distribution is always in the transient state. At large t, the position distribution possesses a scaling behaviour (x/ √ t) which we rigorously derive. Next, we study the first passage time properties with an absorbing wall at the origin. For complete resetting, we find that the mean first passage time (MFPT) is rendered finite by the resetting mechanism. We explicitly derive the expressions for the MFPT and the survival probability at large t. However, in stark contrast, for partial resetting, we find that resetting does not render finite MFPT. This is because even though x is brought to x 0 , the large fluctuation in v (typically of the order ∼ √ t) can take the particle substantially far from the origin. All our analytic results are corroborated by the numerical simulations.

We investigate the run-and-tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise σ(t) drives the particle which changes between ±1 values at certain rates. Denoting the rate of flip from 1 to −1 as R 1 and the converse rate as R 2 , we consider the position-and direction-dependent rates of the form R 1 (x) = |x|

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