A random walk unfolds in discrete time: step one, step two, step three, and so on. If both the time interval and the spatial displacement of each step are made progressively finer, under an appropriate scaling the discrete random walk converges to one of the most important continuous models in stochastic-process theory: Brownian motion, also called the Wiener process.

1. The Definition of Brownian Motion

1.1 Starting from the random walk’s scaling law

The simple random walk Sn=ϵ1+⋯+ϵnS_n=\epsilon_1+\cdots+\epsilon_n (ϵi\epsilon_i independent, taking ±1\pm1 with equal probability) satisfies the central limit theorem Sn/n⇒N(0,1)S_n/\sqrt n\Rightarrow N(0,1): when the time scale grows by a factor of nn, the spatial scale only needs to grow by a factor of n\sqrt n to maintain a stable distribution — this already hints at Brownian motion’s scaling law.

1.2 Brownian motion’s four properties

Standard Brownian motion {Bt}t≥0\{B_t\}_{t\ge0} satisfies: B0=0B_0=0; it has independent increments; for any 0≤s<t0\le s<t, Bt−Bs∼N(0,t−s)B_t-B_s\sim N(0,t-s); and its sample path is almost surely continuous. These four properties almost completely characterize standard Brownian motion.

1.3 Why the variance is t−st-s: worked out

Over a time interval of length Δt\Delta t, the standard deviation of the displacement is Δt\sqrt{\Delta t}, so Brownian motion retains the random walk’s t\sqrt t diffusion scale: the mean displacement is 0, but the range of fluctuation grows with the square root of time. Take t=4t=4: B4∼N(0,4)B_4\sim N(0,4), with standard deviation 2; the probability that the displacement exceeds 3 (that is, 1.5 standard deviations) is P(∣B4∣>3)=2(1−Φ(1.5))≈0.1336P(|B_4|>3)=2(1-\Phi(1.5))\approx0.1336, about 13.4%.

1.4 The reflection principle: how close the maximum’s distribution is to the endpoint’s

Let Mt=max⁡0≤s≤tBsM_t=\max_{0\le s\le t}B_s be the highest point Brownian motion reaches on [0,t][0,t]. Because the sample path is continuous, once BsB_s touches some height a>0a>0, “reflecting” the path from that moment onward (flipping the direction of subsequent ups and downs) is still a valid Brownian motion path with exactly the same probability — this is the continuous-time version of the random walk’s reflection principle. This gives us

P(Mt≥a)=2P(Bt≥a),a>0.P(M_t\ge a)=2P(B_t\ge a),\qquad a>0.

Intuitively: after the path has touched aa, the two ways it can continue — the original path going up, or the reflected version going down — are exactly equally likely, where “original” corresponds to the endpoint still being ≥a\ge a, and “reflected” corresponds to the endpoint being reflected to ≤a\le a; the two have equal probability, and together they make up the full probability of the event “touched aa.” Verify this using the same numbers as Section 1.3: t=4t=4, a=3a=3, so P(M4≥3)=2P(B4≥3)=2(1−Φ(1.5))≈0.1336P(M_4\ge3)=2P(B_4\ge3)=2\big(1-\Phi(1.5)\big)\approx0.1336 — exactly the same as the P(∣B4∣>3)≈0.1336P(|B_4|>3)\approx0.1336 computed in Section 1.3. This is no coincidence: P(∣Bt∣≥a)=P(Bt≥a)+P(Bt≤−a)=2P(Bt≥a)P(|B_t|\ge a)=P(B_t\ge a)+P(B_t\le-a)=2P(B_t\ge a) (by symmetry), so for standard Brownian motion, “the whole path once surged above aa” and “the endpoint alone lands outside ±a\pm a” turn out to have exactly the same probability.

2. Path and Increment Properties

2.1 Continuous but not differentiable

One of Brownian motion’s most counterintuitive properties is that its sample path is continuous, yet almost nowhere differentiable. It doesn’t suddenly break like a jump process, yet it’s full of violent fluctuation at any arbitrarily small time scale, so the ordinary notion of velocity, lim⁡Δt→0(Bt+Δt−Bt)/Δt\lim_{\Delta t\to0}(B_{t+\Delta t}-B_t)/\Delta t, generally doesn’t exist. This is also the fundamental reason stochastic calculus can’t simply reuse the rules of ordinary calculus.

