Brownian motion in a wedge with oblique reflection

S. R.S. Varadhan, R. J. Williams

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Abstract

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties: The state space is an infinite two‐dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side. The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero). Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ1 and θ2 be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (‐½π < θ1, θ2 ½π), and set α = (θ1 + θ2)/ξ. The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147‐225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)‐(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)‐(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2. 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 α, the process arises from diffusion approximations to storage systems and queueing networks. (i) The state space is an infinite two‐dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) 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).

Original languageEnglish (US)
Pages (from-to)405-443
Number of pages39
JournalCommunications on Pure and Applied Mathematics
Volume38
Issue number4
DOIs
StatePublished - Jul 1985

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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