standard Brownian motion

布朗运动

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Other names / variants: Wiener process · 维纳过程

Translation evidence: Literature usage

Meaning in context

A process $B_t$ starting at zero, with almost surely continuous paths and independent increments on disjoint intervals, such that $B_t-B_s\sim N(0,t-s)$ for $0\le s<t$.

从0出发、轨道几乎必然连续、不同时间区间增量独立，且s至t的增量服从均值0方差t−s正态分布的过程。

Example

For standard Brownian motion, $E[B_sB_t]=\min(s,t)$.

Usage and distinctions

Normal marginals alone are insufficient; this definition explicitly includes continuity and uses elapsed time as variance.

Read the encyclopedia explanation

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Gregory F. Lawler — Notes on ProbabilitySection 7, Brownian motion, defining conditions (i)–(iv) · Checked 2026-10-14

Translation sources

1. 李东风 — 应用随机过程，第7章 布朗运动（北京大学）§7.2.2 定义，定义7.3：布朗运动、维纳过程 · Checked 2026-10-14

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