Wiener process

Canonical definition

The Wiener process W is a real-valued continuous-time stochastic process characterized by the following properties:

1. W₀ = 0 almost surely.
2. W has independent increments: for every t > 0, the future increments ${\textstyle W_{t+u}-W_{t},\,u>0,}$ are independent of the past values W_s, s < t. Equivalently, for every t > 0 and ${\textstyle u\geq 0}$, the increment ${\textstyle W_{t+u}-W_{t}}$ is independent of the sigma-algebra ${\textstyle {\mathcal {F}}_{t}^{B}=\sigma (W_{s}:0\leq s\leq t)}$.
3. W has Gaussian increments: for all ${\textstyle u,t\geq 0}$, ${\textstyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).}$ That is, a time step u results in an increment that is normally distributed with mean 0 and variance u.
4. W has almost surely continuous paths: W_t is almost surely continuous in t.

That the process has independent increments means that if 0 ≤ s₁ < t₁ ≤ s₂ < t₂ then W_t1 − W_s1 and W_t2 − W_s2 are independent random variables, and the similar condition holds for n increments.

The Wiener measure is the probability law of the Wiener process on the space of continuous functions g with g(0) = 0 equipped with the Borel σ-algebra. An integral with respect to the Wiener measure may be called a Wiener integral.

Lévy characterization

An alternative characterization of the Wiener process is the so-called Lévy characterization: A continuous (local) martingale W with W₀ = 0 is a Wiener process if and only if its quadratic variation is [W, W]_t = t (which means that W_t² − t is a (local) martingale).

Gaussian process characterization

The Wiener process can be equivalently defined as a Gaussian process ${\displaystyle W_{t}}$, that is, a stochastic process such that for every set of indices ${\displaystyle t_{1},\dots ,t_{n}\geq 0}$, the random vector ${\displaystyle (W_{t_{1}},\dots ,W_{t_{n}})}$ is multivariate Gaussian, which has continuous paths and such that for all ${\displaystyle s,t\geq 0}$ it holds

$${\displaystyle {\begin{aligned}&\operatorname {E} [W_{t}]=0,\\&\mathrm {Cov} (W_{t},W_{s})=\operatorname {E} [W_{s}W_{t}]=\min(s,t).\end{aligned}}}$$

In particular, the law of ${\displaystyle W}$ on the space of continuous functions (equipped with the Borel σ-algebra) is uniquely determined, which is the Wiener measure.

Wiener representation

Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If ${\textstyle \xi _{n}}$ are independent Gaussian variables with mean zero and variance one, then $${\displaystyle W_{t}=\xi _{0}t+{\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \pi nt}{\pi n}}}$$ and $${\displaystyle W_{t}={\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \left(\left(n-{\frac {1}{2}}\right)\pi t\right)}{\left(n-{\frac {1}{2}}\right)\pi }}}$$ represent a Brownian motion on ${\textstyle [0,1]}$. This representation can also be obtained using the Karhunen–Loève theorem.

The scaled process $${\displaystyle {\sqrt {c}}\,W_{t/c}}$$ is a Brownian motion on ${\textstyle [0,c]}$.

White noise representation

In the physics and engineering literature, Brownian motion is often defined (informally) as

$${\displaystyle W_{t}=\int _{0}^{t}\xi (s)\mathrm {d} s}$$

with the white noise process ${\displaystyle \xi }$ satisfying for ${\displaystyle t,s\geq 0}$

$${\displaystyle {\begin{aligned}&\operatorname {E} [\xi (t)]=0,\\&\operatorname {E} [\xi (t)\xi (s)]=\delta (t-s),\end{aligned}}}$$

with the Dirac delta "function" ${\displaystyle \delta }$. However, as shown below, the paths of the Wiener process have unbounded variation and, thus, are not absolutely continuous, that is, ${\displaystyle W_{t}}$ cannot be representated as an integral over a function ${\displaystyle \xi }$.

