5 Brownian motion

Definition 5.1 pre-Brownian process

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Let \(\Omega = \mathbb {R}^{\mathbb {R}_+}\) and consider the probability space \((\Omega , P_B)\) (where \(P_B\) is the measure defined in Definition 3.10). The identity on that space is a function \(\Omega \to \mathbb {R}_+ \to \mathbb {R}\). We reorder the arguments to define a stochastic process \(X : \mathbb {R}_+ \to \Omega \to \mathbb {R}\), which we call the pre-Brownian process.

Lemma 5.2

The pre-Brownian process \(X\) of Definition 5.1 is a Gaussian process.

Proof ▶

Lemma 5.3

The pre-Brownian process \(X\) of Definition 5.1 satisfies the Kolmogorov condition for exponents \((2n,n)\) with some constant \(M_n\) for all \(n \in \mathbb {N}\). That is, for all \(s, t \in \mathbb {R}_+\), we have

\begin{align*} \mathbb {E} \left[ |X_t - X_s|^{2n} \right] \le M_n |t - s|^n \: . \end{align*}

Proof ▶

Definition 5.4 Brownian motion

By Theorem 4.56, there exists a version \(Y\) of the pre-Brownian process such that all the paths of \(Y\) are Hölder continuous of all orders \(\gamma \in (0, 1/2)\). We call \(Y\) the Brownian motion on \(\mathbb {R}_+\).

Lemma 5.5

The Brownian motion is a Gaussian process.

Proof ▶

The pre-Brownian process is a Gaussian process by Lemma 5.2. The Brownian motion is a version of the pre-Brownian process by Definition 5.4. Thus, the Brownian motion is a Gaussian process as well by Lemma 2.32.