---
title: "Wiener process"
firstLetter: "W"
publishDate: 2021-10-29
categories:
- Physics
- Mathematics

date: 2021-10-21T19:40:02+02:00
draft: false
markup: pandoc
---

# Wiener process

The **Wiener process** is a [stochastic process](/know/concept/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](/know/concept/martingale/)
and a [Markov process](/know/concept/markov-process/),
since each increment has mean zero (so it is a martingale),
and all increments are independent (so it is a Markov process).



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