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--- math105-s22:s:mchlxo:start [2022/05/06 08:11]
mchlxo [Definition (Brownian Motion)]
+++ math105-s22:s:mchlxo:start [2026/02/21 14:41] (current)
@@ Line 59: / Line 59: @@
 One last thing before we construct the measure is we have to define the space of Brownian motion paths. We characterize the path by its location at positive rational time t, and the space of all paths is
 $$\mathcal{P} = \prod_{t \in \mathbb{Q}^{+}} \dot{\mathbb{R}}^{n}$$
-where $\dot{\mathbb{R}}^{n} = \mathbb{R}^{n} \cup \{ \infty \}$ is the one-point compactification of $\mathbb{R}^{n}$. Hausdorff's theorem ensures that $\mathcal{P}$ is compact. Now, we are ready to construct our measure, motivated by the following theorem:
+where $\dot{\mathbb{R}}^{n} = \mathbb{R}^{n} \cup \{ \infty \}$ is the one-point compactification of $\mathbb{R}^{n}$. Hausdorff's theorem (Wright 1994) ensures that $\mathcal{P}$ is compact. Now, we are ready to construct our measure, motivated by the following theorem:
 ==== Theorem 1 ====
@@ Line 66: / Line 66: @@
 for all $f \in C(X)$
-**proof:** \\
+**proof:** see Taylor (2006) Theorem 13.5\\
 To use this theorem, we construct a positive linear functional $E: C(\mathcal{P}) \rightarrow \mathbb{R}$. We first define $E$ on the subspace $C^{\#}$ with only continuous functions that depend on only finitely many of the factors in $\mathcal{P}$, with the form
 $$\phi (\omega) = F(\omega(t_1), ... , \omega(t_k)) \text{ , } t_1 < ... <t_k$$
@@ Line 73: / Line 73: @@
 To check this functional is well-defined, we resort to the following proposition:
 ==== Proposition 1 ====
-For $t \neq s$, $\int p(t, x-y)p(s,y)dy = p(t+s, x)$
+For $t \neq s$, $\int p(t, x-y)p(s,y)dy = p(t+s, x)$\\
-**proof:**\\
+**proof:** see Sternberg (2014) slides 10 and 11\\
-Proposition 1 showed that if $F$ does not depend on a given set of $x_i$, we obtain the same functional $E(\phi)$. In addition, we have $E(1) = 1$, so by the Stone-Weierstrass Theorem, $C^{\#}$ is dense in $C(\mathcal{P})$. Furthermore, this functional can be extended to $C(\mathcal{P})$. Therefore, by Theorem 1, we have a measure on the space $\mathcal{P}$:
+Proposition 1 showed that if $F$ does not depend on a given set of $x_i$, we obtain the same functional $E(\phi)$. In addition, we have $E(1) = 1$, so by the Stone-Weierstrass Theorem, $C^{\#}$ is dense in $C(\mathcal{P})$ (see Taylor 2006 Theorem A.23). Furthermore, this functional can be extended to $C(\mathcal{P})$. Therefore, by Theorem 1, we have a measure on the space $\mathcal{P}$:
 ==== Theorem 2 (Wiener Measure) ====
 There exists a unique Borel measure (called the Wiener Measure) $\mu$ on $\mathcal{P}$ such that
 $$ E(\phi) = \int_{\mathcal{P}} \phi(\omega) d \mu(\omega)$$
 for each $\phi(\omega)$ with a continuous $F$ on $\mathcal{P}$
+==== References ====
+Sternberg, Shlomo Z. 2014. “Wiener Measure.” Harvard Math 201a, November 11.\\
+\\
+Taylor, Michael E. 2006. Measure Theory and Integration. Graduate Studies in Mathematics, v. 76. Providence, R.I: American Mathematical Society.\\
+\\
+Wright, David G. 1994. “Tychonoff’s Theorem.” Proceedings of the American Mathematical Society 120 (3): 985–87. https://doi.org/10.1090/S0002-9939-1994-1170549-2.
 ===== Resources =====
 A {{ :math105-s22:s:mchlxo:exception_points.pdf |paper}} that constructs a set which includes points at which the density of the set can take on any values in $[0,1]$

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