Brownian motion is the continuous process whose increments are independent Gaussian fluctuations with variance equal to elapsed time. Its construction illustrates the two-stage method for processes: first prescribe consistent finite-dimensional laws, then prove that a regular modification exists.
Equivalent defining data
Definition. A standard Brownian motion is a real process \(B=(B_t)_{t\ge0}\) satisfying:
- \(B_0=0\) almost surely;
- increments over pairwise disjoint time intervals are independent;
- \(B_t-B_s\sim N(0,t-s)\) for \(0\le s<t\);
- sample paths are continuous almost surely.
Proposition. A centered continuous Gaussian process with covariance \[ \mathbb E[B_sB_t]=s\wedge t \] is Brownian motion.
Proof. The increment variance is \[ \mathbb E(B_t-B_s)^2=t+s-2s=t-s. \] For disjoint intervals \([s,t]\) and \([u,v]\) with \(t\le u\), \[ \operatorname{Cov}(B_t-B_s,B_v-B_u) =(t\wedge v-t\wedge u)-(s\wedge v-s\wedge u)=0. \] Any finite collection of increments is jointly Gaussian. Zero cross-covariances therefore imply independence.
Existence of Wiener measure
Theorem. There is a unique probability measure \(W\) on \[ C_0([0,\infty)) =\{\omega\in C([0,\infty)):\omega(0)=0\} \] such that the coordinate process \(B_t(\omega)=\omega(t)\) is Brownian motion. This probability is Wiener measure.
Proof. On the raw product space \(\mathbb R^{[0,\infty)}\), prescribe centered Gaussian finite-dimensional laws with covariance \(K(s,t)=s\wedge t\). The kernel is positive semidefinite because \[ \sum_{i,j}a_ia_j(t_i\wedge t_j) = \int_0^\infty \left(\sum_ia_i1_{[0,t_i]}(u)\right)^2\mathrm{d} u\ge0. \] Kolmogorov extension gives a coordinate process. The preceding proposition gives independent stationary Gaussian increments. For \(p\ge2\), \[ \mathbb E|B_t-B_s|^p =\mathbb E|Z|^p\,|t-s|^{p/2}, \qquad Z\sim N(0,1). \] Taking \(p>2\), Kolmogorov-Chentsov produces a continuous modification. Its law is supported on \(C_0([0,\infty))\), with that space equipped by uniform convergence on compact intervals. Uniqueness follows because coordinate evaluations generate the Borel sigma-field and the finite-dimensional laws agree.
Independent increment factorization
Proposition. For \(0=t_0<t_1<\cdots<t_n\), the joint density of \((B_{t_1},\ldots,B_{t_n})\) can be written through increments as \[ \prod_{k=1}^n p_{t_k-t_{k-1}}(x_k-x_{k-1}), \qquad p_t(x)=\frac1{\sqrt{2\pi t}}e^{-x^2/(2t)}, \quad x_0=0. \]
Proof. The linear map from positions \((x_1,\ldots,x_n)\) to increments \((x_1,x_2-x_1,\ldots,x_n-x_{n-1})\) has determinant one. The increments are independent with the displayed Gaussian densities.
Symmetries
Theorem (Brownian symmetries). If \(B\) is Brownian motion, then:
- for \(c>0\), \[ \widetilde B_t=c^{-1/2}B_{ct} \] is Brownian motion;
- for fixed \(s\ge0\), \[ \widetilde B_t=B_{s+t}-B_s \] is Brownian motion independent of \(\mathcal F_s\);
- for fixed \(T>0\), \[ \widetilde B_t=B_T-B_{T-t},\qquad0\le t\le T, \] has the law of Brownian motion on \([0,T]\);
- the time-inverted process \[ \widetilde B_t=tB_{1/t},\quad t>0,\qquad \widetilde B_0=0, \] is Brownian motion.
Proof. Each process is centered Gaussian. It therefore suffices to compute covariance and continuity. For scaling, \[ \mathbb E[\widetilde B_s\widetilde B_t] =c^{-1}(cs\wedge ct)=s\wedge t. \] The shift statement follows directly from independent stationary increments. Reversal rearranges increments without changing their independent Gaussian laws. For inversion, when \(0<s\le t\), \[ \mathbb E[sB_{1/s}\,tB_{1/t}] =st(1/s\wedge1/t)=s. \] Continuity at zero for the inverted process follows from the Gaussian tail bound and Borel-Cantelli along dyadic times, then continuity between them.
Corollary. Brownian fluctuations at time scale \(h\) have spatial scale \(\sqrt h\). This parabolic relation is the common scaling of the heat equation, diffusion limits, and stochastic integration.
A series construction
Theorem (Levy-Ciesielski representation). On \([0,1]\), Brownian motion admits an almost surely uniformly convergent expansion \[ B_t=Z_0t+\sum_{j=0}^{\infty} \sum_{k=0}^{2^j-1} Z_{j,k}\,2^{-(j+2)/2}\, \psi_{j,k}(t), \] where the \(Z_0,Z_{j,k}\) are i.i.d. \(N(0,1)\) and \(\psi_{j,k}\) are appropriately normalized tent functions obtained by integrating the Haar basis.
Proof outline. Expand white noise in the orthonormal Haar basis and integrate term by term. Orthogonality gives the covariance \[ \mathbb E[B_sB_t] =\langle1_{[0,s]},1_{[0,t]}\rangle_{L^2} =s\wedge t. \] Gaussian maxima at level \(j\) grow only like \(\sqrt j\), while the integrated Haar coefficients decay like \(2^{-j/2}\). Borel-Cantelli therefore gives uniform convergence. The representation makes Brownian motion an explicit random continuous function rather than only an abstract extension.