The matrix of Brownian values
Each row is one outcome \(\omega_i\); each column is one time \(t = kh\), with \(h = 0.04\). Every entry is a single real number \(W_{kh}(\omega_i)\). Click a row header to fix \(\omega\); click a column header to fix \(t\). Clicking a cell selects both.
The sample space \(\Omega\)
\[ \Omega = \{\text{all possible outcomes}\} = \{\omega_1,\omega_2,\omega_3,\omega_4,\dots\} \]
The experiment is the procedure (run a Brownian particle from 0 and watch it). \(\Omega\) is the collection of possible outcomes of that procedure. One run of the experiment selects one \(\omega \in \Omega\), and that single \(\omega\) determines the entire trajectory.
The probability measure \(\mathbb P\) lives on \(\Omega\): it says how likely various sets of outcomes are. All randomness is in which \(\omega\) gets selected; once \(\omega\) is fixed, nothing is random.
Rigorous picture. In the canonical (Wiener space) construction one takes \(\Omega = \{\omega \in C([0,\infty),\mathbb R) : \omega(0)=0\}\) with Wiener measure, and \(W_t(\omega) = \omega(t)\). So an outcome literally is a continuous path.
\(\Omega\) is a set of outcomes. It is not a vector field, and it is not the table above: the table shows only 6 of its uncountably many elements, observed at 7 times.
Random vector and realization
Choose finitely many times and stack the corresponding random variables. The result is a random vector: each component is a random variable, so the whole object is a function of \(\omega\).
The random vector is the mapping \(X\). It is a function on \(\Omega\), not a list of numbers.
A realization is the concrete numerical vector \(X(\omega)\) obtained when one particular outcome \(\omega\) is selected. It is one row of the matrix.
Each component \(W_{kh}:\Omega\to\mathbb R\) is a random variable (a column). In fact \(X\) is a Gaussian vector with mean \(0\) and \(\operatorname{Cov}(W_s,W_t)=\min(s,t)\). The component \(W_0\equiv 0\) is degenerate.
Click a component to select that time; the matching column in §1 is the random variable behind that component.
Brownian increments
\( W_t \) = the value of Brownian motion at time \(t\).
\( \Delta W = W_{t_2} - W_{t_1} \) is an increment: a difference of two times.
Here \(\Delta W_k := W_{kh} - W_{(k-1)h}\), the increment over the \(k\)-th interval \([(k-1)h,\,kh]\), for \(k = 1,\dots,6\). The bracket is drawn on one fixed path: both endpoints come from the same \(\omega\).
For equally spaced times: \( \Delta W_k \sim \mathcal N(0,h) \), and increments over disjoint intervals are independent. Hence
\[ W_{kh} = \Delta W_1 + \Delta W_2 + \cdots + \Delta W_k , \]
a sum of \(k\) independent \(\mathcal N(0,h)\) variables, so \(W_{kh}\sim\mathcal N(0,kh)\).
Same outcome \(\omega\), two different times. Move along a row.
Same time, two different outcomes. Move down a column: this compares two different sample paths.
For fixed \(\omega_1,\omega_2\) this is just a number; it is not even a function of a single \(\omega\), so it is not a random variable on \(\Omega\), let alone an increment of the process.
All increments vs \(\mathcal N(0,h)\)
Independence: \(\Delta W_1\) against \(\Delta W_2\)
Brownian motion vs. simple symmetric random walk
Simple symmetric random walk
Brownian motion at discrete times
Sampling Brownian motion at discrete times resembles a random walk, but its increments are Gaussian rather than exactly \(+1\) or \(-1\). Both pictures above are rows (one \(\omega\), time varying). Fixing \(t\) does not give a random walk:
\[ \text{Fix } t,\ \text{vary } \omega \;\Rightarrow\; W_t \sim \mathcal N(0,t), \qquad\qquad \text{Fix } \omega,\ \text{vary } t \;\Rightarrow\; \text{one Brownian sample path.} \]
Aside: properly scaled random walks converge to Brownian motion. If \(S\) is a simple symmetric random walk, \(\big(S_{\lfloor nt\rfloor}/\sqrt n\big)_{t\ge0}\) converges in distribution to \((W_t)_{t\ge0}\) as \(n\to\infty\) (Donsker's invariance principle).