Brownian Motion on a Submanifold

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

Limiting Behaviors for Brownian Motion Reflected on Brownian Motion

Suppose that g(t) and Wt are independent Brownian motions starting from g(0) = W0 = 0. Consider the Brownian motion Yt reflected on g(t), obtained from Wt by the means of the Skorohod lemma. The upper and lower limiting behaviors of Yt are presented. The upper tail estimate on exit time is computed via principal eigenvalue.

on Brownian Motion

We present an introduction to Brownian motion, an important continuous-time stochastic process that serves as a continuous-time analog to the simple symmetric random walk on the one hand, and shares fundamental properties with the Poisson counting process on the other hand. Throughout, we use the following notation for the real numbers, the non-negative real numbers, the integers, and the non-n...

Notes on Brownian Motion

On Skew Brownian Motion

We consider the stochastic equation X(t) = W(t) + βlX0(t), where W is a standard Wiener process and lX0(⋅) is the local time at zero of the unknown process X. There is a unique solution X (and it is adapted to the fields of W) if |β| ≤ 1, but no solutions exist if |β| > 1. In the former case, setting α = (β + 1)/2, the unique solution X is distributed as a skew Brownian motion with parameter α....