Brownian Motion and Stochastic Calculus

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/Part 1: Stochastic Processes/Chapter 1: Abstract Wiener Spaces/Section 1.1: Classical Wiener Space

Definition 1.1.5 (Characteristic Function [Lemma 8.1.2, Str24]).label Let $(E, \norm{\cdot})$ be a separable Banach space over $\real$, and $\mu$ be a Borel probability measure on $E$. Define

\[\wh \mu: E^{*} \to \complex \quad \wh x^{*} \mapsto \int_{E} e^{i\angles{x, x^*}_E}d\mu(x)\]

as the characteristic function of $\mu$, then

  1. (1)

    $\wh \mu$ is sequentially[1] continuous with respect to the weak* topology on $E^{*}$.

  2. (2)

    If $\nu$ is a Borel probability measure on $E$ such that $\wh \mu = \wh \nu$, then $\nu = \mu$.

Proof. $(1)$: Dominated Convergence Theorem.

$(2)$: Let $\bracsn{x_j^*}_{1}^{n} \subset E^{*}$ and $\mu_{(x_1^*, \cdots, x_n^*)}$ be the distribution of $(x_{1}^{*}, \cdots, x_{n}^{*}): E \to \real^{n}$ under $\mu$, then the characteristic function $\mu_{(x_1^*, \cdots, x_n^*)}$ is

\begin{align*}\wh \mu_{(x_1^*, \cdots, x_n^*)}(\xi)&= \int_{\real^n}e^{i\angles{\xi, x}_{\real^n}}d\mu_{(x_1^*, \cdots, x_n^*)}(x) \\&= \int_{E}e^{i\angles{x, \sum_{j = 1}^n \xi_j x_j^*}}d\mu(x) = \wh \mu\paren{\sum_{j = 1}^n \xi_jx_j^*}\end{align*}

Thus $\wh \mu$ uniquely determines the joint characteristic function of $(x_{1}^{*}, \cdots, x_{n}^{*})$. So for any $\bracsn{x_j^*}_{1}^{n} \subset E^{*}$ and Borel sets $\seqf{B_j}\subset \cb(\real)$,

\[\mu\paren{\bigcap_{j = 1}^n (x_j^*)^{-1}(B_j)}= \nu\paren{\bigcap_{j = 1}^n (x_j^*)^{-1}(B_j)}\]

As these cylinder sets form a $\pi$ system that generates $\cb(E)$, $\mu = \nu$ by Dynkin’s Uniqueness Theorem.$\square$

  1. Most sources I found only claimed and proved the sequential case. Since Stroock used Dominated Convergence Theorem, I assume it should be sequential too.keyboard_return

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Brownian Motion and Stochastic Calculus

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