The mathematical formalization of the above defines a Gaussian white noise ${\displaystyle G}$ (with Lebesgue intensity) as an isometry from the space of square-integrable functions ${\displaystyle L^{2}(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ to the space of centered Gaussian random variables, that is, for ${\displaystyle f,g\in L^{2}(\mathbb {R} )}$ the evaluation ${\displaystyle G(f)}$ is a Gaussian random variable and

$${\displaystyle {\begin{aligned}&\operatorname {E} [G(f)]=0,\\&\operatorname {E} [G(f),G(g)]=\langle f,g\rangle _{L^{2}(\mathbb {R} )}.\end{aligned}}}$$

Such a function ${\displaystyle G}$ indeed exists. Intuitively (and again informally), the Gaussian white noise can be thought of as ${\displaystyle G(f)=\langle \xi ,f\rangle _{L^{2}(\mathbb {R} )}}$, which means it is testing ${\displaystyle \xi }$ against test functions ${\displaystyle f}$ in the spirit of distribution theory. Using the linearity of ${\displaystyle \langle \cdot ,f\rangle }$ and the properties of ${\displaystyle \xi }$ yields the properties of ${\displaystyle G}$.

This allows to define a pre-Brownian motion as

$${\displaystyle {\tilde {W}}_{t}:=G(1_{[0,t]}),\quad t\geq 0.}$$

Pre-Brownian motions do not necessarily have continuous paths, but by Kolmogorov's continuity criterion, there exists a modification of ${\displaystyle {\tilde {W}}}$ that does. Therefore, the Wiener process ${\displaystyle W}$ can be defined as a pre-Brownian motion that has continuous paths.

Donsker's theorem: Wiener process as a limit of a random walk

The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes).

Let ${\textstyle \xi _{1},\xi _{2},\ldots }$ be i.i.d. random variables with mean 0 and variance 1. For each n, define the constant interpolation of the random walk process

$${\displaystyle W_{n}(t)={\frac {1}{\sqrt {n}}}\sum \limits _{1\leq k\leq \lfloor nt\rfloor }\xi _{k},\qquad t\in [0,1].}$$

This is a random step function. Increments of W_n are independent because the ${\textstyle \xi _{k}}$ are independent. For large n, ${\textstyle W_{n}(t)-W_{n}(s)}$ is close to ${\textstyle N(0,t-s)}$ by the central limit theorem. Donsker's theorem asserts that as ${\textstyle n\to \infty }$, W_n approaches a Wiener process.

This characterization explains mathematically the ubiquity of Brownian motion in natural phenomena.

Basic properties

• Brownian motion is a centered Gaussian process. In particular, for ${\displaystyle t>0}$, ${\displaystyle W_{t}=W_{t}-W_{0}\sim N(0,t)}$, that is, the probability density function of ${\displaystyle W_{t}}$ is

$${\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}.}$$

• The covariance between ${\displaystyle W_{t}}$ and ${\displaystyle W_{s}}$ (${\textstyle s\leq t}$) is$${\displaystyle \operatorname {cov} (W_{s},W_{t})=\operatorname {E} [W_{s}W_{t}]=s.}$$ In particular, the variance is ${\displaystyle \operatorname {Var} (W_{t})=t.}$

• The correlation between ${\displaystyle W_{t}}$ and ${\displaystyle W_{s}}$ (${\textstyle s\leq t}$) is

$${\displaystyle \operatorname {corr} (W_{s},W_{t})={\frac {\operatorname {cov} (W_{s},W_{t})}{\sigma _{W_{s}}\sigma _{W_{t}}}}={\frac {s}{\sqrt {st}}}={\sqrt {\frac {s}{t}}}.}$$

• A useful decomposition for proving martingale properties, also called Brownian increment decomposition, is

$${\displaystyle W_{t}=W_{s}+(W_{t}-W_{s}),\;s\leq t.}$$

• Brownian motion is a Lévy process. In particular, it has stationary and independent increments.
• Brownian motion is a continuous martingale.
• Brownian motion has the strong Markov property.

