ThmDex – An index of mathematical definitions, results, and conjectures.
Distribution of the standard real Wiener process at a given point is gaussian
Formulation 0
Let $W : [0, \infty) \to \text{Random}(\mathbb{R})$ be a D3658: Standard real Wiener process.
Let $t \in [0, \infty)$ be an D4767: Unsigned real number.
Then \begin{equation} W_t \overset{d}{=} \text{Gaussian}(0, \sqrt{t}) \end{equation}
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Comment 0
Note: the second parameter $\sqrt{t}$ in the notation $\text{Gaussian}(0, \sqrt{t})$ is the standard deviation so that $\text{Var} W_t = t$.