Speaker
Kay Wiese
(LPTENS)
Description
Fractional Brownian motion is a Gaussian process x(t) with
zero mean and two-time correlations of the form
<x(t1)x(t2)> =
D(t12H + t22H -
|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)∼tHR+(x
tH). 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)∼yγ
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.
