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