Related papers: Brownian motion between two random trajectories

Bifractional Brownian motion (bfBm) is a centered Gaussian process with covariance \[ R^{(H,K)}(s,t)= 2^{-K} \left( \left(|s|^{2H}+|t|^{2H} \right)^{K}-|t-s|^{2HK}\right), \qquad s,t\in R. \] We study the existence of bfBm for a given pair…

We study a one-dimensional Brownian motion conditioned on a self-repelling behaviour. Given a nondecreasing positive function f(t), consider the measures mu_t obtained by conditioning a Brownian path so that L_s< f(s), for all s<t, where…

We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Sch\"{u}tz-type formula is derived for the transition probability. We investigate an…

A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to…

This work deals with the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. If the potential has no inversion symmetry and its fluctuations are asymmetric and correlated in time, a net flow can be generated…

In this article we study a problem related to the first passage and inverse first passage time problems for Brownian motions originally formulated by Jackson, Kreinin and Zhang (2009). Specifically, define $\tau_X = \inf\{t>0:W_t + X \le…

We investigate the run and tumble particle (RTP), also known as persistent Brownian motion, in one dimension. A telegraphic noise $\sigma(t)$ drives the particle which changes between $\pm 1$ values with some rates. Denoting the rate of…

We have shown recently how to calculate the large deviation function of the position $X_{\max}(t) $ of the right most particle of a branching Brownian motion at time $t$. This large deviation function exhibits a phase transition at a…

We study interacting Brownian particles on the half-line whose interaction occurs through boundary local times at the origin. The particle system is given by \[ X_i^n(t)=X^n_{0,i}+W_i^n(t)+L_i^n(t) +\frac{1}{n-1}\sum_{j\ne…

We study theoretically, experimentally and numerically the probability distribution $F(t_f|x_0,L)$ of the first passage times $t_f$ needed by a freely diffusing Brownian particle to reach a target at a distance $L$ from the initial position…

We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift…

Donsker's theorem shows that random walks behave like Brownian motion in an asymptotic sense. This result can be used to approximate expectations associated with the time and location of a random walk when it first crosses a nonlinear…

We consider the problem of strong existence and uniqueness of a Brownian motion forced to stay in the quadrant by an electrostatic repulsion from the sides that works obliquely. The results are reminiscent of the study of a Brownian motion…

Using extensive numerical studies we demonstrate that absolute negative mobility of a Brownian particle (i.e. the net motion into the direction opposite to a constant biasing force acting around zero bias) does coexist with anomalous…

In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and difusion by suggesting and examining several Brownian functionals which characterize the lifetime and reactivity of such stochastic processes. We…

Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian process with a power-law time dependent diffusion coefficient. Its mean squared displacement is $\langle x^2(t)\rangle\simeq\mathscr{K}(t)t$ with…

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to B.B. Mandelbrot and J.W. van Ness (1968) as a self-similar…

Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the…

We consider a fractional Brownian motion with unknown linear drift such that the drift coefficient has a prior normal distribution and construct a sequential test for the hypothesis that the drift is positive versus the alternative that it…

We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may…