  We consider the non-degenerate second-order parabolic partial differential
equations of non-divergence form with bounded measurable coefficients (not
necessary continuous). Under some assumptions it is known that the fundamental
solution to the equations exists uniquely, has the Gaussian bounds and is
locally H\"older continuous. In the present paper, we concern the Gaussian
bounds and the lower bound of the index of the H\"older continuity with respect
to the initial point. We use the pinned diffusion processes for the
probabilistic representation of the fundamental solutions and apply the
coupling method to obtain the regularity of them. Under some assumptions weaker
than the H\"older continuity of the coefficients, we obtain the Gaussian bounds
and the $(1-\varepsilon)$-H\"older continuity of the fundamental solution in
the initial point.
