Which Brownian motion property is the most important? Which Brownian motion property is the most important?
A standard Brownian motion is a stochastic process $(W_t, t\geqslant 0)$ indexed by nonnegative real numbers t with the following properties:


*

*$W_0=0$;

*With probability 1, the function $t \to W_t$ is continuous in t;

*The process $(W_t, t\geqslant 0)$ has stationary, independent increments;

*The increment $W_{t+s}-W_s$ has the $\mathrm{NORMAL}(0, t)$ distribution.


I would just exclude $W_0$. The initial value is quite irrelevant and is typically chosen to be zero only for normalization purposes. If you compare it to discrete-time, the Brownian motion is the equivalent of an iid $N(0,\sigma)$ process. 
Then, it depends on the context, whether you care most about it being continuous, independent or normal.
