Sampling fractional Brownian motion in presence of absorption: A Markov chain method

Abstract

We numerically study fractional Brownian motion (fBm) with an absorbing boundary at the origin for selected values of the Hurst exponent Hε[0,1]. Using a Monte Carlo sampling technique, we are able to numerically generate these fBm processes at discrete times for up to 10(7) time steps, even for values as small as H=1/4. The results are compatible with previous analytical results that suggest that the distribution of (rescaled) endpoints y follow a power law P(+)(y)~y(φ) with φ=(1-H)/H, even for small values of H. Furthermore, for H=0.5 we study analytically the finite-length corrections to first order, namely a plateau of P(+)(y) for y→0 which decreases with increasing process length. These corrections are compatible with our numerical results.

Scaling Properties of Weakly Self-Avoiding Fractional Brownian Motion in One Dimension

“…Fractional Brownian motion (fBm) has been suggested as a more general model for chain polymers, see e.g. [HMR13]. FBm on R d , d ≥ 1, with "Hurst parameter" H ∈ (0, 1) is a dtuple of independent centered Gaussian processes [BHØZ07,Mis08]…”

“…The same holds for approximative methods using spectral densities, the Paxson method [Pax97], which uses Fourier transform or the Decreusefond-Lavaud method [DL96] which is using the kernel representation of fBm. A more natural way to implement a discrete fractional walk with selfrepellence is to use off-lattice discretization and Monte Carlo methods based on a Metropolis algorithm as in [BGZ00] for the Brownian case and in [HMR13] for the fractional Brownian motion. In the our algorithm the Metropolis routine is used to update "bonds", i.e.…”

Observing symmetry-broken optimal paths of the stationary Kardar-Parisi-Zhang interface via a large-deviation sampling of directed polymers in random media

“…To achieve this, we use a biased distribution [37] of the randomness by modifying the original quenched distribution weight [38], which is a product of independent Gaussians, by an additional exponential Boltzmann factor e −H/Θ , where Θ is an adjustable temperature-like parameter, allowing us to address different regions of values of H. When Θ → ±∞, we restore the original unbiased distribution. For Θ → 0 + , one will focus the sampling on large negative values of H, while for Θ → 0 − the sampling will be in the region of large positive values of H. The fundamental idea of this large-deviation approach is versatile and it has been applied to various problems, e.g., to study large-deviation properties of random graphs [39,40], of random walks [41,42] of energy grids [43,44], of biological sequence alignments [38,45], of nonequilibrium work distributions [46], and of traffic models [47].…”

Critical behavior of the Anderson model on the Bethe lattice via a large-deviation approach

“…Standard large-deviation algorithms rely on sampling of biased distributions and unbiasing the obtained data in the end. Such approaches have been widely used, e.g., to study the large-deviation properties of random-graphs [63,64], biological sequence alignments [65], protein folding [66], random walks [67,68], models of transport [69,70], the Kardar-Parisi-Zhang equation [71], nonequilibrium work processes [72] and many more. We have tried such an approach based on a bias here, but were not able to see convergence of the used Markov chains deep enough in the tails.…”