  Random walks with stochastic resetting provides a treatable framework to
study interesting features about central-place motion. In this work, we
introduce non-instantaneous resetting as a two-state model being a combination
of an exploring state where the walker moves randomly according to a propagator
and a returning state where the walker performs a ballistic motion with
constant velocity towards the origin. We study the emerging transport
properties for two types of reset time probability density functions (PDFs):
exponential and Pareto. In the first case, we find the stationary distribution
and a general expression for the stationary mean square displacement (MSD) in
terms of the propagator. We find that the stationary MSD may increase, decrease
or remain constant with the returning velocity. This depends on the moments of
the propagator. Regarding the Pareto resetting PDF we also study the stationary
distribution and the asymptotic scaling of the MSD for diffusive motion. In
this case, we see that the resetting modifies the transport regime, making the
overall transport sub-diffusive and even reaching a stationary MSD., i.e., a
stochastic localization. This phenomena is also observed in diffusion under
instantaneous Pareto resetting. We check the main results with stochastic
simulations of the process.
