Brownian motion, martingales, Markov Chains - Rosetta Stone 
What are the most
  fundamental/useful/interesting ways in
  which the concepts of Brownian motion,
  martingales and markov chains are
  related?

I'm a graduate student doing a crash course in probability and stochastic analysis. At the moment, the world of probability is a confusing blur, but I'm starting with a grounding in the basic theory of markov chains, martingales and Brownian motion. While I've done a fair amount of analysis, I have almost no experience in these other matters and while understanding the definitions on their own isn't too difficult, the big picture is a long way away.
I would like to gather together results and heuristics, each of which links together two or more of Brownian motion, martingales and Markov chains in some way. Answers which relate probability to real or complex analysis would also be welcome, such as "Result X about martingales is much like the basic fact Y about sequences".
The thread may go on to contain a Big List in which each answer is the posters' favourite as yet unspecified result of the form "This expression related to a markov chain is always a martingale because blah. It represents the intuitive idea that blah".
Because I know little, I can't gauge the worthiness of this question very well so apologies in advance if it is deemed untenable by the MO police.
 A: Hi, 
Regarding Martingales you can see them as fair games 
This means that if the (martingale) process represents your (random) wealth, you should not be able to design a strategy to increase your current wealth, no matter what the outcome of the sample space is.
Brownian Motion can be seen as a limit of rather simple random walks but I'm sure that you know about this. 
Markov processes "disconnect" Future and Past of the process conditionnally on the present value of the process. Where "disconnect" means that functions of past and of future values of the process are independent conditionnally on the present value of the process.
Does it make things more clear ?
A: Levy's characterisation of Brownian motion:
If $X$ is a continuous martingale and $X$ has quadratic variation process $[ X ]_t = t$ then $X$ is a standard Brownian motion.
A: Let $(B_t)_{t \geq 0}$ be a Brownian motion and $\mathcal B_t:=\sigma(\{B_s : s \leq t\})$ its natural filtration. Then


*

*$B_t$ is a martingale,

*$B_t^2-t$ is a martingale,

*$\exp(\theta B_t - \frac{1}{2}\theta^2 t)$ is a martingale, and

*$H_n(t,B_t)$ is a martingale for every $n$, where $H_n(t,x)$ is the Hermite polynomial defined by 
$$ \exp(\theta x - \frac{1}{2}\theta^2 t) = \sum_{n=0}^{\infty} \frac{\theta^n}{n!} H_n(t,x).$$
Note that $H_1(t,x)=x$ and $H_2(t,x)=x^2-t$, so this latter statement implies the first two.


When writing "is a martingale", it is obviously "with respect to the filtration $(\mathcal B_t)_{t\geq 0}$".
A remarkable consequence of the Levy's characterization of Brownian motion is that every continuous martingale is a time-change of Brownian motion.
Source: L.C.G. Rogers and D. Williams, Diffusion, Markov Processes and Martingales, Vol.1 (2000)
A: If X is a continuous martingale of finite variation such that $X_0 = 0$, then $P(X_t = 0 \  \ \forall t) = 1$. 
A: Girsanov's Theorem.
