Brownian Motion and the Riemann Zeta Function

Intriguingly, various constructions related to Brownian motion result in quantities with moments described by the Riemann zeta function. These distributions appear in integral representations used to extend the zeta function to the entire complex plane, as described in an earlier post. Now, I look at how they also arise from processes constructed from Brownian motion such as Brownian bridges, excursions and meanders.

Recall the definition of the Riemann zeta function as an infinite series

\displaystyle  \zeta(s)=1+2^{-s}+3^{-s}+4^{-s}+\cdots

which converges for complex argument s with real part greater than one. This has a unique extension to an analytic function on the complex plane outside of a simple pole at s = 1.

Often, it is more convenient to use the Riemann xi function which can be defined as zeta multiplied by a prefactor involving the gamma function,

\displaystyle  \xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).

This is an entire function on the complex plane satisfying the functional equation ξ(1 - s) = ξ(s).

It turns out that ξ describes the moments of a probability distribution, according to which a random variable X is positive with moments

\displaystyle  {\mathbb E}[X^s]=2\xi(s), (1)

which is well-defined for all complex s. In the post titled The Riemann Zeta Function and Probability Distributions, I denoted this distribution by Ψ, which is a little arbitrary but was the symbol used for its probability density. A related distribution on the positive reals, which we will denote by Φ, is given by the moments

\displaystyle  {\mathbb E}[X^s]=\frac{1-2^{1-s}}{s-1}2\xi(s) (2)

which, again, is defined for all complex s.

As standard, complex powers of a positive real x are defined by xs = eslogx, so (1,2) are equivalent to the moment generating functions of logX, which uniquely determines the distributions. The probability densities and cumulative distribution functions can be given, although I will not do that here since they are already explicitly written out in the earlier post. I will write X ∼ Φ or X ∼ Ψ to mean that random variable X has the respective distribution. As we previously explained, these are closely connected:

  • If X ∼ Ψ and, independently, Y is uniform on [1, 2], then X/Y ∼ Φ.
  • If X, Y ∼ Φ are independent then X2 + Y2 ∼ Ψ.

The purpose of this post is to describe some constructions involving Brownian bridges, excursions and meanders which naturally involve the Φ and Ψ distributions.

Theorem 1 The following have distribution Φ:

  1. 2/πZ where Z = supt|Bt| is the absolute maximum of a standard Brownian bridge B.
  2. Z/√ where Z = suptBt is the maximum of a Brownian meander B.
  3. Z where Z is the sample standard deviation of a Brownian bridge B,

    \displaystyle  Z=\left(\int_0^1(B_t-\bar B)^2\,dt\right)^{\frac12}

    with sample mean  = ∫01Btdt.

  4. π/2Z where Z is the pathwise Euclidean norm of a 2-dimensional Brownian bridge B = (B1, B2),

    \displaystyle  Z=\left(\int_0^1\lVert B_t\rVert^2\,dt\right)^{\frac12}
  5. τπ/2 where τ = inf{t ≥ 0: ‖Bt‖= 1} is the first time at which the norm of a 3-dimensional standard Brownian motion B = (B1, B2, B3) hits 1.

The Kolmogorov distribution is, by definition, the absolute maximum of a Brownian bridge. So, the first statement of theorem 1 is saying that Φ is just the Kolmogorov distribution scaled by the constant factor 2/π. Moving on to Ψ;

Theorem 2 The following have distribution Ψ:

  1. 2/πZ where Z = suptBt – inftBt is the range of a standard Brownian bridge B.
  2. 2/πZ where Z = suptBt is the maximum of a (normalized) Brownian excursion B.
  3. π/2Z where Z is the pathwise Euclidean norm of a 4-dimensional Brownian bridge B = (B1, B2, B3, B4),

    \displaystyle  Z=\left(\int_0^1\lVert B_t\rVert^2\,dt\right)^{\frac12}.

