The space R, locally isometric to an open N-dimensional sphere, is called a Green space of dimensionality N. A Markov process in this space which is locally a Brownian motion of dimensionality N we shall call a Brownian motion in R. It is shown that for each value of the variance parameter there exists synonymously a transition probability for the Brownian motion process in R. This transition probability from the constant $\xi $ to the constant $\eta $, at time t has a density $p(t,\xi ,\eta )$. It is shown that for a reasonable choice of this density the function p, when it is equal to zero for $t \leqq 0$, defines for fixed $\eta $ a superparabolic function on $(t,\xi )$ which is parabolic everywhere except the point $(0,\eta )$; the singularity at this point is the same as for the transition probability density of the corresponding N-dimensional Brownian motion. In addition, \[ p( {t,\xi ,\eta } ) = p( {t,\eta ,\xi } ). \]Let $R( \pm )$ be the direct product of R and the real line. A heat motion proces...