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≤a<1. For -1<a≤0, we also solved it analytically but up to a constant factor κ 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 optimized MFPT T_{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.