Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations of the form <x(t₁)x(t₂)> = D(t₁^2H + t₂^2H - |t₁-t₂|^2H), where H, with 0<H<1 is called the Hurst exponent. For H=½, x(t) is a Brownian motion, while for H≠½, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P+(x, t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P₊(x, t)∼t^HR₊(x t^H). Our objective is to compute the scaling function R₊(y), which up to now was only known for the Markov case H=½. We develop a systematic perturbation theory around this limit, setting H=½+ε, to calculate the scaling function R₊(y) to first order in t. We find that R₊(y) ∼ y as y → 0 (near the absorbing boundary), while R+(y)∼y^γ exp(y²/2) as y → ∞, with φ=1-4ε+O(&epsilon²) and γ=1-2ε+O(&epsilon²). Our t-expansion result confirms the scaling relation φ=(1H)/H. 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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Random Geometry and Applications