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Categories: Mathematics, Physics, Stochastic analysis.

Wiener process

The Wiener process is a stochastic process that provides a pure mathematical definition of the physical phenomenon of Brownian motion, and hence is also called Brownian motion.

A Wiener process $B_t$ is defined as any stochastic process $\{B_t: t \ge 0\}$ that satisfies:

1. Initial condition $B_0 = 0$.
2. Each increment of $B_t$ is independent of the past: given $0 \le s < t \le u < v$, then $B_t \!-\! B_s$ and $B_v \!-\! B_u$ are independent random variables.
3. The increments of $B_t$ are Gaussian with mean $0$ and variance $h$, where $h$ is the time step, such that $B_{t+h} \!-\! B_t \sim \mathcal{N}(0, h)$.
4. $B_t$ is a continuous function of $t$.

There exist stochastic processes that satisfy these requirements, infinitely many in fact. In other words, Brownian motion exists, and can be constructed in various ways.

Since the variance of an increment is expressed in units of time $t$, the physical unit of the Wiener process is the square root of time $\sqrt{t}$.

Brownian motion is self-similar: if we define a rescaled $W_t = \sqrt{\alpha} B_{t/\alpha}$ for some $\alpha$, then $W_t$ is also a valid Wiener process, meaning that there are no fundemental scales. A consequence of this is that: $\mathbf{E}|B_t|^p = \mathbf{E}|\sqrt{t} B_1|^p = t^{p/2} \mathbf{E}|B_1|^p$. Another consequence is invariance under “time inversion”, by defining $\sqrt{\alpha} = t$, such that $W_t = t B_{1/t}$.

Despite being continuous by definition, the Wiener process is not differentiable in general, not even in the mean square, because:

$$\begin{aligned} \frac{B_{t+h} - B_t}{h} \sim \frac{1}{h} \mathcal{N}(0, h) \sim \mathcal{N}\Big(0, \frac{1}{h}\Big) \qquad \quad \lim_{h \to 0} \mathbf{E} \bigg|\mathcal{N}\Big(0, \frac{1}{h}\Big) \bigg|^2 = \infty \end{aligned}$$

Furthermore, the Wiener process is a good example of both a martingale and a Markov process, since each increment has mean zero (so it is a martingale), and all increments are independent (so it is a Markov process).

Recurrence

An important question about the Wiener process is whether it is recurrent or transient: given a hypersphere (interval in 1D, circle in 2D, sphere in 3D) away from the origin, will $B_t$ visit it after a finite time $\tau\!<\!\infty$? It is recurrent if yes, i.e. $P(\tau \!<\! \infty) = 1$, or transient otherwise. The answer to this question turns out to depend on the number of dimenions.

To demonstrate this, we model the $d$-dimensional Wiener process as an Itō diffusion $X_t$, which also allows us to shift the initial condition $X_0$ (or resume a “paused” process):

$$\begin{aligned} X_t = X_0 + \int_0^t \dd{B_s} \end{aligned}$$

Consider two hyperspheres, the inner with radius $R_i$, and the outer with $R_o > R_i$. Let the initial condition $|X_0| \in \: ]R_i, R_o[$, then we define the stopping times $\tau_i$, $\tau_o$ and $\tau$ like so:

$$\begin{aligned} \tau_i \equiv \inf\{ t : |X_t| \le R_i \} \qquad \tau_o \equiv \inf\{ t : |X_t| \ge R_o \} \qquad \tau \equiv \min\{\tau_i, \tau_o\} \end{aligned}$$

We stop when the inner or outer hypersphere is touched by $X_t$, whichever happens first.

Dynkin’s formula is applicable to this situation, if we define $h(x)$ as follows, where the terminal reward $\Gamma$ equals $1$ for $|X_\tau| = R_i$, and $0$ for $|X_\tau| = R_o$, such that $h(X_0)$ equals the probability that we touch $R_i$ before $R_o$ for a given $X_0$:

$$\begin{aligned} h(X_0) = \mathbf{E}\Big[ \Gamma(X_\tau) \Big| X_0 \Big] = P\Big[|X_\tau| \!=\! R_i \:\Big|\: X_0\Big] \end{aligned}$$

Dynkin’s formula then tells us that $h(x)$ is given by the following equation, with the boundary conditions $h(R_i) = 1$ and $h(R_o) = 0$:

$$\begin{aligned} 0 = \hat{L}\{h(x)\} = \frac{1}{2} \nabla^2 h(x) \end{aligned}$$

Thanks to this problem’s spherical symmetry, $h$ only depends on the radial coodinate $r$, so the Laplacian $\nabla^2$ can be written as follows in $d$-dimensional spherical coordinates:

$$\begin{aligned} 0 = \nabla^2 h(r) = \pdvn{2}{h}{r} + \frac{d - 1}{r} \pdv{h}{r} \end{aligned}$$

For $d = 1$, the solution $h_1(r)$ is as follows, of which we take the limit for $R_o \to \infty$:

$$\begin{aligned} h_1(r) = \frac{r - R_o}{R_i - R_o} \quad\underset{R_o \to \infty}{\longrightarrow}\quad 1 \end{aligned}$$

The outer hypersphere becomes harder to reach for larger $R_o$, and for $R_o \to \infty$ we are left with the probability of hitting $R_i$ only. This turns out to be $1$, so in 1D the Wiener process is recurrent: it always comes close to the origin in finite time.

For $d = 2$, the solution $h_2(r)$ is as follows, whose limit turns out to be $1$, so the Wiener process is also recurrent in 2D:

$$\begin{aligned} h_2(r) = 1 - \frac{\log(r/R_i)}{\log(R_o/R_i)} \quad\underset{R_o \to \infty}{\longrightarrow}\quad 1 \end{aligned}$$

However, for $d \ge 3$, the solution $h_d(r)$ does not converge to $1$ for $R_o \to \infty$, meaning the Wiener process is transient in 3D or higher:

$$\begin{aligned} h_d(r) = \frac{R_o^{2 - d} - r^{2 - d}}{R_o^{2 - d} - R_i^{2 - d}} \quad\underset{R_o \to \infty}{\longrightarrow}\quad \frac{R_i^{d - 2}}{r^{d - 2}} < 1 \end{aligned}$$

This is a major qualitative difference. For example, consider a situation where some substance is diffusing from a localized infinite source: in 3D, the substance can escape and therefore a steady state can exist, while in 2D, the substance never strays far from the source, so no steady state is ever reached as long as the source continues to emit.

References

1. U.H. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.

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