  Dynamics of DNA bubbles are of interest for both statistical physics and
biology. We present exact solutions to the Fokker-Planck equation governing
bubble dynamics in the presence of a long-range entropic interaction. The
complete meeting time and meeting position probability distributions are
derived from the solutions. Probability distribution functions reflect the
value of the loop exponent of the entropic interaction. Our results extend
previous results which concentrated mainly on the tails of the probability
distribution functions and open a way to determining the strength of the
entropic interaction experimentally which has been a matter of recent
discussions. Using numerical integration, we also discuss the influence of the
finite size of a DNA chain on the bubble dynamics. Analogous results are
obtained also for the case of subdiffusive dynamics of a DNA bubble in a
heteropolymer, revealing highly universal asymptotics of meeting time and
position probability functions.
