Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

Details Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

2010-11-22

arxiv.org/abs/1011.4807v2

Physics

Condensed Matter

Statistical Mechanics

16 pages, 8 figures; revised version 2 adds discussion on spatial small-distance cutoff

Scientific paper

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0 0 (near the absorbing boundary), while R(y) ~ y^gamma exp(-y^2/2) as y -> oo, with phi = 1 - 4 epsilon + O(epsilon^2) and gamma = 1 - 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.

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