Covariance and correlation

The covariance and correlation between ${\displaystyle W_{t}}$ and ${\displaystyle W_{s}}$ follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that ${\textstyle t_{1}\leq t_{2}}$.

$${\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[(W_{t_{1}}-\operatorname {E} [W_{t_{1}}])\cdot (W_{t_{2}}-\operatorname {E} [W_{t_{2}}])\right]=\operatorname {E} \left[W_{t_{1}}\cdot W_{t_{2}}\right].}$$

Substituting $${\displaystyle W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}}$$ we arrive at: $${\displaystyle {\begin{aligned}\operatorname {E} [W_{t_{1}}\cdot W_{t_{2}}]&=\operatorname {E} \left[W_{t_{1}}\cdot ((W_{t_{2}}-W_{t_{1}})+W_{t_{1}})\right]\\&=\operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]+\operatorname {E} \left[W_{t_{1}}^{2}\right].\end{aligned}}}$$

Since ${\textstyle W_{t_{1}}=W_{t_{1}}-W_{t_{0}}}$ and ${\textstyle W_{t_{2}}-W_{t_{1}}}$ are independent, $${\displaystyle \operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]=\operatorname {E} [W_{t_{1}}]\cdot \operatorname {E} [W_{t_{2}}-W_{t_{1}}]=0.}$$

Thus $${\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[W_{t_{1}}^{2}\right]=t_{1}.}$$

A corollary useful for simulation is that we can write, for t₁ < t₂: $${\displaystyle W_{t_{2}}=W_{t_{1}}+{\sqrt {t_{2}-t_{1}}}\cdot Z}$$ where Z is an independent standard normal variable.

Infinitesimal generator

The infinitesimal generator of Brownian motion is given by

$${\displaystyle Af(x)=\lim _{t\to 0}{\frac {\operatorname {E} [f(W_{t}+x)]-f(x)}{t}}={\frac {1}{2}}f''(x),\quad x\in \mathbb {R} ,}$$

for all ${\displaystyle f\in C^{2}(\mathbb {R} )}$ that vanish at infinity. This is an almost direct consequence of Itô's formula.

Running maximum

The joint distribution of the running maximum

$${\displaystyle M_{t}=\max _{0\leq s\leq t}W_{s}}$$

and W_t is

$${\displaystyle f_{M_{t},W_{t}}(m,w)={\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}},\qquad m\geq 0,w\leq m.}$$

To get the unconditional distribution of ${\textstyle f_{M_{t}}}$, integrate over −∞ < w ≤ m: $${\displaystyle {\begin{aligned}f_{M_{t}}(m)&=\int _{-\infty }^{m}f_{M_{t},W_{t}}(m,w)\,dw=\int _{-\infty }^{m}{\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}}\,dw\\[5pt]&={\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}},\qquad m\geq 0,\end{aligned}}}$$

the probability density function of a Half-normal distribution. The expectation is $${\displaystyle \operatorname {E} [M_{t}]=\int _{0}^{\infty }mf_{M_{t}}(m)\,dm=\int _{0}^{\infty }m{\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}}\,dm={\sqrt {\frac {2t}{\pi }}}}$$

If at time t the Wiener process has a known value ${\textstyle W_{t}}$, it is possible to calculate the conditional probability distribution of the maximum in interval ${\textstyle [0,t]}$ (cf. Probability distribution of extreme points of a Wiener stochastic process). The cumulative probability distribution function of the maximum value, conditioned by the known value ${\textstyle W_{t}}$, is: $${\displaystyle \,F_{M_{W_{t}}}(m)=\Pr \left(M_{W_{t}}=\max _{0\leq s\leq t}W_{s}\leq m\mid W_{t}\right)=\ 1-\ e^{-2{\frac {m(m-W_{t})}{t}}}\ \,,\,\ \ m>\max(0,W_{t})}$$

Arcsine laws

There are multiple (random) quantities ${\displaystyle T}$ related to Brownian motion that follow the arcsine distribution:

$${\displaystyle \Pr(T\leq t)={\frac {2}{\pi }}\arcsin({\sqrt {t}}),\quad t\in [0,1].}$$

These statements are referred to as the arcsine laws of Brownian motion.

Self-similarity

Stopping times

A class of Brownian martingales

If a polynomial p(x, t) satisfies the partial differential equation $${\displaystyle \left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t)=0}$$ then the stochastic process $${\displaystyle M_{t}=p(W_{t},t)}$$ is a martingale.

Example: ${\textstyle W_{t}^{2}-t}$ is a martingale, which shows that the quadratic variation of W on [0, t] is equal to t. It follows that the expected time of first exit of W from (−c, c) is equal to c².

