Definition1.1. *We say that the real-valued process $(B(t))*{t≥0}$ is a one-dimensional Brownianmotion started from $0$ if:_

- $B(0) = 0$ almost surely, and for all $t≥0$, the law of $B(t)$ is $\mathcal{N}(0,t)$ (centered Gaussian with variance $t$)
- For all positive integerkand all $0≤t_1< t_2<···< t_k$, thekincrements $B(t_1)−B(0),B(t_2)−B(t_1),...,B(t_k)−B(t_{k−1})$ are independent random variables.
- For each $t≥0$ and $h >0$, the law of $B(t+h)−B(t)$ is the same as the law of $B(h)−B(0)$.
- There exists a measurable set $A$ with probability $1$ , such that for all $\omega\in A$, the map $t\to B(t)$ is continuous on $R_+$.

*its existence*. The proof is constructive and gives an iterative way to produce a Brownian motion on dyadic intervals and then extend it to continuous domain.

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```<img src="bm_iterative_construction.png">

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