A standard Brownian motion (or Wiener process) is a stochastic process $(W_t)_{t \ge 0}$ on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ satisfying:
1. Initial condition: $W_0 = 0$ almost surely.
2. Independent increments: For any $0 \le s < t$, the increment $W_t - W_s$ is independent of $\sigma(W_r : r \le s)$.
3. Gaussian increments: For any $0 \le s < t$, $W_t - W_s \sim \mathcal{N}(0, t - s)$.
4. Continuous paths: Almost surely, the map $t \mapsto W_t(\omega)$ is continuous.