More generally, for every polynomial p(x, t) the following stochastic process is a martingale: $${\displaystyle M_{t}=p(W_{t},t)-\int _{0}^{t}a(W_{s},s)\,\mathrm {d} s,}$$ where a is the polynomial $${\displaystyle a(x,t)=\left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t).}$$

Example: ${\textstyle p(x,t)=\left(x^{2}-t\right)^{2},}$ ${\textstyle a(x,t)=4x^{2};}$ the process $${\displaystyle \left(W_{t}^{2}-t\right)^{2}-4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s}$$ is a martingale, which shows that the quadratic variation of the martingale ${\textstyle W_{t}^{2}-t}$ on [0, t] is equal to $${\displaystyle 4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s.}$$

About functions p(xa, t) more general than polynomials, see local martingales.

Properties of sample paths

The set of all functions w with the following properties has probability one under the Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely:

Information rate

The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by $${\displaystyle R(D)={\frac {2}{\pi ^{2}D\ln 2}}\approx 0.29D^{-1}.}$$ Therefore, it is impossible to encode ${\textstyle \{w_{t}\}_{t\in [0,T]}}$ using a binary code of less than ${\textstyle TR(D)}$ bits and recover it with expected mean squared error less than D. On the other hand, for any ${\textstyle \varepsilon >0}$, there exists T large enough and a binary code of no more than ${\textstyle 2^{TR(D)}}$ distinct elements such that the expected mean squared error in recovering ${\textstyle \{w_{t}\}_{t\in [0,T]}}$ from this code is at most ${\textstyle D-\varepsilon }$.

In many cases, it is impossible to encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals ${\textstyle T_{s}}$ before applying a binary code to represent these samples, the optimal trade-off between code rate ${\textstyle R(T_{s},D)}$ and expected mean square error D (in estimating the continuous-time Wiener process) follows the parametric representation $${\displaystyle R(T_{s},D_{\theta })={\frac {T_{s}}{2}}\int _{0}^{1}\log _{2}^{+}\left[{\frac {S(\varphi )-{\frac {1}{6}}}{\theta }}\right]d\varphi ,}$$ $${\displaystyle D_{\theta }={\frac {T_{s}}{6}}+T_{s}\int _{0}^{1}\min \left\{S(\varphi )-{\frac {1}{6}},\theta \right\}d\varphi ,}$$ where ${\textstyle S(\varphi )=(2\sin(\pi \varphi /2))^{-2}}$ and ${\textstyle \log ^{+}[x]=\max\{0,\log(x)\}}$. In particular, ${\textstyle T_{s}/6}$ is the mean squared error associated only with the sampling operation (without encoding).

Definition

A process ${\displaystyle W=(W^{1},\dots ,W^{d})}$ is a ${\displaystyle d}$-dimensional Wiener process, if ${\displaystyle W^{1},\dots ,W^{d}}$ are independent one-dimensional Wiener processes.

Lévy characterization

Let ${\displaystyle W=(W^{1},\dots ,W^{d})}$ be an adapted stochastic process. The following are equivalent:

• ${\displaystyle W}$ is a ${\displaystyle d}$-dimensional Wiener process.
• ${\displaystyle W^{1},\dots ,W^{d}}$ are continuous local martingales and for ${\displaystyle i,j\in \{1,\dots ,d\}}$, the covariation fulfills ${\displaystyle \langle W^{i},W^{j}\rangle _{t}=\delta _{ij}t}$ for all ${\displaystyle t\geq 0}$, where ${\displaystyle \delta _{ij}}$ is the Kronecker delta.

Generator representation

The Wiener process can also be defined as the unique continuous Markov process starting in zero almost surely and whose generator is ${\displaystyle A={\frac {1}{2}}\Delta }$ with domain ${\displaystyle \mathrm {Dom} (A)=\{f\in C^{2}(\mathbb {R} ^{d}):f,\partial _{ij}^{2}f\in C_{0}(\mathbb {R} ^{d})\}}$, where ${\displaystyle C_{0}(\mathbb {R} ^{d})}$ denotes the space of continuous functions vanishing at infinity.

