Brownian Motion

A New Kind of Calculus

This page opens a new track, devoted to the mathematics of randomness that evolves in continuous time. Ordinary calculus describes quantities whose change is governed by a rate. Given an initial condition, a well-posed differential equation determines the entire future. Many systems are not like this. A microscopic particle suspended in water is buffeted by molecular collisions, a transmitted signal drifts under noise, and an image dissolves into static. In such systems the change over each instant carries an irreducible random component. The object of study is then a stochastic process, and its law is a probability distribution over entire paths rather than over single values.

The track follows a single arc. This page constructs its central object, Brownian motion, the mathematical model of pure noise accumulating in time. We then establish how its paths behave. They turn out to be so irregular that ordinary calculus fails on them. The failure is precise and quantifiable, and it dictates the shape of the remedy. The pages that follow build that remedy. First comes an integral against Brownian motion with its own chain rule, the Itô calculus. Differential equations driven by noise come next, and with them the equations that govern how probability densities evolve under such dynamics.

Two earlier threads of the curriculum meet here. In the analysis track, the study of the heat equation showed that its fundamental solution is, after a constant rescaling of time, the density of a one-dimensional Brownian motion started at the origin. The connection was left there as a bridge to be crossed in either direction. This track is the crossing from the probabilistic side. On the machine-learning side, flow matching manipulates a process \(\mathbf{w}_t\) and its increments \(d\mathbf{w}_t\) as given, while diffusion models invoke Brownian motion as the noise driving the stochastic differential equation that their forward chain becomes in continuous time. Both defer the construction of these objects. That debt begins to be repaid on this page and is settled over the course of the track.

Constructing a Process from its Finite-Dimensional Distributions

We have built the measure-theoretic apparatus of probability. A random variable is a measurable map on a probability space, its law is a pushforward measure, and the independence of random variables is the independence of the \(\sigma\)-algebras they generate. All of this was static, a fixed, finite (or countable) collection of random quantities on one space. We now confront a new object, a family of random variables \(\{\mathbf{X}_t\}_{t \geq 0}\) indexed by a continuum of times, whose defining feature is not any single marginal law but the way its values at different times relate to one another.

A stochastic process with values in \(\mathbb{R}^n\) is such a family \(\{\mathbf{X}_t\}_{t \in T}\) of random variables on a common probability space \((\Omega, \mathcal{F}, \mathbb{P})\), where the index set \(T\) is here the half-line \([0, \infty)\). There are two complementary ways to look at it. Fixing \(t\), the map \(\omega \mapsto \mathbf{X}_t(\omega)\) is an ordinary random variable, a snapshot. Fixing \(\omega\) instead, the map \[ t \mapsto \mathbf{X}_t(\omega) \] is a single function of time, called a path (or a sample path) of the process. The intuition to keep in mind is that \(t\) is time and each \(\omega\) is one "experiment" or "particle". The value \(\mathbf{X}_t(\omega)\) records the state of that particle at time \(t\).

How does one build a process indexed by an uncountable set? We cannot list its values, and it is far from clear how to specify a measure on the space of all paths directly. What we can specify are the finite-dimensional distributions. For every finite tuple of distinct times \(t_1, \ldots, t_k\), we prescribe a probability measure \(\nu_{t_1, \ldots, t_k}\) on \(\mathbb{R}^{nk}\) intended to be the joint law of \((\mathbf{X}_{t_1}, \ldots, \mathbf{X}_{t_k})\). This is a dramatic reduction. The law of an object over a continuum of times, on the events decided by countably many of its values, is pinned down by a consistent system of finite-dimensional data. The word "consistent" is doing real work, and two conditions make it precise.

