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 BtB_t is defined as any stochastic process {Bt:t0}\{B_t: t \ge 0\} that satisfies:

  1. Initial condition B0=0B_0 = 0.
  2. Each increment of BtB_t is independent of the past: given 0s<tu<v0 \le s < t \le u < v, then Bt ⁣ ⁣BsB_t \!-\! B_s and Bv ⁣ ⁣BuB_v \!-\! B_u are independent random variables.
  3. The increments of BtB_t are Gaussian with mean 00 and variance hh, where hh is the time step, such that Bt+h ⁣ ⁣BtN(0,h)B_{t+h} \!-\! B_t \sim \mathcal{N}(0, h).
  4. BtB_t is a continuous function of tt.

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 tt, the physical unit of the Wiener process is the square root of time t\sqrt{t}.

Brownian motion is self-similar: if we define a rescaled Wt=αBt/αW_t = \sqrt{\alpha} B_{t/\alpha} for some α\alpha, then WtW_t is also a valid Wiener process, meaning that there are no fundemental scales. A consequence of this is that: EBtp=EtB1p=tp/2EB1p\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 α=t\sqrt{\alpha} = t, such that Wt=tB1/tW_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:

Bt+hBth1hN(0,h)N(0,1h)limh0EN(0,1h)2=\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).


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 BtB_t visit it after a finite time τ ⁣< ⁣\tau\!<\!\infty? It is recurrent if yes, i.e. P(τ ⁣< ⁣)=1P(\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 dd-dimensional Wiener process as an Itō diffusion XtX_t, which also allows us to shift the initial condition X0X_0 (or resume a “paused” process):

Xt=X0+0tdBs\begin{aligned} X_t = X_0 + \int_0^t \dd{B_s} \end{aligned}

Consider two hyperspheres, the inner with radius RiR_i, and the outer with Ro>RiR_o > R_i. Let the initial condition X0]Ri,Ro[|X_0| \in \: ]R_i, R_o[, then we define the stopping times τi\tau_i, τo\tau_o and τ\tau like so:

τiinf{t:XtRi}τoinf{t:XtRo}τmin{τi,τo}\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 XtX_t, whichever happens first.

Dynkin’s formula is applicable to this situation, if we define h(x)h(x) as follows, where the terminal reward Γ\Gamma equals 11 for Xτ=Ri|X_\tau| = R_i, and 00 for Xτ=Ro|X_\tau| = R_o, such that h(X0)h(X_0) equals the probability that we touch RiR_i before RoR_o for a given X0X_0:

h(X0)=E[Γ(Xτ)X0]=P[Xτ ⁣= ⁣RiX0]\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)h(x) is given by the following equation, with the boundary conditions h(Ri)=1h(R_i) = 1 and h(Ro)=0h(R_o) = 0:

0=L^{h(x)}=122h(x)\begin{aligned} 0 = \hat{L}\{h(x)\} = \frac{1}{2} \nabla^2 h(x) \end{aligned}

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

0=2h(r)=2hr2+d1rhr\begin{aligned} 0 = \nabla^2 h(r) = \pdvn{2}{h}{r} + \frac{d - 1}{r} \pdv{h}{r} \end{aligned}

For d=1d = 1, the solution h1(r)h_1(r) is as follows, of which we take the limit for RoR_o \to \infty:

h1(r)=rRoRiRoRo1\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 RoR_o, and for RoR_o \to \infty we are left with the probability of hitting RiR_i only. This turns out to be 11, so in 1D the Wiener process is recurrent: it always comes close to the origin in finite time.

For d=2d = 2, the solution h2(r)h_2(r) is as follows, whose limit turns out to be 11, so the Wiener process is also recurrent in 2D:

h2(r)=1log(r/Ri)log(Ro/Ri)Ro1\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 d3d \ge 3, the solution hd(r)h_d(r) does not converge to 11 for RoR_o \to \infty, meaning the Wiener process is transient in 3D or higher:

hd(r)=Ro2dr2dRo2dRi2dRoRid2rd2<1\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.


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