In fact, given the above generator ${\displaystyle A}$, there exist a unique Feller semigroup ${\displaystyle (P_{t})_{t\geq 0}}$, that is, ${\displaystyle P_{t}}$ is a Markov kernel with

1. ${\displaystyle \forall x\in \mathbb {R} ^{d}:P_{0}(x,\cdot )=\delta _{x}}$, the Dirac measure,
2. ${\displaystyle \forall s,t\geq 0,\,A\in {\mathcal {B}}(\mathbb {R} ^{d}):P_{t+s}(x,A)=\int _{\mathbb {R} ^{d}}P_{t}(y,A)P_{s}(x,\mathrm {d} y)}$,
3. ${\displaystyle \forall f\in C_{0}(\mathbb {R} ^{d}):P_{t}f\equiv \int _{\mathbb {R} ^{d}}f(y)P_{t}(\cdot ,\mathrm {d} y)\in C_{0}(\mathbb {R} ^{d})}$,
4. ${\displaystyle \forall f\in C_{0}(\mathbb {R} ^{d}):\lim _{t\to 0}\lVert P_{t}f-f\rVert _{L^{\infty }(\mathbb {R} ^{d})}=0}$.

such that

$${\displaystyle Af=\lim _{t\to 0}{\frac {P_{t}f-f}{t}},\qquad f\in \mathrm {Dom} (A).}$$

In this case, for ${\displaystyle f\in \mathrm {Dom} (A)}$, the correspondence is precisely

$${\displaystyle P_{t}f(x)=e^{At}f(x)=\int _{\mathbb {R} ^{d}}{\frac {1}{(2\pi t)^{d/2}}}\exp \left(-{\frac {\lVert x-y\rVert ^{2}}{2t}}\right)f(y)\mathrm {d} y,\quad x\in \mathbb {R} ^{d}.}$$

At the same time, there exist a unique Markov process ${\displaystyle W}$ on ${\displaystyle (\mathbb {R} ^{[0,\infty )},{\mathcal {B}}(\mathbb {R} )^{\otimes [0,\infty )})}$ starting in zero such that

$${\displaystyle \mathrm {Pr} (W_{t}+x\in \cdot )=P_{t}(x,\cdot ),\qquad \forall t\geq 0,\,x\in \mathbb {R} ^{d},}$$

that is, the transition probabilities of ${\displaystyle W_{t}+x}$ are given by ${\displaystyle P_{t}(x,\cdot )}$. Finally, by Kolmorov's continuity criterion, there exists a modification of ${\displaystyle W}$ with continuous paths, which is the Wiener process.

Path properties

The following path properties hold almost surely (some of them follow directly from those of the one-dimensional Wiener process):

• The paths are everywhere continuous, but nowhere differentiable.
• For any ${\textstyle \epsilon >0}$, the paths are nowhere ${\textstyle ({\tfrac {1}{2}}+\epsilon )}$-Hölder continuous, but locally ${\textstyle ({\tfrac {1}{2}}-\epsilon )}$-Hölder continuous.
• For ${\displaystyle d=2}$, the Wiener process is recurrent, that is, for every Borel set ${\displaystyle A}$, if the first hitting time ${\displaystyle \inf\{t>0:W_{t}\in A\}}$ is not almost surely infinite, then ${\displaystyle \operatorname {Pr} (W_{t}\in A{\text{ for infinitely many }}t>0)=1}$. In particular, the Wiener process returns arbitrarily close to the origin infinitely often almost surely.
• For ${\displaystyle d\geq 3}$, every compact set ${\displaystyle K}$ is transient, that is, the last hitting time ${\displaystyle \sup\{t\geq 0:W_{t}\in K\}}$ is almost surely finite. In particular, the Wiener process returns in a neighborhood of the origin only finitely many times almost surely.

Infinitesimal generator

Similarly to the one-dimensional case, the infinitesimal generator of Brownian motion is given by

$${\displaystyle Af(x)=\lim _{t\to 0}{\frac {\operatorname {E} [f(W_{t}+x)]-f(x)}{t}}={\frac {1}{2}}\Delta f(x),\quad x\in \mathbb {R} ^{d},}$$

where ${\displaystyle \Delta }$ is the Laplace operator, for all ${\displaystyle f\in C^{2}(\mathbb {R} ^{d})}$ that vanish at infinity.