For a family \(\{\nu_{t_1, \ldots, t_k}\}\) to be the finite-dimensional distributions of an actual process, it must be internally coherent under the two operations that relate different finite tuples: permuting the times, and forgetting some of them. These are the Kolmogorov consistency conditions. Writing \(\sigma\) for a permutation of \(\{1, \ldots, k\}\), the first demands that reordering the times relabels the coordinate sets consistently, \[ \begin{align*} \nu_{t_{\sigma(1)}, \ldots, t_{\sigma(k)}}&(F_1 \times \cdots \times F_k) \\\\ &= \nu_{t_1, \ldots, t_k}\bigl(F_{\sigma^{-1}(1)} \times \cdots \times F_{\sigma^{-1}(k)}\bigr), \end{align*} \] and the second demands that appending extra times and then integrating them out over all of \(\mathbb{R}^n\) recovers the lower-dimensional law, \[ \begin{align*} \nu_{t_1, \ldots, t_k}&(F_1 \times \cdots \times F_k) \\\\ &= \nu_{t_1, \ldots, t_{k+m}}\bigl(F_1 \times \cdots \times F_k \times \underbrace{\mathbb{R}^n \times \cdots \times \mathbb{R}^n}_{m}\bigr). \end{align*} \] These are exactly the compatibilities that any genuine joint law automatically satisfies. The content of the following theorem is that they are also sufficient.

The proof of this theorem applies the Carathéodory extension theorem that produced product measures to the set function on cylinder sets described above. The subtlety is that the index set is now infinite. The construction belongs to the measure-theoretic foundations of probability, so we state it here without proof. What matters is the license it grants. To build a process, it suffices to write down a consistent system of finite-dimensional laws. The infinite product measure that underwrote i.i.d. sequences in the laws of large numbers was one special case. Brownian motion is another.

The Wiener Process

In 1827 the Scottish botanist Robert Brown observed that particles released from pollen grains suspended in water executed an irregular motion, later understood as the aggregate effect of molecular collisions. To model the displacement of such a particle we want a process \(\{\mathbf{w}_t\}_{t \geq 0}\) in \(\mathbb{R}^n\) whose increments are random, independent over disjoint time intervals, and statistically homogeneous in time. We now turn these physical desiderata into three axioms and then verify, via Kolmogorov's extension theorem and a continuity criterion, that a process meeting them exists.

The first axiom fixes the marginal and joint laws. Recall that a Gaussian process is a collection of random variables, any finite subcollection of which has a joint multivariate normal distribution, here allowed to be degenerate. Its covariance matrix need only be positive semidefinite, and the singular case is one the linked definition of the multivariate normal sets aside.

For a process \(\{\mathbf{w}_t\}\) with values in \(\mathbb{R}^n\), the Gaussian-process condition is applied to the scalar coordinates, so that for any times \(t_1, \ldots, t_k\) the vector \((\mathbf{w}_{t_1}, \ldots, \mathbf{w}_{t_k})\) is jointly Gaussian on \(\mathbb{R}^{nk}\). Likewise, the independence of random vectors in \(\mathbb{R}^n\) is defined by the same product formula as the independence of random variables, with Borel sets of \(\mathbb{R}^n\) in place of Borel sets of \(\mathbb{R}\). Brownian motion is the Gaussian process whose index set is time and whose mean and covariance are the simplest ones compatible with independent, time-homogeneous increments starting from a point.

Existence.
Axioms (W1)-(W2) determine every finite-dimensional distribution explicitly. The vector \((\mathbf{w}_{t_1}, \ldots, \mathbf{w}_{t_k})\) is the multivariate normal with mean \((x, \ldots, x)\) and covariance block matrix \([\min(t_i, t_j)\, I_n]_{i,j}\). This is well defined because \(\min(s, t)\) is a Mercer kernel. From the identity \[ \min(s, t) = \int_0^\infty \mathbf{1}_{[0,s]}(u)\, \mathbf{1}_{[0,t]}(u)\, du , \] it follows that \[ \begin{align*} \sum_{i,j=1}^{k} c_i c_j \min(t_i, t_j) &= \int_0^\infty \Bigl(\sum_{i=1}^{k} c_i\, \mathbf{1}_{[0,t_i]}(u)\Bigr)^{2} du \\\\ &\geq 0 , \end{align*} \] so every matrix \([\min(t_i, t_j)]_{i,j}\) is positive semidefinite. For distinct times \(0 \lt t_1 \lt \cdots \lt t_k\) it is even positive definite. If the integral vanishes, the step function \(\sum_i c_i\, \mathbf{1}_{[0,t_i]}\) is zero almost everywhere, and reading its values on \((t_{k-1}, t_k)\), then on \((t_{k-2}, t_{k-1})\), and so on downward (with \(t_0 = 0\)), forces every \(c_i = 0\). Applied to each of the \(n\) coordinates separately, the same computation shows that the block matrix \([\min(t_i, t_j)\, I_n]_{i,j}\) is positive definite, so the prescribed law has a genuine multivariate normal density. A tuple that contains the time \(0\) adjoins the constant coordinate \(x\) to such a vector.

