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I wish to define the Wiener process and some associated diffusions. For this, I thought to work with the space $C([0,\infty),\mathbb{R})$ of continuous functions, endowed with the sup-norm $||f||=\sup(|f(s)|:s\geq 0)$, $f\in C([0,\infty),\mathbb{R})$. The norm induces a metric and topology and as I understand, this induces the Borel $\sigma$-field generated by the open sets. However, in the book Brownian motion and siffusion by David Freedman, the space is indeed $C([0,\infty),\mathbb{R})$ but what he calls the Borel $\sigma$-field is $\sigma(X(t),t\geq 0)$. It's tough for me to tell if these definitions are the same or not, I have a feeling they are not, but I'm not sure. To me, Freedman's definition sounds not well defined because he doesn't define the topology, so why use the name Borel. Is his $\sigma$-field the Borel $\sigma$-field that corresponds to the discrete topology on $C([0,\infty),\mathbb{R})$?

As you can see, I'm quite confused even in these basic definitions. Thanks for your help.

Yonatan
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1 Answers1

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The topology of uniform convergence on $C(ℝ_+, ℝ)$ is not the one used in the context of the Wiener process, since it is too fine. The definition in your book, $\mathcal C := σ(X(t), t \geq 0)$, refers to the σ-algebra on $C(ℝ_+, ℝ)$ generated by the canonical projections $$ X(t): C(ℝ_+, ℝ) \ni w ↦ w(t) ∈ ℝ, $$ which happens to coincide with the restriction of the product-σ-algebra $\mathcal B(ℝ)^{⊗ℝ_+}$ to $C(ℝ_+,ℝ)$. In particular $\mathcal C$ contains all the cylinder sets in $C(ℝ_+, ℝ)$. It is indeed the case that $\mathcal C$ is generated by a topology on $C(ℝ_+, ℝ)$, so it is really a Borel-σ-algebra. This generator is not the uniform topology, but rather the topology of compact convergence (i.e. uniform convergence on all compact subsets). The proof of this is sketched in this answer.

jro
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