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+---
+title: "Wiener process"
+date: 2021-10-29
+categories:
+- Physics
+- Mathematics
+- Stochastic analysis
+layout: "concept"
+---
+
+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).
+
+
+## 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](/know/concept/ito-calculus/) $X_t$,
+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](/know/concept/dynkins-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](/know/concept/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.