Nonergodic Brownian Motion

The long-time behavior of a system is suggested to confirm nonergodicity of non-Markovian Brownian dynamics, namely, whether the stationary probability density function （PDF） of the system characterized mainly by low moments of variables depends on the initial preparation. Thus we classify nonergodic Brownian motion into two classes： the class-I is that the PDF of a force-free particle depends on the initial velocity and the equilibration can be recovered through a bounded potential; while the PDF in the class-H depends on the initial coordinate and the equilibration can not be approached by introducing any potential. We also compare our result with the conditions of three kinds for ergodicity.     

On the Asymptotic Behavior of the Hyperbolic Brownian Motion

The main results in this paper concern large and moderate deviations for the radial component of a $n$ -dimensional hyperbolic Brownian motion (for $n\ge 2$ ) on the Poincaré half-space. We also investigate the asymptotic behavior of the hitting probability $P_\eta (T_{\eta _1}^{(n)}<\infty )$ of a ball of radius $\eta _1$ , as the distance $\eta $ of the starting point of the hyperbolic Brownian motion goes to infinity.     

Quantization of Brownian Motion

Following our work on the quantization of nonconservative systems using fractional calculus, the canonical quantization of a system with Brownian motion is carried out according to the Dirac method. A suitable Lagrangian corresponding to the Langevin equation is set up. Further, a Hamiltonian is constructed and is transformed to Schrödinger's equation which is solved.     

Loop-Erasure of Planar Brownian Motion

We use a coupling technique to prove that there exists a loop-erasure of the time-reversal of a planar Brownian motion stopped on exiting a simply connected domain, and that the loop-erased curve is a radial SLE₂ curve. This result extends to Brownian motions and Brownian excursions under certain conditioning in a finitely connected plane domain, and the loop-erased curve is a continuous LERW curve.     

New Type of Brownian Motion

We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision (Brownian motion with adhesion for a harmonically bound particle). This is another form of a stochastic oscillator, different from oscillator usually studied that is subject to a random force or having random frequency or random damping. Calculation of the first two stationary moments shows that for white multiplicative noise of week strength the second moment coincides with that of usual Brownian motion, but for symmetric dichotomous noise, the second moment may appear the same type of the “energetic” instability, which exists for white noise random frequency or damping coefficient.     

Testing of Multifractional Brownian Motion

Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index 0<H<1. In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena.     

Quantum Brownian Motion. II

This paper generalizes some previous resultspresented in Gaioli et al. [Int. J. Theor. Phys. 36,2167 (1997)]. We evaluate the autocorrelation functionof the stochastic acceleration and study the asymptotic evolution of the mean occupation number of aharmonic oscillator playing the role of a Brownianparticle. We also analyze some deviations from the Bosepopulation at low temperatures and compare it with the deviations from the exponential decay lawof an unstable quantum system.     

Brownian Motion with Dry Friction

A small object (Solid or droplet) is placed on a horizontally vibrating plate, imposing an acceleration γ(t) in the form of a white noise. The object experiences dry friction (due to soild/solid interaction, or to contact angle hysteresis in the case of a droplet). The object is driven by a force γ(t) – Δσ(t) where σ(t), =±1, depending on the sign of the velocity. We discuss the motion at two levels: (i) in terms of simple scaling laws, (ii) by a propagator technique. (a) When Δ is below a certain crossover value Δ*, we expect an unperturbed (Langevin) Brownian motion. (b) When Δ > Δ*, we expect a reduced diffusion coefficient proportional to Δ⁻⁴ for small Δ.     

Nonlinear Theory of Quantum Brownian Motion

A nonlinear theory of quantum Brownian motion in classical environment is developed based on a thermodynamically enhanced nonlinear Schrödinger equation. The latter is transformed via the Madelung transformation into a nonlinear quantum Smoluchowski-like equation, which is proven to reproduce key results from the quantum and classical physics. The application of the theory to a free quantum Brownian particle results in a nonlinear dependence of the position dispersion on time, being quantum generalization of the Einstein law of Brownian motion. It is shown that the time of decoherence from quantum to classical diffusion is proportional to the square of the thermal de Broglie wavelength divided by the classical Einstein diffusion constant.     

Survival of Near-Critical Branching Brownian Motion

Consider a system of particles performing branching Brownian motion with negative drift \(\mu= \sqrt{2 - \varepsilon}\) and killed upon hitting zero. Initially there is one particle at x>0. Kesten (Stoch. Process. Appl. 7:9–47, 1978) showed that the process survives with positive probability if and only if ε>0. Here we are interested in the asymptotics as ε→0 of the survival probability Q _μ (x). It is proved that if \(L=\pi/\sqrt{\varepsilon}\) then for all x∈?, lim? _ε→0 Q _μ (L+x)=θ(x)∈(0,1) exists and is a traveling wave solution of the Fisher-KPP equation. Furthermore, we obtain sharp asymptotics of the survival probability when x<L and L?x→∞. The proofs rely on probabilistic methods developed by the authors in (Berestycki et al. in arXiv:1001.2337, 2010). This completes earlier work by Harris, Harris and Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat. 42:125–145, 2006) and confirms predictions made by Derrida and Simon (Europhys. Lett. 78:60006, 2007), which were obtained using nonrigorous PDE methods.     

Noncolliding Brownian Motion and Determinantal Processes

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin–McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin–McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.     

Reinforced Brownian Motion: A Prototype

We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of \(S_N\) (the location of the walker at the \(N^{th}\) step) and \(A_N\) , the maximum location the walker achieved in \(N\) steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its “environment”, both parametrized by the amplitude \(\delta \) of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of \(-1< \delta < \infty \) .     

利用Matlab模拟布朗运动测量实验

利用matlab工具模拟了布朗运动测量的实验。通过一正态随机数产生函数模拟从而产生布朗运动步距。在假定粒子所受拖曳力满足斯托克斯关系的情况下,通过拟合多个粒子的均方位移随时间的变化曲线得到斜率,从而进一步可得出扩散系数和波尔兹曼常数。同时,根据模拟结果也对如何减小实验误差作了分析。     

Noncommutative Brownian Motion in Monotone Fock Space

An example of noncommutative Brownian motion is constructed on the monotone Fock space which is a kind of “Fock space” generated by all the decreasing finite sequences of positive real numbers. The probability distribution at time associated to this Brownian motion is shown to be the arcsine law normalized to mean 0 and variance t. Received: 15 March 1996\,/\,Accepted: 2 July 1996     

Slowdown for Time Inhomogeneous Branching Brownian Motion

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to T ^1/3. We conjecture that this is the worse case correction possible.     

Cooper Pair Formation by Quantizing Brownian Motion

We derive a new quantum interaction by quantizing the Brownian motion based on the Nelson equations. By applying the canonical quantization for the equations, interaction as the connection of Brownian and quantum motions appears. Interesting aspect is that it can overcome the Coulomb repulsion if the diffusion coefficient is large enough. As the parameters are mass, diffusion coefficient, and probability density, we can calculate and predict the Cooper pair formation with measurable variables.