One checks directly that this family of Gaussian laws satisfies the two consistency conditions. Permutation invariance is symmetry of the covariance matrix under simultaneous row/column permutation, and marginalization is the fact that a coordinate projection of a Gaussian vector is again Gaussian with the corresponding sub-mean and sub-covariance. This is an instance of the closure of the Gaussian family under linear maps, which we take for granted here and use throughout the page. Kolmogorov's extension theorem then furnishes a process with precisely these finite-dimensional distributions. This settles (W1)-(W2).

The continuity axiom (W3) is of a different nature and cannot come from the finite-dimensional data alone, as we now explain.

Two processes \(\{\mathbf{X}_t\}\) and \(\{\mathbf{Y}_t\}\) on the same probability space are called versions (or modifications) of one another if \(\mathbb{P}[\mathbf{X}_t = \mathbf{Y}_t] = 1\) for every fixed \(t\). Versions share all finite-dimensional distributions, yet, because the exceptional null set is allowed to depend on \(t\) and there are uncountably many \(t\), their sample paths may differ radically. The continuity of Brownian motion is the statement that among the versions of any process with the Gaussian finite-dimensional distributions above, there is one with continuous paths. Such a version is delivered by a general regularity criterion that controls path oscillation through a single moment estimate.

We take the criterion as given. Its proof is a dyadic chaining argument that bounds the maximal oscillation of the path over successively finer partitions. It belongs to the measure-theoretic development of continuous-time processes rather than to the analytic thread we are following.

To apply it to Brownian motion we need only a single moment bound on the increments. For \(s \neq t\) the increment \(\mathbf{w}_t - \mathbf{w}_s\) is, by (W1)-(W2), a centered Gaussian in \(\mathbb{R}^n\) with covariance \(|t - s|\, I_n\), and its fourth moment reduces to a short computation. The coordinates \(Z_i\) of \(\mathbf{Z} = \mathbf{w}_t - \mathbf{w}_s\) are uncorrelated components of a jointly Gaussian vector, hence independent \(\mathcal{N}(0, |t-s|)\) variables. Like the closure of the Gaussian family under linear maps, this fact about Gaussian vectors, already invoked in reading the covariance, is taken for granted throughout the page. Part (b) of the theorem on the moments of the centered normal distribution gives \(\mathbb{E}[Z_i^2] = |t-s|\) and \(\mathbb{E}[Z_i^4] = 3\,|t-s|^2\). For \(i \neq j\) the squares \(Z_i^2\) and \(Z_j^2\) are independent, since independence is preserved by Borel functions. Both squares have finite expectation, so the product rule for expectations gives \(\mathbb{E}[Z_i^2 Z_j^2] = \mathbb{E}[Z_i^2]\,\mathbb{E}[Z_j^2]\). Expanding \(\|\mathbf{Z}\|^4 = \bigl(\sum_i Z_i^2\bigr)^2\) yields \[ \begin{align*} \mathbb{E}\bigl[\|\mathbf{w}_t - \mathbf{w}_s\|^{4}\bigr] &= \sum_{i} \mathbb{E}[Z_i^4] + \sum_{i \neq j} \mathbb{E}[Z_i^2]\, \mathbb{E}[Z_j^2] \\\\ &= 3n\, |t-s|^2 + n(n-1)\, |t-s|^2 \\\\ &= n(n + 2)\, |t - s|^{2} . \end{align*} \] For \(s = t\) the increment is zero, so both sides of the identity vanish and it holds for all \(s, t\). Thus the hypothesis holds, with constants independent of \(T\), for \(\alpha = 4\), \(D = n(n+2)\), and \(\beta = 1\). Brownian motion therefore has a continuous version, and we henceforth work with it. This is the content of axiom (W3).

