| This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties. (1) The state space is an infinite two-dimensional wedge, and the process behaves in the interior like an ordinary Brownian motion. (2) The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side. (3) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero). Hereafter, let (xi) be the angle of the wedge (0 < (xi) < 2(pi)), let (theta)(,1) and (theta)(,2) be the angles of reflection on the two sides of the wedge, measured from the inward normals with positive angles being toward the corner.;and set (alpha) = (0(,1) + (theta)(,2))/(xi).;The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundard conditions. It is shown that no solution exists if (alpha) (GREATERTHEQ) 2. In this case, there is a unique continuous strong Markov process satisfying (1) and (2) above; it reaches the corner of the wedge almost surely and it adsorbs there. If (alpha) < 2, however, then there is a unique continuous strong Markov process satisfying (1)-(3). It is shown that starting from x (NOT=) 0 this process is a semimartingale and does not reach the corner of the wedge if (alpha) (LESSTHEQ) 0, is a semimartingale and does reach the corner if 0 < (alpha) < 1, and is not a semimartingale and does reach the corner if 1 (LESSTHEQ) (alpha) < 2. If (alpha) < 0, the process is transient. If 0 (LESSTHEQ) (alpha) < 2, the process is null recurrent and the density of its unique (up to a scalar multiple) (sigma)-finite invariant measure is given.;The general theory of multi-dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of (alpha), the process arises from diffusion approximations to storage systems and queueing networks.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). |
