Journal of Applied Probability

A geometric drift inequality for a reflected fractional Brownian motion process on the positive orthant

Chihoon Lee

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We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R+d, with drift r0Rd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Z̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set CS such that ΔV(x):= Ex[V(Z̆(1))] - V(x) ≤ -βV(x) + b1C(x), xS, for an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

Article information

J. Appl. Probab., Volume 48, Number 3 (2011), 820-831.

First available in Project Euclid: 23 September 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G22: Fractional processes, including fractional Brownian motion
Secondary: 90B18: Communication networks [See also 68M10, 94A05] 60G15: Gaussian processes 60G18: Self-similar processes

Reflected fractional Brownian motion heavy traffic theory geometric drift inequality return time


Lee, Chihoon. A geometric drift inequality for a reflected fractional Brownian motion process on the positive orthant. J. Appl. Probab. 48 (2011), no. 3, 820--831. doi:10.1239/jap/1316796917.

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