Wiener Process and White Noise

The Wiener process is the workhorse of stochastic modeling far beyond physics. Single-particle-tracking microscopy fits the mean-square-displacement law $\langle x^2(t)\rangle=2Dt$ to extract diffusion coefficients of membrane proteins and colloids; deviations from linearity flag subdiffusion in crowded cells or active transport. In electronics, Johnson–Nyquist thermal noise across a resistor has the flat spectral density $S_V=4k_BTR$ over a wide band — the white-noise signature — with the accumulated charge performing a Wiener random walk. And in quantitative finance, geometric Brownian motion driven by $\mathrm dW$ is the substrate of the Black–Scholes model. Numerically, every stochastic-differential-equation solver generates Wiener increments as $\Delta W=\sqrt{\Delta t}\,\mathcal N(0,1)$ — the $\sqrt{\Delta t}$, not $\Delta t$, being the crucial and frequently mishandled detail. The $(\mathrm dW)^2=\mathrm dt$ rule dictates the error structure of the Euler–Maruyama and Milstein schemes and is the reason higher-order integrators carry an explicit $\tfrac12 b\,b'[(\Delta W)^2-\Delta t]$ correction.

Take a symmetric random walker who steps a distance $\sqrt{\Delta t}$ every $\Delta t$ seconds. As $\Delta t\to0$ the trajectory approaches a continuous but infinitely jagged curve. The walker covers a net distance of order $\sqrt t$ in time $t$ — the diffusive signature, growing more slowly than the ballistic $t$ of a body with constant velocity — yet the instantaneous velocity is undefined, because each step shrinks like $\sqrt{\Delta t}$ while the time to take it shrinks faster, like $\Delta t$. The ratio blows up. This is the intuitive content of "continuous but nowhere differentiable": you can always get from here to there without lifting your pen, but you can never assign a well-defined direction of travel at any instant. The quadratic variation is the structural heart of the subject. Chop $[0,t]$ into $N$ pieces and add up the squared increments $\sum_i(\Delta W_i)^2$. Each term has mean $\Delta t_i$, so the sum has mean $t$; and because each term's variance is $2(\Delta t_i)^2$, the total variance is $\sum 2(\Delta t_i)^2 \sim 2t\,\Delta t \to 0$ as the mesh refines. The sum therefore converges to the deterministic number $t$ — even though the sum of the unsquared magnitudes $\sum|\Delta W_i|$ diverges. The "area of wiggle" is finite while the "arc length" is infinite. That finite, nonzero quadratic variation is precisely what ordinary calculus assumes to be zero, and it is why $(\mathrm dW)^2$ must be kept as $\mathrm dt$.

The second figure is the thesis. White noise is flat: $S_\xi(\omega)=1$ for all $\omega$, an idealization in which the fluctuating force decorrelates faster than any timescale in the problem. But a real Brownian particle has mass, and its velocity cannot respond instantly; its autocorrelation decays over a finite time $\tau_c=m/\gamma$, and its spectrum is a Lorentzian $S(\omega)\propto 1/(\omega^2+\tau_c^{-2})$ — flat below the cutoff $\omega\approx1/\tau_c$ and rolling off above it. The flat white plateau is what you get in the $\tau_c\to0$ limit; the roll-off is the fingerprint of inertia. (The Ornstein–Uhlenbeck process that carries this velocity is treated in its own article; we recall it here only by name, to locate white noise as its high-cutoff idealization.) Scaling is built in. Because the increments are Gaussian with variance equal to the interval, $W(ct)$ has exactly the same law as $\sqrt c\,W(t)$: the process is statistically self-similar, a fractal whose graph has Hausdorff dimension $3/2$. This self-similarity is why a single diffusion constant captures all of normal diffusion, and why "zooming out" a microscopic random walk reproduces the same universal curve.

