Accumulation time of stochastic processes with resetting

One of the characteristic features a stochastic process under resetting is that probability density converges to nonequilibrium stationary state (NESS). In addition, approach exhibits dynamical phase transition, which can be interpreted as traveling front separating spatial regions for has relaxed NESS from those where it not. establish existence transition by carrying out an asymptotic expansion exact solution. this paper we develop alternative, direct method characterizing with based on calculation so-called accumulation time. The latter analog mean first passage time search process, in survival replaced fraction density. case one-dimensional Brownian motion Poissonian resetting, derive formula $|x-x_0|\approx \sqrt{4rD}T(x)$ $|x-x_0|\gg\sqrt{D/r}$, $T(x)$ at $x$, $r$ constant rate, $D$ diffusivity and $x_0$ reset point. This identical form condition transition. We also analogous result diffusion higher dimensions non-Poissonian resetting. then consider effects delays such refractory periods finite return times. both cases behavior independent delays. Finally, extend analysis run-and-tumble particle thus useful quantity (if exists) relatively straightforward calculate.