Brownian Motion

Book 6A met Brownian motion as the random path whose position at time $t$ is distributed by the heat kernel (6A.3 The Heat Equation on ℝⁿ). This book takes the random path seriously. A Brownian path is continuous but nowhere differentiable; it has finite quadratic variation, $\sum(\Delta B)^2 \to t$, which is the fact that makes stochastic calculus different from ordinary calculus; and averaging functions over Brownian paths solves the heat equation, the Laplace equation and, with Feynman and Kac, every linear parabolic equation with a potential. Those averages are also prices: the value of a financial contract is an expectation over random paths, and the equations of mathematical finance are heat equations in disguise (6D.6 Black–Scholes).

We construct Brownian motion from a sequence of independent Gaussians, prove its basic path properties, and use it to solve the Dirichlet problem. The probability needed is modest: Gaussian random variables, independence, expectation as a Lebesgue integral on a probability space (3A.4 Measures, Probability and Weights), and the Borel–Cantelli lemma. Where a proof needs heavier machinery (the strong Markov property, the existence of continuous versions in general), we state it.

A convention. Probabilists normalise Brownian motion so that $B_t$ has variance $t$; its generator is $\frac12\Delta$, and $\mathbb{E}[g(x + B_t)]$ solves $u_t = \frac12\Delta u$. The heat equation $u_t = \Delta u$ of Books 6A–6C corresponds to $B_{2t}$ (6A.3 The Heat Equation on ℝⁿ). We use the probabilists' convention throughout Book 6D.

By the end of this chapter you will be able to:

• explain the random walk scaling and define Brownian motion as a Gaussian process with covariance $\min(s, t)$;
• construct Brownian motion on $[0, 1]$ from Haar and Schauder functions, and prove that the series converges uniformly almost surely;
• prove that quadratic variation converges to $t$ in $L^2$, and deduce that paths have infinite length;
• use the reflection principle to find the law of the running maximum and of hitting times;
• prove that $x \mapsto \mathbb{E}_x[g(B_\tau)]$ is harmonic, compute exit probabilities, and decide recurrence and transience by dimension.

1.1Speculation before physics

1.2Random walks and the definition

Toss a fair coin every $\frac1n$ seconds and step $\pm\frac{1}{\sqrt n}$. After time $t$ the position is $W^{(n)}_t = \frac{1}{\sqrt n}S_{\lfloor nt\rfloor}$, with $S_k = \xi_1 + \cdots + \xi_k$ a sum of independent $\pm1$ steps. Then $\mathbb{E}W^{(n)}_t = 0$ and $\operatorname{Var}W^{(n)}_t = \frac{\lfloor nt\rfloor}{n} \to t$, and by the central limit theorem $W^{(n)}_t$ converges in distribution to $N(0, t)$. The step size $n^{-1/2}$, the square root of the time step, is forced: smaller steps give a limit that does not move, larger ones a limit that explodes. This diffusive scaling, space like the square root of time, is the scaling of the heat equation. Increments over disjoint time intervals use disjoint sets of coin tosses, so they are independent. Monroe Donsker proved in 1951 that the whole random path $W^{(n)}$ converges in distribution, as a random continuous function, to the following object.

Properties 1–3 say exactly that $B$ is a Gaussian process (every finite collection $(B_{t_1}, \ldots, B_{t_k})$ is jointly Gaussian) with mean zero and covariance

(Exercise 1.8). Property 4 is the real content: it is not implied by 1–3 (one can modify each $B_t$ on a null set depending on $t$ and destroy continuity) and needs a construction.

1.3The Lévy–Ciesielski construction

On $[0, 1]$ use the Haar functions: $h_0 = 1$, and for $m \geq 0$ and $0 \leq k < 2^m$,

They form an orthonormal basis of $L^2(0, 1)$ (orthonormality is a direct check; completeness holds because their span contains the indicator of every dyadic interval, and these span a dense subspace; 4A.5 Hilbert Spaces and Lax–Milgram). Their integrals are the Schauder functions

tents supported on $[k2^{-m}, (k + 1)2^{-m}]$ with height $2^{-m/2 - 1}$; for fixed $m$ the tents have disjoint supports.

To get Brownian motion on $[0, \infty)$, take independent copies $B^{(1)}, B^{(2)}, \ldots$ on $[0, 1]$ and glue them: $B_t = B^{(1)}_1 + \cdots + B^{(j)}_1 + B^{(j+1)}_{t - j}$ for $j \leq t < j + 1$. The construction is also an algorithm: the level-$m$ terms refine the path at the dyadic points of spacing $2^{-m}$ by adding independent Gaussian bumps at the midpoints, which is how Brownian paths are drawn on a computer (the "Brownian bridge" or midpoint displacement method).

