  Inspired by many examples in nature, stochastic resetting of random processes
has been studied extensively in the past decade. In particular, various models
of stochastic particle motion were considered where upon resetting the particle
is returned to its initial position. Here we generalize the model of diffusion
with resetting to account for situations where a particle is returned only a
fraction of its distance to the origin, e.g., half way. We show that this model
always attains a steady-state distribution which can be written as an infinite
sum of independent, but not identical, Laplace random variables. As a result,
we find that the steady-state transitions from the known Laplace form which is
obtained in the limit of full resetting to a Gaussian form which is obtained
close to the limit of no resetting. A similar transition is shown to be
displayed by drift-diffusion whose steady-state can also be expressed as an
infinite sum of independent random variables. Finally, we extend our analysis
to capture the temporal evolution of drift-diffusion with partial resetting,
providing a bottom-up probabilistic construction that yields a closed form
solution for the time dependent distribution of this process in Fourier-Laplace
space. Possible extensions and applications of diffusion with partial resetting
are discussed.