Path Regularity and Irregularity

With the continuous version secured, we now examine how rough those continuous paths are. Almost surely, Brownian paths are continuous everywhere but differentiable nowhere, and their variation over any interval is infinite. We specialize to one dimension, where these phenomena already appear in full and the notation stays light. The precise carrier of this roughness is the quadratic variation, which, unlike the ordinary (first-order) variation, is finite, deterministic, and equal to the elapsed time.

Fix a time \(t \gt 0\) and consider a partition \(\pi = \{0 = s_0 \lt s_1 \lt \cdots \lt s_N = t\}\) of \([0, t]\), with mesh \(\|\pi\| = \max_i (s_i - s_{i-1})\). For the one-dimensional standard Brownian motion, the quadratic variation along \(\pi\) is the random sum \[ Q_\pi = \sum_{i=1}^{N} \bigl(w_{s_i} - w_{s_{i-1}}\bigr)^2 . \] Contrast this with the first-order sum \(\sum_i |w_{s_i} - w_{s_{i-1}}|\). Taking its supremum over partitions leads to a classical notion from real analysis.

For a continuously differentiable function the quadratic sum vanishes as \(\|\pi\| \to 0\) (the squares are second order) while the first-order sums converge to the total variation \(\int_a^b |f'(s)|\, ds\). For Brownian motion the roles invert.

Part (i) admits a suggestive shorthand. The accumulated squared increments of Brownian motion over \([0, t]\) converge in \(L^2\) to \(t\), as if each infinitesimal increment contributed its own duration. We record this as the informal identity \[ (dw_t)^2 = dt, \] to be understood strictly through the \(L^2\) limit of part (i), never as a pathwise statement. This is the single algebraic rule that, carried through a second-order Taylor expansion, will later distinguish the calculus of Brownian motion from ordinary calculus. The term that would be negligible for a smooth path is here of the same order as \(dt\) and cannot be discarded.

From Pathwise Failure to an L² Theory

We wanted to model the displacement of a particle buffeted by noise as a differential relation \(d\mathbf{x}_t = \boldsymbol{\mu}_t\, dt + \sigma_t\, d\mathbf{w}_t\), and to give meaning to the accumulated noise \(\int_0^t \sigma_s\, d\mathbf{w}_s\). The results of this page show that the obvious route is closed. A pathwise Stieltjes integral \(\int \sigma_s\, d\mathbf{w}_s\) would exist if either the integrand or the integrator had bounded variation on each path (the other factor being continuous). But the integrands we need typically depend on the same driving noise \(\mathbf{w}\), and so are themselves as rough as \(\mathbf{w}\), while the unbounded variation of Brownian motion established above, applied to any coordinate of \(\mathbf{w}\), removes the other possibility. With neither factor of bounded variation, the classical existence criterion no longer applies, and the failure is genuine. Already for the most basic integrand, the path itself in one dimension, the Riemann-Stieltjes sums for \(\int_0^t w_s\, dw_s\) evaluated at the right and at the left endpoint of each subinterval differ by exactly \(Q_\pi\). Suppose both sums converged to a common limit on an event \(B\) of positive probability. Then \(Q_\pi \to 0\) on \(B\), and the Fatou argument in the proof of (ii), with \(B\) in place of \(A\) and \(t\) in place of \(b - a\), would force \(t^2\, \mathbb{P}(B) = 0\), a contradiction. The pathwise integral therefore fails on almost every sample path, not merely in exceptional cases.

The escape route is dictated by the one piece of good news we found. While the first-order variation diverges, the quadratic variation converges, and it does so in \(L^2\). This points away from pathwise integration and toward a construction in the Hilbert space \(L^2\) of the underlying probability space, where "convergence" means convergence in mean square.

The plan, to be carried out next, is to define the integral first for simple (piecewise-constant) integrands, where no analysis is needed, and then to extend to general integrands by continuity. Such an extension depends on one property, the completeness of \(L^2\), which ensures that Cauchy sequences of approximate integrals actually converge. Consider simple integrands whose square-integrable value on each subinterval depends only on the path up to the left endpoint of that subinterval. On simple integrands of this kind, the variance identity \(\mathbb{E}[(w_t - w_s)^2] = t - s\) of one-dimensional increments, combined with the independence of each increment from the path before it, makes the integral an isometry into \(L^2\). Completeness then guarantees that the isometric extension has somewhere to land.