Diffusion under time-dependent resetting

We study a Brownian particle diffusing under a time-modulated stochastic
resetting mechanism to a fixed position. The rate of resetting r(t) is a function
of the time t since the last reset event. We derive a sufficient condition on r(t)
for a steady-state probability distribution of the position of the particle to exist.
We derive the form of the steady-state distributions under some particular
choices of r(t) and also consider the late time relaxation behavior of the
probability distribution. We consider first passage time properties for the
Brownian particle to reach the origin and derive a formula for the mean first
passage time (MFPT). Finally, we study optimal properties of the MFPT and
show that a threshold function is at least locally optimal for the problem of
minimizing the MFPT.