### On Two Dimensional Markov Processes with Branching Property

Shinzo Watanabe
1969 Transactions of the American Mathematical Society
Introduction. A continuous state branching Markov process (C.B.P.) was introduced by Jirina  and recently Lamperti  determined all such processes on the half line. (A quite similar result was obtained independently by the author.) This class of Markov processes contains as a special case the diffusion processes (which we shall call Feller's diffusions) studied by Feller  . The main objective of the present paper is to extend Lamperti's result to multi-dimensional case. For simplicity
more » ... se. For simplicity we shall consider the case of 2-dimensions though many arguments can be carried over to the case of higher dimensions(2). In Theorem 2 below we shall characterize all C.B.P.'s in the first quadrant of a plane and construct them. Our construction is in an analytic way, by a similar construction given in Ikeda, Nagasawa and Watanabe , through backward equations (or in the terminology of  through S-equations) for a simpler case and then in the general case by a limiting procedure. A special attention will be paid to the case of diffusions. We shall show that these diffusions can be obtained as a unique solution of a stochastic equation of Ito (Theorem 3). This fact may be of some interest since the solutions of a stochastic equation with coefficients Holder continuous of exponent 1/2 (which is our case) are not known to be unique in general. Next we shall examine the behavior of sample functions near the boundaries (xj-axis or x2-axis). We shall explain, for instance, the case of xraxis. There are two completely different types of behaviors. In the first case Xj-axis acts as a pure exit boundary: when a sample function reaches the Xj-axis then it remains on it moving as a one-dimensional Feller diffusion up to the time when it hits the origin and then it is stopped. In the second case, there is a point x0 on x^-axis such that 22 = (0, *o) acts as a reflecting boundary and Ex= (x0, oo) acts as a pure entrance boundary (Theorem 4 and Corollaries).