Continue reading “Brownian Motion and the Riemann Zeta Function”

Brownian Excursions

Brownian excursions
Figure 1: Excursions of a Brownian motion

A normalized Brownian excursion is a nonnegative real-valued process with time ranging over the unit interval, and is equal to zero at the start and end time points. It can be constructed from a standard Brownian motion by conditioning on being nonnegative and equal to zero at the end time. We do have to be careful with this definition, since it involves conditioning on a zero probability event. Alternatively, as the name suggests, Brownian excursions can be understood as the excursions of a Brownian motion X away from zero. By continuity, the set of times at which X is nonzero will be open and, hence, can be written as the union of a collection of disjoint (and stochastic) intervals (σ, τ).

In fact, Brownian motion can be reconstructed by simply joining all of its excursions back together. These are independent processes and identically distributed up to scaling. Because of this, understanding the Brownian excursion process can be very useful in the study of Brownian motion. However, there will by infinitely many excursions over finite time periods, so the procedure of joining them together requires some work. This falls under the umbrella of ‘excursion theory’, which is outside the scope of the current post. Here, I will concentrate on the properties of individual excursions.

In order to select a single interval, start by fixing a time T > 0. As XT is almost surely nonzero, T will be contained inside one such interval (σ, τ). Explicitly,

\displaystyle  \begin{aligned} &\sigma=\sup\left\{t\le T\colon X_t=0\right\},\\ &\tau=\inf\left\{t\ge T\colon X_t=0\right\}, \end{aligned} (1)

so that σ < T < τ < ∞ almost surely. The path of X across such an interval is t ↦ Xσ + t for time t in the range [0, τ - σ]. As it can be either nonnegative or nonpositive, we restrict to the nonnegative case by taking the absolute value. By invariance, S-1/2XtS is also a standard Brownian motion, for each fixed S > 0. Using a stochastic factor S = τ – σ, the width of the excursion is normalised to obtain a continuous process {Bt}t ∈ [0, 1] given by

\displaystyle  B_t=(\tau-\sigma)^{-1/2}\lvert X_{\sigma+t(\tau-\sigma)}\rvert. (2)

By construction, this is strictly positive over 0 < t < 1 and equal to zero at the endpoints t ∈ {0, 1}.

Brownian excursion
Figure 2: Constructing a Brownian excursion

The only remaining ambiguity is in the choice of the fixed time T.

Lemma 1 The distribution of B defined by (2) does not depend on the choice of the time T > 0.

Proof: This follows from scaling invariance of Brownian motion. Consider any other fixed positive time , and use the construction above with , σ̃, τ̃,  in place of T, σ, τ, B respectively. We need to show that and B have the same distribution. Using the scaling factor S = /T, then Xt = S-1/2XtS is a standard Brownian motion. Also, σ′= σ̃/S and τ′= τ̃/S are random times given in the same way as σ and τ, but with the Brownian motion X′ in place of X in (1). So,

\displaystyle  \tilde B_t=(\tau^\prime-\sigma^\prime)^{-1/2}\lvert X^\prime_{\sigma^\prime+t(\tau^\prime-\sigma^\prime)}\rvert

has the same distribution as B. ⬜

This leads to the definition used here for Brownian excursions.

Definition 2 A continuous process {Bt}t ∈ [0, 1] is a Brownian excursion if and only it has the same distribution as (2) for a standard Brownian motion X and time T > 0.

In fact, there are various alternative — but equivalent — ways in which Brownian excursions can be defined and constructed.

  • As a normalized excursion away from zero of a Brownian motion. This is definition 2.
  • As a normalized excursion away from zero of a Brownian bridge. This is theorem 6.
  • As a Brownian bridge conditioned on being nonnegative. See theorem 9 below.
  • As the sample path of a Brownian bridge, translated so that it has minimum value zero at time 0. This is a very interesting and useful method of directly computing excursion sample paths from those of a Brownian bridge. See theorem 12 below, sometimes known as the Vervaat transform.
  • As a Markov process with specified transition probabilities. See theorem 15 below.
  • As a transformation of Bessel process paths, see theorem 16 below.
  • As a Bessel bridge of order 3. This can be represented either as a Bessel process conditioned on hitting zero at time 1., or as the vector norm of a 3-dimensional Brownian bridge. See lemma 17 below.
  • As a solution to a stochastic differential equation. See theorem 18 below.