The Wiener process is fixed by its first two moments together with the Gaussian, independent-increment structure. We state the covariance first, because everything else — the diffusion kernel, the white-noise correlation, the quadratic-variation rule — follows from differentiating or exponentiating it.

where: $W(t)$ — the standard Wiener process at time $t$; $\langle\cdot\rangle$ — ensemble average over noise realizations; $\min(t,s)$ — the smaller of the two times, the covariance kernel. Setting $s=t$ gives $\langle W(t)^2\rangle=t$: the variance grows linearly with elapsed time, the defining diffusive scaling. The increments are stationary Gaussians whose variance is the interval length, and disjoint increments are statistically independent. This is the property that makes $W$ a Markov process and the universal random-walk limit.

where: $\mathcal N(\mu,\sigma^2)$ — the normal distribution of mean $\mu$ and variance $\sigma^2$; $t-s$ — the length of the interval, equal to the increment variance. Independence of increments over non-overlapping windows completes the definition. The probability density of finding the walker at $x$ at time $t$ having started at $x_0$ is the Gaussian diffusion kernel — the fundamental solution of the diffusion equation, which the Wiener density satisfies. We keep a general diffusion constant $D$; the standard $W$ has $D=\tfrac12$.

where: $P$ — transition probability density; $D$ — diffusion coefficient; the width $\sqrt{2D(t-t_0)}$ gives $\langle(x-x_0)^2\rangle=2D(t-t_0)$. This density obeys $\partial_t P = D\,\partial_x^2 P$, the pure-diffusion Fokker–Planck equation (the drift-free case treated in the diffusion-equation article). White noise is the formal time-derivative of $W$. It is defined by its mean and autocorrelation, which follow from differentiating the covariance $(1)$ twice.

where: $\xi(t)$ — Gaussian white noise, the distributional derivative of $W$; $\delta(t-s)$ — the Dirac delta enforcing zero correlation at distinct times. Because $\xi$ has infinite variance at coincident times, it exists only as a generalized process, never as an ordinary function. The Wiener–Khinchin theorem states that the power spectral density is the Fourier transform of the autocorrelation. Applied to the delta correlation of $(4)$, it produces the flat "white" spectrum — the punchline of the article.

where: $S_\xi(\omega)$ — the power spectral density at angular frequency $\omega$. A flat spectrum means every frequency carries equal power; the sifting property of the delta makes the integral exactly $1$, independent of $\omega$. Finally, the quadratic-variation rule and its immediate corollary, Itô's lemma, encode the roughness of the paths. These are stated here as the bridge to the stochastic calculus; their full development belongs to the Itô-versus-Stratonovich article, which we forward to by name.

where: $\mathrm{d}W$ — the infinitesimal Wiener increment over $\mathrm{d}t$, Gaussian with variance $\mathrm{d}t$. The first identity is the statement that quadratic variation equals elapsed time.

Because $W(ct)\overset d=\sqrt c\,W(t)$, the process is statistically self-similar under the diffusive rescaling $x\to\sqrt c\,x,\ t\to ct$.

where: $\overset d=$ — equality in law (same distribution, not the same realization); $c>0$ — the scale factor. This is the fixed-point structure that makes the Wiener process the universal attractor of finite-variance random walks under coarse-graining.

The Wiener process is an exactly solvable corner of the space of stochastic processes; the neighboring models are reached by adding structure to it.

Example 1 — The flat spectrum, and where it stops being flat. Compute the power spectrum of ideal white noise and compare it to a physical Ornstein–Uhlenbeck velocity with correlation time $\tau_c$, then find the frequency where the physical spectrum has fallen to half its low-frequency value.

Step 1. For white noise with $\langle\xi(t)\xi(0)\rangle=\delta(t)$, Wiener–Khinchin gives $S_\xi(\omega)=\int\delta(\tau)e^{-i\omega\tau}\mathrm d\tau=1$ for every $\omega$ — a genuinely flat spectrum with no cutoff.

