Gaussian Increment Moments and Path Regularity

Why Increment Moments Matter

Brownian path regularity is studied by comparing values at nearby times. The distribution of a single increment gives more than its variance: it gives moments of every order. Higher moments penalize large increments more strongly, and their dependence on the time gap provides estimates that can be used in continuity and Hölder-regularity arguments. The crucial calculation is exact: an increment over a time interval of length \(h\) has its \(2m\)-th absolute moment proportional to \(h^m\).

Throughout, let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space and let \((B_t)_{t\ge0}\) be a real-valued standard Brownian motion with state space \(S=\mathbb{R}\) and continuous-time index \(t\ge0\). We use the defining increment law: for \(0\le s\le t\), \(B_t-B_s\) is Gaussian with mean \(0\) and variance \(t-s\); when \(t=s\), the increment is identically zero. The estimates below concern the distribution of increments and do not by themselves prove a path-regularity theorem.

Gaussian Moments and Brownian Scaling

Proof. For \(Z\) standard normal, its density is \(\phi(x)=(2\pi)^{-1/2}e^{-x^2/2}\). Set \(I_m=\int_{-\infty}^{\infty}x^{2m}\phi(x)\,dx\), so \(I_m=\mathbb{E}[Z^{2m}]\). Since \(\phi'(x)=-x\phi(x)\), integration by parts gives, for \(m\ge1\), \[ I_m=-\int_{-\infty}^{\infty}x^{2m-1}\phi'(x)\,dx =-(x^{2m-1}\phi(x))\big|_{-\infty}^{\infty} +(2m-1)\int_{-\infty}^{\infty}x^{2m-2}\phi(x)\,dx. \] The boundary term is zero because a polynomial times \(e^{-x^2/2}\) tends to zero at both ends of the real line. Thus \(I_m=(2m-1)I_{m-1}\). Since \(I_0=1\), repeated application gives \(I_m=(2m-1)(2m-3)\cdots1\). The density is symmetric, so the integral of any odd power over the real line is zero. If \(h>0\), write \(Y=\sqrt{h}Z\) in distribution; then \(\mathbb{E}|Y|^{2m}=h^m\mathbb{E}|Z|^{2m}\). If \(h=0\), \(Y=0\) almost surely, and both sides are zero. \(\square\)

Applying this result to a Brownian increment gives the central formula. The absolute value can be omitted for an even power, but retaining it makes the nonnegativity of the moment explicit.

The moment order and time exponent are linked: a moment of order \(2m\) scales as the \(m\)-th power of the interval length. In particular, multiplying the time gap by \(c\ge0\) multiplies this moment by \(c^m\). This is the moment form of Brownian scaling; it follows directly from the Gaussian increment law.

From Moments to Increment-Probability Bounds

Proof. Set \(Y=|B_t-B_s|^{2m}\), which is nonnegative. If \(|B_t-B_s|>a\), then \(Y>a^{2m}\). Therefore \[ a^{2m}\mathbf{1}_{\{|B_t-B_s|>a\}}\le |B_t-B_s|^{2m}. \] Taking expectations and using the even-moment formula gives \[ a^{2m}\mathbb{P}(|B_t-B_s|>a) \le \mathbb{E}|B_t-B_s|^{2m} =(2m-1)!!(t-s)^m. \] Division by \(a^{2m}>0\) proves the strict-event bound. Since the same pointwise inequality holds with the event \(|B_t-B_s|\ge a\), it proves the non-strict version as well. \(\square\)

For regularity arguments, the threshold is often chosen in relation to the interval length. Fix \(\alpha\ge0\) and \(c>0\), and take \(a=c(t-s)^\alpha\). Writing \(h=t-s>0\), the proposition yields \[ \mathbb{P}(|B_t-B_s|>c h^\alpha) \le \frac{(2m-1)!!}{c^{2m}}h^{m-2m\alpha}. \] Thus the exponent of \(h\) is \(m(1-2\alpha)\). If one compares increments on a time grid with spacing \(h\), there are on the order of \(h^{-1}\) neighboring intervals in a fixed time range. The exponent exceeds \(1\) precisely when the single-interval bound decays faster than this count grows. This exponent comparison motivates moment-based regularity methods, but it is not by itself a proof of a uniform pathwise bound.

Choosing the Moment Order

The Gaussian law supplies moments of every even order, so the moment order can be chosen to match a desired threshold exponent. If the target threshold is proportional to \(h^\alpha\), the exponent in the probability bound is \(m(1-2\alpha)\). For a fixed \(\alpha<\tfrac12\), the factor \(1-2\alpha\) is positive. One can therefore choose an integer \(m\) large enough that \(m(1-2\alpha)>1\). At \(\alpha=\tfrac12\), this exponent is \(0\) for every \(m\); for \(\alpha>\tfrac12\), it is negative. This describes the strength of this particular moment-bound method, not a proof of the sharp path regularity of Brownian motion.

Common Pitfall: Scaling the Variance Incorrectly

A frequent error is to use the time gap itself as the scale of a fourth moment. The variance of \(B_t-B_s\) is \(h=t-s\), but a fourth moment scales as the square of that variance, not as the variance alone. For example, if \(h=2\), the incorrect calculation \(3h=3\cdot2=6\) gives the wrong fourth moment. The correct calculation is \[ \mathbb{E}|B_t-B_s|^4=3h^2=3\cdot2^2=12. \] The error is visible by standardizing: dividing the correct fourth moment by \(h^2=4\) gives \(3\), the fourth moment of a standard normal variable; dividing the incorrect value \(6\) by \(4\) does not.

There is a second distinction to keep in view: an estimate for one fixed pair \(s,t\) is not automatically a statement holding simultaneously for every pair of times. Path regularity requires control across many intervals and, ultimately, a uniform argument. The moment formula provides the quantitative input; additional reasoning is needed to convert that input into a pathwise conclusion. It is also important to use \(h=t-s\ge0\), not \(t+s\): the increment variance depends on the length of the interval between the times.