Optimization and growth in first-passage resetting

We combine the processes of resetting and first-passage to define \emph{first-passage resetting}, where a random walk fixed position is triggered by event itself. In an infinite domain, isotropic diffusion non-stationary, with number events growing time as $\sqrt{t}$. calculate resulting spatial probability distribution particle analytically, also obtain this geometric path decomposition. finite interval, we optimization problem that controlled resetting; scenario motivated reliability theory. The goal operate system close its maximum capacity without experiencing too many breakdowns. However, when breakdown occurs reset minimal operating point. optimize objective function maximizes reward (being operation) minus penalty for each breakdown. investigate extensions basic model include delay after two dimensions. Finally, study growth dynamics domain in which boundary recedes specified amount whenever diffusing reaches occurs. determine rate semi-infinite line interval find wide range behaviors depend on how much recession hits boundary.