Time-averaged MSD of Brownian motion

On time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

First Passage Time of Skew Brownian Motion

Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper, we first generalize results of Pitman and Yor (2001) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use this result to derive the first passage time distribut...

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 . We conjecture that this is the worse case correction possible.

Brownian motion at short time scales

Continuous time process and Brownian motion

Consider a complete probability space (Ω,F ,P;F) equipped with the Þltration F = {Ft; 0 ≤ t <∞}. A stochastic process is a collection of random variables X = {Xt; 0 ≤ t <∞} where, for every t, Xt : Ω → Rd is a random variable. We assume the space Rd is equipped with the usual Borel σ-algebra B(Rd). Every Þxed ω ∈ Ω corresponds to a sample path (or, trajectory), that is, the function t 7→ Xt(ω) ...