We consider linear parabolic equations of the non-divergence type, and
assume the ellipticity and the continuity on the coefficients of the
second order derivatives and the boundedness on all the coefficients.
Under the assumptions we show the Hölder continuity of the solution in
the spatial component. Furthermore, adding an assumption on the
continuity of the coefficient of the second order derivative, we have
the Lipschitz continuity of the solution. In the proof, we use a
probabilistic method, in particular the coupling method. As a corollary,
under an additional assumption we obtain the Hölder and Lipschitz
continuity of the fundamental solution in a component.