--- title: "Brownian Motion and Its Defining Properties" description: "Independent normal increments, continuous paths, and no derivative anywhere. It is the scaling limit of a random walk, and it is the process most continuous-time models are written against." canonical: https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z source: https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Brownian Motion and Its Defining Properties Independent normal increments, continuous paths, and no derivative anywhere. It is the scaling limit of a random walk, and it is the process most continuous-time models are written against. Subject: Mathematics · Area: Probability · Ages 23 to 24 Page: https://lightmysky.com/learn/mathematics/brownian-motion-and-its-defining-properties-mt_T5KtWIk07Z ## Ready when they can - State the defining properties and compute a covariance from them - Describe the scaling that turns a random walk into this process - Say why the paths are continuous and yet nowhere differentiable ## Lesson: A random motion with normal steps Standard Brownian motion starts pinned at the origin, so B of 0 is 0. Its increments over disjoint intervals are independent and normal, with variance equal to elapsed time. Over 4 units of time the variance is 4, since unit variances add: position at a fixed time is a familiar bell curve. **Example.** Covariances come from overlap: the covariance of B of s and B of t is the smaller of s and t. So the covariance of B of 5 and B of 7 is 5, and the covariance of B of 5 minus B of 2 with B of 7 is 5 minus 2, which is 3. Rescaling works the same way: B of 9 behaves like 3 times a standard normal. The process is the scaling limit of a random walk: many small independent pieces turn normal, and whole walks settle into Brownian paths. That upgrade of the central limit idea is called Donsker in textbooks. Sample means settling down is the one dimensional preview of walks settling into motion. **Tip.** The paths are continuous everywhere yet differentiable nowhere, which is what scaling implies about roughness. Zoom in and the jaggedness never smooths out. When a model assumes smooth velocity for this motion, that assumption is the first thing to distrust. **Recap.** Brownian motion starts at zero, spreads with variance equal to time, arises from scaled walks, and stays rough at every zoom. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [Martingales and Optional Stopping](https://lightmysky.com/learn/mathematics/martingales-and-optional-stopping-mt_198uXX3jRJ) - [The Law of Large Numbers and the Central Limit Theorem](https://lightmysky.com/learn/mathematics/the-law-of-large-numbers-and-the-central-limit-theorem-mt_NknG3L5rQp) - [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL) - [Joint Distributions, Covariance and Independence](https://lightmysky.com/learn/mathematics/joint-distributions-covariance-and-independence-mt_YyTFTAqhSF) ## Opens up - [Fluctuations, Linear Response and the Fluctuation-Dissipation Relation](https://lightmysky.com/learn/science/fluctuations-linear-response-and-the-fluctuation-dissipation-relation-mt_Ibsvdyx7uP) - [Diffusion Models: Learning to Undo Noise](https://lightmysky.com/learn/computing/diffusion-models-learning-to-undo-noise-mt_kZdQJ96iZ3)