Resetting by rescaling: exact results for a diffusing particle in one-dimension

In this paper, we study a simple model of a diffusive particle on a line, undergoing a stochastic resetting with rate $r$, via rescaling its current position by a factor $a$, which can be either positive or negative. For $|a|<1$, the position distribution becomes stationary at long times and we compute this limiting distribution exactly for all $|a|<1$. This symmetric distribution has a Gaussian shape near its peak at $x=0$, but decays exponentially for large $|x|$. We also studied the mean first-passage time (MFPT) $T(0)$ to a target located at a distance $L$ from the initial position (the origin) of the particle. As a function of the initial position $x$, the MFPT $T(x)$ satisfies a nonlocal second order differential equation and we have solved it explicitly for $0 \leq a < 1$. For $-1<a\leq 0$, we also solved it analytically but up to a constant factor $\kappa$ whose value can be determined independently from numerical simulations. Our results show that, for all $-1<a<1$, the MFPT $T(0)$ (starting from the origin) shows a minimum at $r=r^*(a)$. However, the optimised MFPT $T_{\rm opt}(a)$ turns out to be a monotonically increasing function of $a$ for $-1<a<1$. This demonstrates that, compared to the standard resetting to the origin ($a=0$), while the positive rescaling is not beneficial for the search of a target, the negative rescaling is. Thus resetting via rescaling followed by a reflection around the origin expedites the search of a target in one dimension.