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Variational problem in the non-negative orthant of \(\mathbb{R}^{3}\): reflective faces and boundary influence cones. (English) Zbl 1275.60030

Summary: In this paper we consider the variational problem in the non-negative orthant of \(\mathbb{R}^{3}\). The solution of this problem gives the large deviation rate function for the stationary distribution of an SRBM (semimartingal reflecting Brownian motion). F. Avram, J.G. Dai and J.J. Hasenbein [Queueing Syst. 37, No. 1–3, 259–289 (2001; Zbl 0970.60027)] provided an explicit solution of this problem in the non-negative quadrant. Building on this work, we characterize reflective faces of the non-negative orthant of \(\mathbb{R}^{d}\), we construct boundary influence cones and we provide an explicit solution of several constrained variational problems in \(\mathbb{R}^{3}\). Moreover, we give conditions under which certain spiraling paths to a point on an axis have a cost which is strictly less than the cost of every direct path and path with two pieces.

MSC:

60F10 Large deviations
60J60 Diffusion processes
60J65 Brownian motion
60K25 Queueing theory (aspects of probability theory)

Citations:

Zbl 0970.60027
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References:

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