An overview of Brownian Motion in Probability Theory

I will try providing a qualitative overview of Brownian motion as it may be taught in a post-graduate measure theoretic probability class. I will exactly follow Morters and Peres. Additional notes may come from Karatzas if I have the time.

Definition to provide context

BM is infinitesimally random normally incremented motion. Denoted $(B_t)_{t \ge 0}$, has 3 defining tenets:

1. $B_0 \in \RR$ (deterministic starting point)
2. $B_t - B_s \sim \N(0,t-s)$ independently of $B_s$ $\forall t,s$ (independent normal increments)
3. function $t \to B_t$ is continuous (continuity of almost all paths)

The 1st tenet, really, follows from the 2nd because of the stability of $\N$ family.

Now if a particle moves at every instant, does it even move, or does it remain stagnant due to "chronic indecision" (every attempted move is instantly discarded by the whim of the immediate next moment)! Surprisingly, math deems it legal to create a stochastic process whose no matter how small a timespan you assess, you shall always find a random normal increment independent of the past. And it says that the process does move. We will discuss why.

2 levels of studying BM

1. Calculus aspect
2. Measure theoretic aspect

These levels are indeed interconnected. For example:

Defining stop time requires measurability notions. But once the definition is well understood, many applications of stop time and its related theorems flow without explicit reference to measurability conditions anymore. Mostly because "is this object measurable with respect to this field" is answered implicitly, heuristically. We will see how.

Measure theory portions of BM discussion can be significantly more challenging because they will often involve lots of real analytic probability (an actual domain; see Real Analysis and Probability by Richard Dudley).

To prepare you (and myself) honestly for these looming, risky parts, I will mention explicitly, how much you need to be scared before you begin every section. This is same as the percentage of measure theory there is.

1. Definition and first properties

Constructive Definition: 80% measure

Path properties: 80% measure