Brownian Motion on a Green Space

Abstract

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 process with the initial point $(s,\xi )$ in $R( \pm )$ is defined as the process $\{ [s - t,z(t)],t \geqq 0\} $, where the process $z(t)$ is a Brownian motion in R with the initial point $\xi $. It is proved that for any $\eta $ the function $p( \cdot , \cdot ,\eta )$ has a limit equal to zero on almost every path of the heat motion process beginning at any point of $R( \pm )$. These paths exceed the bounds of any compact subset $R( \pm )$ according to the measure of the increase of their parameters. (The upper bound of the values of the parameter depends on the path.) The transition function of the more complex, but also asymmetric, form has analogous properties for any non-empty open subset in $R( \pm )$. The functions described are Green’s functions of Green spaces. The function $\int_0^\infty {p(t, \cdot , \cdot )dt} $ is a Green’s function on R if R has a positive boundary, i.e., if there exist non-constant, bounded, subharmonic functions in R. The last result for the case where R is a subset of an N-dimensional Euclidean space was obtained by Hunt. The Dirichlet problem is solved in probability terms both for parabolic functions defined on open subsets in $R( \pm )$ and for harmonic functions in R. The construction cited is possible also when the boundary in the space considered is not fixed.