Brownian Motion In Space Research Articles

Second order Lyapunov exponents for parabolic and hyperbolic Anderson models

In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension $d$, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures: (i) the Fourier transform of the spectral measure of the noise is a non-negative locally-integrable function; (ii) $d=1$ and the noise is a fractional Brownian motion in space with index $1/4<H<1/2$. In both cases, we show that there is striking similarity between the Laplace transforms of the second moment of the solutions to these two models. Building on this connection and the recent powerful results of [Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 1305–1340] for the parabolic model, we compute the second order (upper) Lyapunov exponent for the hyperbolic model. In case (i), when the spatial covariance of the noise is given by the Riesz kernel, we present a unified method for calculating the second order Lyapunov exponents for the two models.

Highly Resolved Brownian Motion in Space and in Time

Since the discovery of Brownian motion in bulk fluids by Robert Brown in 1827 , Brownian motion at long timescales has been extensively studied both theoretically and experimentally for over a century. The theory for short-timescale Brownian motion was also well established in the last century, while experimental studies were not accessible until this decade. This article reviews experimental progress on short-timescale Brownian motion and related applications. The ability to measure instantaneous velocity enables new fundamental tests of statistical mechanics of Brownian particles, such as the Maxwell–Boltzmann velocity distribution and the energy equipartition theorem. In addition, Brownian particles can be used as probes to study boundary effects imposed by a solid wall, wettability at solid–fluid interfaces, and viscoelasticity. We propose future studies of fluid compressibility and nonequilibrium physics using short-duration pulsed lasers. Lastly, we also propose that an optically trapped particle can serve as a new testing ground for nucleation in a supersaturated vapor or a supercooled liquid.

Inertial and bias effects in the rotational Brownian motion of rodlike molecules in a uniaxial potential

Inertial effects in the rotational brownian motion in space of a rigid dipolar rotator (needle) in a uniaxial potential biased by an external field giving rise to asymmetry are treated via the infinite hierarchy of differential-recurrence relations for the statistical moments (orientational correlation functions) obtained by averaging the Euler-Langevin equation over its realizations in phase space. The solutions of this infinite hierarchy for the dipole correlation function and its characteristic times are obtained using matrix continued fractions showing that the model simultaneously predicts both slow overbarrier (or interwell) relaxation at low frequencies accompanied by intermediate frequency Debye relaxation due to fast near-degenerate motion in the wells of the potential (intrawell relaxation) as well as the high frequency resonance (Poley) absorption due to librations of the dipole moments. It is further shown that the escape rate of a brownian particle from a potential well as extended to the Kramers turnover problem via the depopulation factor yields a close approximation to the longest (overbarrier) relaxation time of the system. For zero and small values of the bias field parameter h, both the dipole moment correlation time and the longest relaxation time have Arrhenius behavior (exponential increase with increasing barrier height). While at values of h in excess of a critical value however far less than that required to achieve nucleation, the Arrhenius behavior of the correlation time disappears.

Simulation of Stochastic Reaction-Diffusion Processes on Unstructured Meshes

Stochastic chemical systems with diffusion are modeled with a reaction-diffusion master equation. On a macroscopic level, the governing equation is a reaction-diffusion equation for the averages of the chemical species. On a mesoscopic level, the master equation for a well stirred chemical system is combined with Brownian motion in space to obtain the reaction-diffusion master equation. The space is covered by an unstructured mesh and the diffusion coefficients on the mesoscale are obtained from a finite element discretization of the Laplace operator on the macroscale. The resulting method is a flexible hybrid algorithm in that the diffusion can be handled either on the meso- or on the macroscale level. The accuracy and the efficiency of the method are illustrated in three numerical examples inspired by molecular biology.

The exact distribution of the oscillation period in the underdamped one-dimensional Sinai model

We consider the Newtonian dynamics of a massive particle in a one dimemsional random potential which is a Brownian motion in space. This is the zero temperature nondamped Sinai model. As there is no dissipation the particle oscillates between two turning points where its kinetic energy becomes zero. The period of oscillation is a random variable fluctuating from sample to sample of the random potential. We compute the probability distribution of this period exactly and show that it has a power law tail for large period, P(T)\sim T^{-5/3} and an essential singluarity P(T)\sim \exp(-1/T) as T\to 0. Our exact results are confirmed by numerical simulations and also via a simple scaling argument.

The trace of spatial brownian motion is capacity-equivalent to the unit square

We show that with probability 1, the trace B[0, 1] of Brownian motion in space, has positive capacity with respect to exactly the same kernels as the unit square. More precisely, the energy of occupation measure on B[0, 1] in the kernel f(∣x−y∣), is bounded above and below by constant multiples of the energy of Lebesgue measure on the unit square. (The constants are random, but do not depend on the kernel.) As an application, we give almost-sure asymptotics for the probability that an α-stable process approaches within ɛ of B[0, 1], conditional on B[0, 1]. The upper bound on energy is based on a strong law for the approximate self-intersections of the Brownian path. We also prove analogous capacity estimates for planar Brownian motion and for the zero-set of one-dimensional Brownian motion.

A Representation for the Intersection Local Time of Brownian Motion in Space

We present a "Tanaka-like" representation for $\alpha(x, B)$, the local time of intersection for Brownian motion in 2 and 3 dimensions, where $\alpha(x, B)$ is formally $\int_B \int \delta_x(\omega_t - \omega_s) ds dt$.

First Passage times and Sojourn Times for Brownian Motion in Space and the Exact Hausdorff Measure of the Sample Path

First Passage times and Sojourn Times for Brownian Motion in Space and the Exact Hausdorff Measure of the Sample Path

First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the definition of 4-measure. In the past the only successful method has been to use the connection between Hausdorff measures and generalized capacities. The first result of this kind appears in [16] where it is proved that for k >2, the Hausdorff measure with respect to ta is infinite for all a 3 we obtain a law of the iterated logarithm for the total time Tk(a, co) spent by the path X in a sphere of radius a as a-* 0+. We also determine the local behavior of the first passage time Pk(a, co) out of a sphere of radius a for k > 1. In order to obtain the required asymptotic results we required good estimates of the distribution functions for the random variables Tk(a, co), Pk(a, c). We use the method developed by Mark Kac to compute these dis-