Invariances. Each of the following is again a Brownian motion (Exercise 1.9): $-B_t$; $B_{s+t} - B_s$ for fixed $s$ (and it is independent of $(B_r)_{r\leq s}$, the Markov property); the rescaled path $c^{-1/2}B_{ct}$ for $c > 0$ (Brownian scaling); and the time-inverted path $tB_{1/t}$ (extended by $0$ at $t = 0$). Scaling says Brownian motion looks the same at every scale, statistically: zooming in by a factor $c$ in time and $\sqrt c$ in space gives the same picture. It is a random fractal.

1.4Quadratic variation

For a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_k = t\}$ of $[0, t]$ with mesh $|\Pi| = \max(t_{i+1} - t_i)$, the quadratic variation of the path along $\Pi$ is $Q_\Pi = \sum_i(B_{t_{i+1}} - B_{t_i})^2$.

For a $C^1$ function $f$, $\sum(f(t_{i+1}) - f(t_i))^2 \leq \max|f'|^2\sum\delta_i^2 \to 0$. Brownian paths are therefore nothing like $C^1$ functions: increments over a time $\delta$ are of size $\sqrt\delta$, not $\delta$, and their squares add up to something finite and deterministic. In shorthand, $(dB_t)^2 = dt$, which is the rule behind Itô's formula (6D.2 Martingales and the Itô Integral).

In particular Brownian paths are not differentiable on any interval (a function differentiable everywhere on an interval with bounded derivative has finite variation). Much more is true: Raymond Paley, Norbert Wiener and Antoni Zygmund proved in 1933 that almost every Brownian path is differentiable at no point (statement). Infinite variation is also why the integral $\int f\,dB$ cannot be defined path by path as a Riemann–Stieltjes integral, and why Itô had to define it as an $L^2$ limit.

1.5The reflection principle

For $a > 0$ let $\tau_a = \inf\{t \geq 0 : B_t = a\}$, the first time the path reaches $a$, and $M_t = \max_{s\leq t}B_s$ the running maximum. Then $\{M_t \geq a\} = \{\tau_a \leq t\}$.

The tool is the strong Markov property (statement): if $\tau$ is a stopping time (whether $\tau \leq t$ is decided by the path up to time $t$; 6D.2 Martingales and the Itô Integral) with $\tau < \infty$ almost surely, then $\tilde B_s = B_{\tau + s} - B_\tau$ is a Brownian motion independent of the path up to time $\tau$. For deterministic $\tau$ this is the Markov property; for hitting times it is a theorem (Øksendal, section 7.2; Mörters and Peres, section 2.1).

The name describes the proof: every path that reaches $a$ before time $t$ and ends below $a$ corresponds, by reflecting its part after $\tau_a$ in the line $y = a$, to a path that ends above $a$, and reflection preserves probabilities. Bachelier used this argument in 1900. Consequences (Exercise 1.10): $\tau_a$ is finite almost surely (Brownian motion reaches every level), but $\mathbb{E}\tau_a = \infty$; and $\tau_a$ has density $\frac{a}{\sqrt{2\pi t^3}}e^{-a^2/(2t)}$.

1.6Harmonic measure and the Dirichlet problem

Let $U \subset \mathbb{R}^n$ be a bounded domain and $\tau = \inf\{t : B_t \notin U\}$ the exit time, finite almost surely (Exercise 1.11). For a continuous function $g$ on $\partial U$, define

the expected value of $g$ at the point where Brownian motion started at $x$ first leaves $U$. The law of $B_\tau$ under $\mathbb{P}_x$ is the harmonic measure of $U$ seen from $x$.

So harmonic measure is the answer to the question "where does a random walker leave the room?", and the Dirichlet problem is a gambling problem: if you receive $g(z)$ when the walk exits at $z$, the fair price of the game started at $x$ is $u(x)$. Shizuo Kakutani found this connection in 1944. In one dimension, harmonic functions are linear, and the exit problem from $(a, b)$ is gambler's ruin: $\mathbb{P}_x(\text{exit at } b) = \frac{x - a}{b - a}$ (Exercise 1.12).

1.7Recurrence and transience

Apply the same idea to the annulus $A = \{r < |x| < R\}$ in $\mathbb{R}^n$ and the event that the walk reaches the inner sphere before the outer one. The probability $p(x)$ is harmonic in $A$, equal to $1$ on $|x| = r$ and $0$ on $|x| = R$, and the boundary is smooth, so $p$ is the radial harmonic function with these values (6A.2 Harmonic Functions):

Now let $R \to \infty$ (the walk reaches the outer sphere in finite time almost surely, and $R \to \infty$ makes this ever later). In two dimensions $p(x) \to 1$: Brownian motion in the plane comes back arbitrarily close to every point, infinitely often (neighbourhood recurrence). In three or more dimensions $p(x) \to \big(\frac{r}{|x|}\big)^{n-2} < 1$: with positive probability the walk never comes back to the ball, and in fact $|B_t| \to \infty$ almost surely (transience). George Pólya had proved the same dichotomy for random walks on the integer lattice in 1921. The dividing line is the dimension at which the Newtonian potential $|x|^{2-n}$ stops being a logarithm: recurrence is a statement about the fundamental solution of the Laplacian.