2.2 Independent and stationary increments, yet not a stationary process

Brownian motion has independent increments (changes over non-overlapping intervals are independent of each other) and stationary increments (Bt+h−Bt∼N(0,h)B_{t+h}-B_t\sim N(0,h) depends only on the length hh, not the starting point). But Brownian motion itself is not a stationary process: Var⁡(Bt)=t\operatorname{Var}(B_t)=t keeps growing over time. Stationary increments say the distribution of “change” doesn’t vary with time, not that the distribution of “absolute position” itself stays unchanged — these are two entirely different properties, easily conflated.

2.3 The scaling property

Brownian motion satisfies a beautiful scaling property: {Bct}=d{c Bt}\{B_{ct}\}\overset{d}{=}\{\sqrt c\,B_t\} — scaling time by a factor of cc is equivalent to scaling space by a factor of c\sqrt c, exactly the continuous-time version of the random walk’s n\sqrt n scale.

3. The Continuous Limit and Diffusion

3.1 From random walk to Brownian motion

It’s not just the endpoint Sn/nS_n/\sqrt n that tends toward a Gaussian distribution — if the entire random walk trajectory is rescaled in both time and space, the whole process converges to Brownian motion. This is the functional central limit theorem, often stated as Donsker’s theorem: Brownian motion can be understood as the limiting process obtained by continuously accumulating a large number of small, independent random perturbations, and is the final destination of the random walk → central limit theorem → Brownian motion storyline.

3.2 Brownian motion and the diffusion equation

Brownian motion originally came from a physical description of random particle motion: the direction of a huge number of microscopic collisions is unpredictable, yet a stable diffusion law emerges at the macroscopic scale. The equation its probability density satisfies is closely related to the heat equation, ∂p/∂t=12∂2p/∂x2\partial p/\partial t=\frac12\partial^2p/\partial x^2, establishing a very deep connection between stochastic processes and partial differential equations — this is also why probability theory and analytical tools can be translated into each other when later studying stochastic differential equations.

4. Connections to Other Processes

4.1 Brownian motion is also a Gaussian process, worked out

For any finite collection of time points t1,…,tnt_1,\dots,t_n, the random vector (Bt1,…,Btn)(B_{t_1},\dots,B_{t_n}) follows a multivariate Gaussian distribution, so Brownian motion is also a Gaussian process, with covariance function Cov⁡(Bs,Bt)=min⁡(s,t)\operatorname{Cov}(B_s,B_t)=\min(s,t). Take s=2s=2, t=7t=7: Cov⁡(B2,B7)=min⁡(2,7)=2\operatorname{Cov}(B_2,B_7)=\min(2,7)=2, giving a correlation coefficient of 2/2×7=2/14≈0.5352/\sqrt{2\times7}=2/\sqrt{14}\approx0.535. The farther apart two time points are (t−st-s larger, with the smaller value ss held fixed), the more the correlation tends to drop, because the ratio of min⁡(s,t)\min(s,t) to st\sqrt{st} shrinks.

4.2 Why it’s so important

Brownian motion is the basic source of noise for many continuous stochastic models. Stochastic differential equations later on are usually written dXt=μ(Xt,t) dt+σ(Xt,t) dBtdX_t=\mu(X_t,t)\,dt+\sigma(X_t,t)\,dB_t, where dBtdB_t comes from Brownian motion — widely used in diffusion modeling, financial mathematics, statistical physics, stochastic control, and Bayesian stochastic modeling.

5. Summary

Brownian motion is the bridge from discrete random walks into continuous stochastic processes: a random walk, via the central limit theorem, converges under appropriate scaling to Brownian motion, and Brownian motion is in turn the source of the noise term in stochastic differential equations. It is simultaneously a process with independent increments, a nonstationary process (since its variance grows over time), and a Gaussian process — these three identities together make Brownian motion the most fundamental and most frequently cited building block in continuous-time stochastic-process theory.