14-JUL-2026
We study diffusions subject to Poissonian resetting: at rate $r$ the process is returned instantaneously to a fixed point $x_0$. Two consequences separate this from the diffusions of classical stochastic analysis. First, the reset generator $\mathcal{L}_r f = \mathcal{L}f + r\,[f(x_0) – f(x)]$ is non-local and, even when $\mathcal{L}$ is self-adjoint in its natural weight, $\mathcal{L}_r$ is not; the stationary state is consequently a genuine **nonequilibrium steady state** carrying a nonzero probability current, and cannot be written as a Gibbs measure. For Brownian motion we obtain the stationary density in closed form — a cusped exponential $p_{\mathrm{ss}}(x) = \tfrac{1}{2}\sqrt{r/D}\,e^{-\sqrt{r/D}\,|x – x_0|}$ — and show that its current jumps by exactly $r$ across the reset point, the teleported flux closing the balance. Second, resetting changes first-passage times qualitatively. A renewal argument gives the mean first-passage time under restart as $\langle T_r\rangle = (1 – \tilde{T}(r))/(r\,\tilde{T}(r))$, where $\tilde{T}$ is the Laplace transform of the un-restarted passage time, and we prove the resulting sharp criterion: **restart reduces the mean first-passage time if and only if the un-restarted passage time has coefficient of variation exceeding one**, and at the optimal rate the restarted passage time has coefficient of variation exactly one. Numerical experiments — using exact stationary sampling and an exact renewal simulation, so that no time-discretisation bias enters — confirm the criterion, locate the optimum, and verify the CV $=1$ signature to within $3.4 \times 10^{-3}$.