1.8History

Robert Brown observed the motion of pollen particles in 1827. Louis Bachelier (1900), Albert Einstein (1905) and Marian Smoluchowski (1906) gave the first theories, and Jean Perrin's experiments (1908–09) confirmed Einstein's (6A.3 The Heat Equation on ℝⁿ). Norbert Wiener constructed Brownian motion as a measure on continuous paths in 1923, which is why it is also called the Wiener process. Paul Lévy developed the theory of its paths in the 1930s and 1940s, including the midpoint construction, and Zbigniew Ciesielski gave the Haar-function form in 1961. Paley, Wiener and Zygmund proved nowhere differentiability in 1933, Kakutani connected Brownian motion to the Dirichlet problem in 1944, and Donsker proved the invariance principle in 1951.

1.9Exercises

Solution

$\mathbb{E}X_0^2 = 0$, so $X_0 = 0$ almost surely. $\operatorname{Var}(X_t - X_s) = t - 2\min(s, t) + s = t - s$. For $s_1 < t_1 \leq s_2 < t_2$, $\mathbb{E}[(X_{t_1} - X_{s_1})(X_{t_2} - X_{s_2})] = t_1 - t_1 - s_1 + s_1 = 0$. All increments are jointly Gaussian (linear combinations of a Gaussian vector), so pairwise uncorrelated increments are independent; the same computation applies to any finite family.

Solution

All three are mean-zero Gaussian processes with continuous paths (for $t > 0$). $\mathbb{E}[X_sX_t] = c^{-1}\min(cs, ct) = \min(s, t)$. $\mathbb{E}[Y_uY_v] = \min(s + u, s + v) - s - s + s = \min(u, v)$. $\mathbb{E}[Z_uZ_v] = uv\min(\frac1u, \frac1v) = \frac{uv}{\max(u, v)} = \min(u, v)$.

Solution

(a) $\frac{d}{dt}2\big(1 - \Phi(at^{-1/2})\big) = 2\varphi(at^{-1/2})\cdot\frac a2t^{-3/2} = \frac{a}{\sqrt{2\pi}}t^{-3/2}e^{-a^2/(2t)}$. As $t \to \infty$, $\mathbb{P}(\tau_a \leq t) \to 2(1 - \Phi(0)) = 1$. (b) $tf(t) \sim \frac{a}{\sqrt{2\pi}}t^{-1/2}$ as $t \to \infty$, not integrable. (c) $2(1 - \Phi(1)) \approx 2 \times 0.1587 = 0.317$.

Solution

Let $1 - q = \mathbb{P}(|B^1_1| > 2R) > 0$. If $|B^1_{j+1} - B^1_j| > 2R$, the path cannot have stayed in $B_R$ during $[j, j + 1]$. These increments are independent, so $\mathbb{P}_x(\tau > k) \leq \mathbb{P}(|B^1_{j+1} - B^1_j| \leq 2R \text{ for } j < k) = q^k$. Then $\mathbb{P}(\tau = \infty) = 0$ and $\mathbb{E}\tau \leq \sum_k\mathbb{P}(\tau > k) \leq \frac{1}{1 - q}$.

Solution

(a) $u(x) = \mathbb{P}_x(\text{exit at } b)$ is harmonic on $(a, b)$, i.e. linear, with $u(a) = 0$, $u(b) = 1$ (all boundary points are regular in one dimension). (b) $\frac{10}{30} = \frac13$. (c) $10\times20 = 200$ units of time.

Solution

(a) $\Delta|x|^2 = 2n$, so $\frac12\Delta\frac{R^2 - |x|^2}{n} = -1$, and $v = 0$ on $|x| = R$. (b) $\frac12v'' = -1$ with $v(a) = v(b) = 0$ gives $v = (x - a)(b - x)$. (c) $\frac13$.

Solution

In $\mathbb{R}^3$, $(r/|x|)^{n-2} = \frac12$. In $\mathbb{R}^2$, $\frac{\log100 - \log2}{\log100 - \log1} = \frac{\log50}{\log100} \approx 0.849$.

Solution

(a) $-2t\log p_t = nt\log(2\pi t) + |x - y|^2 \to |x - y|^2$, since $t\log t \to 0$. (b) $p_{2\tau}(0, x) = (4\pi\tau)^{-n/2}e^{-|x|^2/(4\tau)}$, with exponent $\frac{|x|^2}{4\tau} = \ell$.