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From a old handwritten note without references cited, the first canonical process of a stochastic process $\Omega \times T \to S$ is defined as the identity mapping on $S^T$. I was wondering if there are concepts such as the second canonical process, the third canonical process, ...? Thanks and regards!

Tim
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    I would call it "the canonical random element representation", rather that the process (which is VERY often a "sequence"/"collection" of random elements) or "the first ..." since it does not seem there are "second canonical ..." you often meet in math. – SBF Feb 20 '13 at 09:04

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Second canonical process [Meyer, Probability and Potentials, 1966, Definition IV.1.10]

"Let $E$ be a compact space, with the Baire $\sigma$-field $\mathcal{B}_0(E^T)$, and let $(X_t)_{t\in T}$ be a process with values in $(E,\mathcal{B}_0(E))$. Let $(E^T,\mathcal{E}^T,\mathbb{P},(Y_t)_{t\in T})$ be the first canonical process associated with the process $(X_t)$. Since the law $\mathbb{P}$ is a Radon law on the compact space $E^T$, it is possible to extend it to a law $\hat{\mathbb{P}}$ on the Borel $\sigma$-field $\mathcal{B}(E^T)$, which satisfies the condition of theorem II.T35. We thus define a new process equivalent to the process $(X_t)$: $$(E^T,\mathcal{B}(E^T),\hat{\mathbb{P}},(Y_t)_{t\in T})$$ which we call the second canonical process associated with $(X_t)$. It should be remarked that the random variables $(Y_t)_{t\in T}$ are now measurable when $E$ is given the Borel $\sigma$-field."

Theorem II.T35 is devoted to the classical extension of probability from Baire $\sigma$-field to Borel $\sigma$-field. The corresponding probability is unique which satisfies $$\hat{\mathbb{P}}\left(\cap_{i}K_i\right)=\inf_i \hat{\mathbb{P}}(K_i)$$ for every family of compact sets $K_i$, $i\in I$, which is filtering to the left.

user997881
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