Harmonic measure on the sphere

The harmonic measure of Brownian motion on the sphere ${\displaystyle \mathbb {S} _{d-1}}$, that is, the distribution of ${\displaystyle W_{\tau }}$, where ${\displaystyle \tau =\inf\{t\geq 0:W_{t}\in \mathbb {S} _{d-1}\}}$ is the first hitting time of the unit sphere, when the process is started in ${\displaystyle x\in B_{1}(0)}$, is

$${\displaystyle \mu ^{x}(A)\equiv \operatorname {Pr} (W_{\tau }+x\in A)=\int _{\mathbb {S} _{d-1}}1_{A}(y)q(x,y)\mathrm {d} \sigma (y),\qquad A\in {\mathcal {B}}(\mathbb {R} ^{d}),}$$

where ${\displaystyle \sigma }$ is the surface measure and

$${\displaystyle q(x,y)={\frac {1}{\sigma (\mathbb {S} _{d-1})}}{\frac {1-\lVert x\rVert ^{2}}{\lVert x-y\rVert ^{d}}},\quad x\in B_{1}(0),\,y\in \mathbb {S} _{d-1},}$$

is the Poisson kernel for the unit ball.

Occupation time formula

Let ${\displaystyle D\subset \mathbb {R} ^{d}}$ be an open, bounded set and ${\displaystyle \tau _{D}^{x}=\inf\{t\geq 0:W_{t}+x\in \partial D\}}$ be the first exit time of Brownian motion, when started in ${\displaystyle x\in D}$. Then, the occupation time of a set ${\displaystyle A\in {\mathcal {B}}(\mathbb {R} ^{d})}$ is given by

$${\displaystyle \operatorname {E} \left[\int _{0}^{\tau _{D}^{x}}1_{A}(W_{t}+x)\mathrm {d} t\right]=\int _{\mathbb {R} ^{d}}1_{A}(y)G(x,y)\mathrm {d} y,}$$

where ${\displaystyle G}$ is the Green function of the Laplace operator for the domain ${\displaystyle D}$, that is, the (distributional) solution of the equation

$${\displaystyle {\begin{aligned}-\Delta G(x,\cdot )&=\delta _{x},\quad \ \ \,{\text{in }}D\\G(x,\cdot )&=0,\qquad {\text{on }}\partial D\end{aligned}}}$$

where ${\displaystyle \delta _{x}}$ is the Dirac measure in ${\displaystyle x}$. More generally, for a measurable function ${\displaystyle f}$ that is either non-negative or integrable with respect to ${\displaystyle G(x,\cdot )\mathrm {d} \lambda }$, it holds

$${\displaystyle \operatorname {E} \left[\int _{0}^{\tau _{D}^{x}}f(W_{t}+x)\mathrm {d} t\right]=\int _{\mathbb {R} ^{d}}f(y)G(x,y)\mathrm {d} y.}$$

Representation of harmonic functions

Let ${\displaystyle D\subset \mathbb {R} ^{d}}$ be a bounded open set. Let ${\displaystyle u}$ be harmonic (${\displaystyle \Delta u=0}$) in ${\displaystyle D}$ and continuous in ${\displaystyle {\bar {D}}}$. Then,

$${\displaystyle u(x)=\operatorname {E} [u(W_{\tau _{D}^{x}}+x)],\quad x\in {\bar {D}},}$$

where ${\displaystyle \tau _{D}^{x}=\inf\{t\geq 0:W_{t}+x\in D^{c}\}}$ is the first exit time of the Wiener process from ${\displaystyle D}$ when started in ${\displaystyle x}$.

Stopping times

Q-Wiener process

Given a non-negative, symmetric, trace-class operator ${\displaystyle Q}$, a stochastic process ${\displaystyle W^{Q}=(W_{t}^{Q})_{t\geq 0}}$ with values in ${\displaystyle H}$ is a ${\displaystyle Q}$-Wiener process if

1. ${\displaystyle W_{0}^{Q}=0}$,
2. ${\displaystyle W^{Q}}$ has continuous trajectories, i.e., for all ${\displaystyle \omega \in \Omega }$ and ${\displaystyle t\geq 0}$