Brownian Scaling and Self-Similarity

The Time–Space Balance of Brownian Motion

Brownian motion has fluctuations of order \(\sqrt{t}\), not \(t\). Consequently, changing the time scale by a factor \(c\) requires changing the space scale by \(\sqrt{c}\) to preserve the same distribution. This balance is more than a statement about the distribution at one time: after rescaling, the entire continuous process has the same law.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Thus \(B_0=0\), its paths are continuous, and for \(0\le s<t\), the increment \(B_t-B_s\) is independent of \(\mathcal{F}_s\) and has distribution \(N(0,t-s)\). For a fixed \(c>0\), define the rescaled process and its filtration by

The rescaling changes both the time argument and the units of space. Its importance is that \(\widetilde B\) is again a standard Brownian motion, as the next result proves directly from the defining properties.

The Brownian Scaling Theorem

Proof. Fix \(c>0\). The paths of \(\widetilde B\) are continuous because they are obtained from continuous paths of \(B\) by a continuous change of time and a constant multiplication. Also, \(\widetilde B_0=c^{-1/2}B_0=0\). For \(0\le s<t\),

The increment \(B_{ct}-B_{cs}\) is independent of \(\mathcal{F}_{cs}=\widetilde{\mathcal{F}}_s\), so the rescaled increment is independent of the rescaled process's past. Thus \(\widetilde B\) has continuous paths, starts at zero, and has independent Gaussian increments with variance equal to elapsed time. These are the defining properties of standard Brownian motion.

This argument also gives equality of the finite-dimensional distributions of \(\widetilde B\) and \(B\). To obtain equality in law on \(C([0,T])\), the space of continuous paths on \([0,T]\), note that its Borel sigma-field is generated by evaluations at rational times in \([0,T]\). The two continuous processes agree in all joint distributions at those times, so their probability laws on this path space agree. \(\square\)

Proof. The theorem says that \((c^{-1/2}B_{ct})_{t\ge0}\) has the law of \(B\). Multiplying each path value by \(\sqrt{c}\) gives the first identity in law. For the second, take \(c=a^{-2}\), so that \(c^{-1/2}=a\) and \(ct=t/a^2\). \(\square\)

For example, on \([0,1]\), the path \(t\mapsto B_{ct}\) has the same distribution as \(t\mapsto\sqrt{c}B_t\). A change in the time horizon by \(c\) therefore corresponds to a change in the typical spatial scale by \(\sqrt{c}\). The same identity can be applied to measurable functions of an entire path, not just to its value at one time.

Worked Consequences of Scaling

Scaling a Functional of the Whole Path

Equality of path laws lets us transfer distributions of path functionals. One example is the ordinary time integral of a Brownian path. Define \(A_T=\int_0^T B_s\,ds\), where this is a usual Lebesgue integral along the continuous path, not an Ito integral.

Proof. The path-scaling theorem on \([0,1]\) gives \((B_{Tu})_{0\le u\le1}\) the same law as \((\sqrt{T}B_u)_{0\le u\le1}\). Using the substitution \(s=Tu\) in the path integral therefore gives

It remains to identify the law of \(A_1\). Riemann sums for the integral are linear combinations of jointly Gaussian Brownian values, hence are centered Gaussian random variables. They converge in \(L^2\) to \(A_1\): for example, the Brownian covariance \(\mathbb{E}[B_sB_t]=\min(s,t)\) and continuity of this covariance show that the second moments of the Riemann-sum errors tend to zero. The \(L^2\) limit is therefore centered Gaussian. Its variance is

The diagonal \(s=t\) has zero area, so splitting the square into its two triangles gives the factor \(2\). Consequently \(A_1\sim N(0,1/3)\), and the scaling identity implies \(A_T\sim N(0,T^3/3)\). \(\square\)

Common Mistake: Scaling Time Without Scaling Space

A tempting but incorrect claim is that \((B_{ct})_{t\ge0}\) has the same law as \(B\) without any change in space. At time \(t\), however, \(B_{ct}\) has variance \(ct\), while \(B_t\) has variance \(t\). For example, \(B_4\sim N(0,4)\), whereas \(B_1\sim N(0,1)\); they cannot have the same distribution. The correction is to divide by \(\sqrt{c}\): \(B_{ct}/\sqrt{c}\) has variance \(t\) and, by the theorem, the full Brownian law.

The scaling identity is useful whenever a problem involves a different time horizon or a different spatial threshold. It can simplify endpoint probabilities, joint increment calculations, and path functionals. It does not by itself evaluate every path event; it transfers the event to a rescaled version with the same probability, after which an appropriate distributional calculation may still be needed.