Brownian motion on a smash line algebra (a smash or braided version of the algebra resulting by tensoring the real line and the generalized paragrassmann line algebras), is constructed by means of its Hopf algebraic structure. Further, statistical moments, non stationary generalizations and its diffusion limit are also studied. The ensuing diffusion equation posseses triangular matrix realizations.
We define kinetic Brownian motion on the diffeomorphism group of a closed Riemannian manifold, and prove that it provides an interpolation between the hydrodynamic flow of a fluid and a Brownian-like flow.
We consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale $eta_* geq N^{-2/3}$ and no outliers, then after times $t gg sqrt{ eta_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-ErdH{o}s-Yau-Yin] and the energy estimate of [Bourgade-ErdH{o}s-Yau].
We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two additional terms and the one of them can be regarded as the the quantum potential. By choosing a parameter space, various important states are obtained. Further, the equations contain other interesting phenomena such as general dynamics of diffusion process, collapse of the soliton, the nonlinear extension of the Schrdinger equation, and the dynamics of phase transition.
Tempered fractional Brownian motion is revisited from the viewpoint of reduced fractional Ornstein-Uhlenbeck process. Many of the basic properties of the tempered fractional Brownian motion can be shown to be direct consequences or modifications of the properties of fractional Ornstein-Uhlenbeck process. Mixed tempered fractional Brownian motion is introduced and its properties are considered. Tempered fractional Brownian motion is generalised from single index to two indices. Finally, tempered multifractional Brownian motion and its properties are studied.
Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent $Hin [0,1]$, generalising standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for non-equilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many non-trivial observables analytically: We generalize the celebrated three arcsine-laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in $epsilon = H-tfrac{1}{2}$, up to second order. We find that the three probabilities are different, except for $H=tfrac{1}{2}$ where they coincide. Our results are confirmed to high precision by numerical simulations.