Lévy's modulus of continuity theorem¶

Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10381
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Probability Theory, Stochastic Processes → Mathematics

Core Idea¶

Lévy's modulus of continuity theorem is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion. Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as.

Scope of Application¶

Documented setting. Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used.
Statement of the result. Let B : [0, 1] \times \Omega \to \mathbb{R} be a standard Wiener process.
Statement of the result. \lim{h \to 0} \sup{t, t'\leq 1; |t-t'|\leq h } \frac {\sqrt{2 h \log (1 / h)}} = 1.
Statement of the result. In other words, the sample paths of Brownian motion have modulus of continuity.
Statement of the result. \omega{B} (\delta) = c\sqrt{2 \delta \log (1 / \delta)}.

Clarity¶

A clear use of Lévy's modulus of continuity theorem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion.

Manages Complexity¶

Lévy's modulus of continuity theorem compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion.—and the practical consequence—with probability one, for.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion.
Check operation and conditions. \lim{h \to 0} \sup{t, t'\leq 1; |t-t'|\leq h } \frac {\sqrt{2 h \log (1 / h)}} = 1. 4.

Knowledge Transfer¶

Within the home domain. Knowledge about Lévy's modulus of continuity theorem transfers literally when a new case preserves the same carrier type, relation, and recognition test. Lévy's modulus of continuity theorem is a theorem that gives a result about an almost sure behaviour of an estimate of the modulus of continuity for Wiener process, that is used to model what's known as Brownian motion. Let B : [0, 1] \times \Omega \to \mathbb{R} be a standard Wiener process. Beyond the home domain. No canonical parent is asserted for Lévy's modulus of continuity theorem.

Neighborhood in Abstraction Space¶

Lévy's modulus of continuity theorem sits in a sparse region of the domain-specific corpus (70^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Named Analytic Theorems & Operators (39 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08