  We investigate the persistence probability of a Brownian particle in a
harmonic potential, which decays to zero at long times -- leading to an
unbounded motion of the Brownian particle. We consider two functional forms for
the decay of the confinement, an exponential and an algebraic decay. Analytical
calculations and numerical simulations show, that for the case of the
exponential relaxation, the dynamics of Brownian particle at short and long
times are independent of the parameters of the relaxation. On the contrary, for
the algebraic decay of the confinement, the dynamics at long times is
determined by the exponent of the decay. Finally, using the two-time
correlation function for the position of the Brownian particle, we construct
the persistence probability for the Brownian walker in such a scenario.
