Impact of the memory of a Brownian particle on the resetting-induced steady state

In this study, we generalize the theory of stochastic resetting for the non-Markovian dynamics of a free Brownian particle (BP) with arbitrary damping strength and correlation time of the thermal noise. To apply this theory, we calculate the properties of the BP under thermal Ornstein-Uhlenbeck noise and compare the results with those from the Markovian dynamics case. We find that, at long times, the evolution of the distribution function toward the steady state in non-Markovian dynamics can resemble that observed in Markovian systems. In the asymptotic limit, the stationary distribution function reveals that the probability density at a given position can increase exponentially with memory time. Notably, the difference in probability density between the two cases tends to be maximal at an intermediate stage of the dynamics for a fixed memory time. Using the exact distribution function, we also calculate the variance of the position to explore a central question: How the nature of diffusion for a free particle is influenced by resetting, which can lead to a steady state. This analysis suggests that, although ballistic diffusion and memory effects may not significantly impact the long-time behavior of free Brownian motion or equilibrium in the presence of an external conservative force field, they play a crucial role in the formation of a resetting-induced localized stationary state. Additionally, we observe that the survival probability decreases exponentially at all times. Finally, we compute the mean first-passage time and uncover interesting results that provide further insights into the system's behavior.