The strong Markov property for fermion Brownian motion

The Strong Markov Property of the Brownian Motion

The Strong Markov Property

Throughout, X := {X ï¿¿ } ï¿¿≥0 denotes a Lévy process on R ï¿¿ with triple (ï¿¿ ï¿¿ σï¿¿ ï¿¿), and exponent Ψ. And from now on, we let {ï¿¿ ï¿¿ } ï¿¿≥0 denote the natural filtration of X, all the time remembering that, in accord with our earlier convention, {ï¿¿ ï¿¿ } ï¿¿≥0 satisfies the usual conditions. Definition 1. The transition measures of X are the probability measures P ï¿¿ (ï¿¿ ï¿¿ A)...

Strong Approximation of Brownian Motion

Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation ...

Occupation Times for Markov-Modulated Brownian Motion

We determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. They will be derived in terms of their Laplace transforms, distinguishing occupation times in different phases too. For MMBMs with strictly positive variation parameters we further propose scale functions.

Ergodic Property of the Brownian Motion Process.

This work was carried out while the author was a National Science Foundation predoctoral fellow. The author expresses his appreciation to D. C. Spencer for suggesting this problem and for his help and advice in the preparation of this paper. 1 M. P. Gaffney, "The Heat Equation Method of Milgram and Rosenbloom for Open Riemannian Manifolds," Ann. Math. (to appear). 2 D. C. Spencer, "The Heat Equ...