  We investigate the Brownian diffusion of particles in one spatial dimension
and in the presence of finite regions within which particles can either
evaporate or be reset to a given location. For open boundary conditions, we
highlight the appearance of a Brownian yet non-Gaussian diffusion: at long
times, the particle distribution is non-Gaussian but its variance grows
linearly in time. Moreover, we show that the effective diffusion coefficient of
the particles in such systems is bounded from below by $(1-2/\pi)$ times their
bare diffusion coefficient. For periodic boundary conditions, i.e., for
diffusion on a ring with resetting, we demonstrate a "gauge invariance" of the
spatial particle distribution for different choices of the resetting
probability currents, in both stationary and non-stationary regimes. Finally,
we apply our findings to a stochastic biophysical model for the motion of RNA
polymerases during transcriptional pauses, deriving analytically the
distribution of the length of cleaved RNA transcripts and the efficiency of RNA
cleavage in backtrack recovery.