Continue reading “Brownian Excursions”

Brownian Bridge Fourier Expansions

Sine series
Figure 1: Sine series approximations to a Brownian bridge

Brownian bridges were described in a previous post, along with various different methods by which they can be constructed. Since a Brownian bridge on an interval {[0,T]} is continuous and equal to zero at both endpoints, we can consider extending to the entire real line by partitioning the real numbers into intervals of length T and replicating the path of the process across each of these. This will result in continuous and periodic sample paths, suggesting another method of representing Brownian bridges. That is, by Fourier expansion. As we will see, the Fourier coefficients turn out to be independent normal random variables, giving a useful alternative method of constructing a Brownian bridge.

There are actually a couple of distinct Fourier expansions that can be used, which depends on precisely how we consider extending the sample paths to the real line. A particularly simple result is given by the sine series, which I describe first. This is shown for an example Brownian bridge sample path in figure 1 above, which plots the sequence of approximations formed by truncating the series after a small number of terms. This tends uniformly to the sample path, although it is quite slow to converge as should be expected when approximating such a rough path by smooth functions. Also plotted, is the series after the first 100 terms, by which time the approximation is quite close to the target. For simplicity, I only consider standard Brownian bridges, which are defined on the unit interval {[0,1]}. This does not reduce the generality, since bridges on an interval {[0,T]} can be expressed as scaled versions of standard Brownian bridges.

Theorem 1 A standard Brownian bridge B can be decomposed as

\displaystyle  B_t=\sum_{n=1}^\infty\frac{\sqrt2Z_n}{\pi n}\sin(\pi nt) (1)

over {0\le t\le1}, where {Z_1,Z_2,\ldots} is an IID sequence of standard normals. This series converges uniformly in t, both with probability one and in the {L^p} norm for all {1\le p < \infty}.

Continue reading “Brownian Bridge Fourier Expansions”

Brownian Bridges

Brownian bridges
Figure 1: Brownian bridges on subintervals of Brownian motion

A Brownian bridge can be defined as standard Brownian motion conditioned on hitting zero at a fixed future time T, or as any continuous process with the same distribution as this. Rather than conditioning, a slightly easier approach is to subtract a linear term from the Brownian motion, chosen such that the resulting process hits zero at the time T. This is equivalent, but has the added benefit of being independent of the original Brownian motion at all later times.

Lemma 1 Let X be a standard Brownian motion and {T > 0} be a fixed time. Then, the process

\displaystyle  B_t = X_t - \frac tTX_T (1)

over {0\le t\le T} is independent from {\{X_t\}_{t\ge T}}.

Proof: As the processes are joint normal, it is sufficient that there is zero covariance between them. So, for times {s\le T\le t}, we just need to show that {{\mathbb E}[B_sX_t]} is zero. Using the covariance structure {{\mathbb E}[X_sX_t]=s\wedge t} we obtain,

\displaystyle  {\mathbb E}[B_sX_t]={\mathbb E}[X_sX_t]-\frac sT{\mathbb E}[X_TX_t]=s-\frac sTT=0

as required. ⬜

This leads us to the definition of a Brownian bridge.

Definition 2 A continuous process {\{B_t\}_{t\in[0,T]}} is a Brownian bridge on the interval {[0,T]} if and only it has the same distribution as {X_t-\frac tTX_T} for a standard Brownian motion X.

In case that {T=1}, then B is called a standard Brownian bridge.

There are actually many different ways in which Brownian bridges can be defined, which all lead to the same result.

  • As a Brownian motion minus a linear term so that it hits zero at T. This is definition 2.
  • As a Brownian motion X scaled as {tT^{-1/2}X_{T/t-1}}. See lemma 9 below.
  • As a joint normal process with prescribed covariances. See lemma 7 below.
  • As a Brownian motion conditioned on hitting zero at T. See lemma 14 below.
  • As a Brownian motion restricted to the times before it last hits zero before a fixed positive time T, and rescaled to fit a fixed time interval. See lemma 15 below.
  • As a Markov process. See lemma 13 below.
  • As a solution to a stochastic differential equation with drift term forcing it to hit zero at T. See lemma 18 below.

There are other constructions beyond these, such as in terms of limits of random walks, although I will not cover those in this post. Continue reading “Brownian Bridges”