Maximum of a Fractional Brownian Motion: Probabilities of Small Values

Let b _γ (t), b _γ(0)= 0 be a fractional Brownian motion, i.e., a Gaussian process with the structure function , 0 < γ < 2. We study the logarithmic asymptotics of P _T = P{b _γ (t) < 1,□t∈TΔ} as T→∞, where Δ is either the interval (0,1) or a bounded region that contains a vicinity of 0 for the case of multidimensional time. It is shown that ln P _T = - D ln T(1 + o(1)), where D is the dimension of zeroes of b _γ (t) in the former case and the dimension of time in the latter. Received: 28 September 1998 / Accepted: 19 February 1999     

Self-avoiding Fractional Brownian Motion��The Edwards Model

In this work we extend Varadhan??s construction of the Edwards polymer model to the case of fractional Brownian motions in ? ^d , for any dimension d??2, with arbitrary Hurst parameters H??1/d.     

A Renormalized Rough Path over Fractional Brownian Motion

We construct in this article a rough path over fractional Brownian motion with arbitrary Hurst index by (i) using the Fourier normal ordering algorithm introduced in (Unterberger, Commun Math Phy 298(1):1–36, 2010) to reduce the problem to that of regularizing tree iterated integrals and (ii) applying the Bogolioubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization algorithm to Feynman diagrams representing tree iterated integrals.     

Persistence Exponents for Gaussian Random Fields of Fractional Brownian Motion Type

Journal of Statistical Physics - The fractional Brownian motion of index 0?&lt;?H?&lt;?1, H-FBM, with d-dimensional time is considered on an expanding set $$ T\Delta...     

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.     

On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal \(m = m_{0} \epsilon \), the reduced Planck constant to equal \(\hbar = \epsilon \) and the cutoff frequency to equal \(\varLambda = E_{\varLambda }/\epsilon \), where \(m_0\) and \(E_{\varLambda }\) are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as \(\epsilon \rightarrow 0\). We study the limit as \(\epsilon \rightarrow 0\) of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.     

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.     

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.     

Free Energies and Fluctuations for the Unitary Brownian Motion

We show that the Laplace transforms of traces of words in independent unitary Brownian motions converge towards an analytic function on a non trivial disc. These results allow one to study the asymptotic behavior of Wilson loops under the unitary Yang–Mills measure on the plane with a potential. The limiting objects obtained are shown to be characterized by equations analogue to Schwinger–Dyson’s ones, named here after Makeenko and Migdal.     

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.     

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.     

Brownian Motion of the Quantum Dissipation in the RWA

A microscopic approach treating the quantum dissipation process presented by Sun and Yu (Phys. Rev. A49 (1994) 592; A51 (1995) 1845) is invoked to construct the wavefunction of the composite system——the model for a harmonic oscillator interacting with a many-oscillator bath under the rotating wave approximation. It shows the back-action of the system on the bath. In particular, the dynamic evolution of the wavefunction for the composite system maintains a factorized form in its wavefunction. In the limited temperature, the reduced density matrix for the system is also calculated to clarify the influence of Brownian motion on the system.     

A Comparative Study of the Determination of Ferrofluid Particle Size by Means of Rotational Brownian Motion and Translational Brownian Motion

Two methods, the Toroidal Technique and the Forced Rayleigh Scattering (FRS) method, were used in the determination of the size of magnetic particles and their aggregates in magnetic fluids. The toroidal technique was used in the determination of the complex, frequency dependent magnetic susceptibility, x(w)=x'(w) - ix"(w) of magnetic fluids consisting of two colloidal suspensions of cobalt ferrite in hexadecene and a colloidal suspension of magnetite in isopar m with corresponding saturation magnetisation of 45.5 mT, 20 mT and 90 mT, respectively. Plots of the susceptibility components against frequency f over the range 10 Hz to 1 MHz, are shown to have approximate Debye-type profiles with the presence of relaxation components being indicated by the frequency, f _max, of the maximum of the loss-peak in the x"(w) profiles. The FRS method (the interference of two intense laser beams in the thin film of magnetic fluid) was used to create the periodical structure of needle like clusters of magnetic particles. This creation is caused by a thermodiffusion effect known as the Soret effect. The obtained structures are indicative of as a self diffraction effect of the used primary laser beams. The relaxation phenomena arising from the switching off of the laser interference field is discussed in terms of a spectrum of relaxation times. This spectrum is proportional to the hydrodynamic particle size distribution. Corresponding calculations of particle hydrodynamic radius obtained by both mentioned methods indicate the presence of aggregates of magnetic particles.     

Anomalies in Multifractal Formalism for Local Time of Brownian Motion

The Renyi function for the logical time measure of Brownian motion is found. It is shown that this function, the Legendre transform of the multifractal spectrum of and the -function derived by the reciprocal measure formalism are not identical. More examples of having similar anomalies are discussed.     

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。