Step 2. For the OU velocity with $C(\tau)=\sigma^2 e^{-|\tau|/\tau_c}$, the transform is $S(\omega)=2\sigma^2\tau_c/(1+\omega^2\tau_c^2)$. The zero-frequency value is $S(0)=2\sigma^2\tau_c$.

Step 3. The half-power point solves $1+\omega^2\tau_c^2=2$, i.e. $\omega_{1/2}=1/\tau_c$. Taking a colloidal bead with $\tau_c=m/\gamma\approx 10^{-8}\,\text{s}$, the flat band extends only up to $\omega_{1/2}=10^{8}\,\text{rad/s}$ ($\approx16\,\text{MHz}$); above that the spectrum rolls off as $\omega^{-2}$.

Result: white noise is flat at all $\omega$; a real inertial velocity is flat only below the cutoff $\omega_{1/2}=1/\tau_c$, here about $\mathbf{10^{8}\,rad/s}$. The flat plateau is the white idealization; the roll-off is inertia.

Example 2 — Quadratic variation from a coarse partition. Verify numerically that the sum of squared increments approaches the elapsed time while the sum of absolute increments grows without bound as the mesh refines.

Step 1. Over $[0,1]$ with $N$ equal steps, each increment has $\langle(\Delta W_i)^2\rangle=\Delta t=1/N$, so $\big\langle\sum_i(\Delta W_i)^2\big\rangle=N\cdot(1/N)=1$, independent of $N$.

Step 2. The variance of the sum is $\sum_i 2(\Delta t)^2=N\cdot 2/N^2=2/N\to0$: the squared-increment sum concentrates on its mean of $1$ as $N\to\infty$. This is the deterministic quadratic variation.

Step 3. The absolute increments behave differently: $\langle|\Delta W_i|\rangle=\sqrt{2\Delta t/\pi}=\sqrt{2/(\pi N)}$, so $\sum_i\langle|\Delta W_i|\rangle=N\cdot\sqrt{2/(\pi N)}=\sqrt{2N/\pi}\to\infty$. The path has infinite total length.

Result: the quadratic variation converges to the elapsed time $\mathbf{t=1}$ (variance $\to0$) while the total variation diverges as $\mathbf{\sqrt{2N/\pi}}$ — finite wiggle-area, infinite arc-length, the mathematical face of $(\mathrm dW)^2=\mathrm dt$.

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 15 (Fluctuations and stochastic processes) · Kardar, Statistical Physics of Particles, Ch. 9 · Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, Ch. 2 & 10 · Risken, The Fokker–Planck Equation, 2nd ed., Ch. 3 · Gardiner, Handbook of Stochastic Methods, Ch. 3–4 · van Kampen, Stochastic Processes in Physics and Chemistry, Ch. VIII.

Provenance. The defining axioms, covariance $\min(t,s)$, and diffusion kernel follow Gardiner Ch. 3–4 and Risken Ch. 3; the white-noise delta correlation, flat spectrum, and $(\mathrm dW)^2=\mathrm dt$ rule are standard (van Kampen Ch. VIII, Kardar Ch. 9). A verifier recomputes: (i) $\partial_t\partial_s\min(t,s)=\delta(t-s)$ and the flat spectrum $S_\xi=1$; (ii) the OU-autocorrelation-to-Lorentzian transform $2\tau_c/(1+\omega^2\tau_c^2)$ and its $\tau_c\to0$ white limit; (iii) the Gaussian kernel solving $\partial_tP=D\partial_x^2P$ with $\langle x^2\rangle=2Dt$; (iv) the Donsker rescaling giving variance $t$; (v) the quadratic vs. total variation scaling.

Builds on: Brownian Motion · Gaussian distributions and the central limit theorem · Fourier analysis and the Dirac delta. Leads to: The Langevin Equation · Stochastic Calculus: Itô versus Stratonovich · The Ornstein–Uhlenbeck Process · The Fokker–Planck Equation · The Diffusion Equation.