Related papers: Matrix valued Brownian motion and a paper by Polya
In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…
In this paper by calculating carefully the capacities (defined by high order Sobolev norms on the Wiener space) for some functions of Brownian motion, we show that the dyadic approximations of the sample paths of the Brownian motion…
This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…
Individual movements of a rod-like self-propelled particle on a flat substrate are quantified. Biological systems that fit into this description may be the Gram-negative delta-proteobacterium Myxococcus xanthus, Gram-negative bacterium…
The interest in the concept of entropic forces has risen considerably since E. Verlinde proposed to interpret the force in Newton s second law and Gravity as entropic forces [1]. Brownian motion, the motion of a small particle (pollen)…
Let (B_t : t > 0) be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point x in the plane such that $H^{\phi_\alpha}({t > 0 :…
We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.
Brownian motion has served as a pilot of studies in diffusion and other transport phenomena for over a century. The foundation of Brownian motion, laid by Einstein, has generally been accepted to be far from being complete since the late…
The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution…
In this note, we combine the two approaches of Billingsley (1998) and Cs\H{o}rg\H{o} and R\'ev\'esz (1980), to provide a detailed sequential and descriptive for creating s standard Brownian motion, from a Brownian motion whose time space is…
In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are…
We consider a class of linear Volterra transforms of Brownian motion associated to a sequence of M\"untz Gaussian spaces and determine explicitly their kernels; some interesting links with M\"untz-Legendre polynomials are provided. This…
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…
Active Brownian motion commonly assumes spherical overdamped particles. However, self-propelled particles are often neither symmetric nor overdamped yet underlie random fluctuations from their surroundings. Active Brownian motion has…
In the first paper of this series, I investigated whether a wavefunction model of a heavy particle and a collection of light particles might generate "Brownian-Motion-Like" trajectories of the heavy particle. I concluded that it was…
We solve the problem of formulating Brownian motion in a relativistically covariant framework in 1+1 and 3+1 dimensions. We obtain covariant Fokker-Planck equations with (for the isotropic case) a differential operator of invariant…
In this paper we investigate the class of grey Brownian motions $B_{\alpha,\beta}$ ($0<\alpha<2$, $0<\beta\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional…
We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(\theta)$ which depends only on the angle $\theta$ of the particle. We…
We interpret a formula for meromorphic functions on foliations by Riemann surfaces as an analogue to the product formula of valuations in algebraic number theory.
We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.