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Multi-Occupation Field Generates the Borel-Sigma-Field of Loops

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Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 2123)

Abstract

In this article, we consider the space of càdlàg loops on a Polish space S. The loop space can be equipped with a “Skorokhod” metric. Moreover, it is Polish under this metric. Our main result is to prove that the Borel-σ-field on the space of loops is generated by a class of loop functionals: the multi-occupation field. This result generalizes the result in the discrete case, see (Le Jan, Markov Paths, Loops and Fields, vol. 2026, Springer, Heidelberg, 2011).

Keywords

  • Markov Paths
  • Polish Space
  • Loop Space
  • Linear Time Scaling
  • Countable Topological Base

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    The terminology “càdlàg” is short for right-continuous with left hand limits.

  2. 2.

    The countability is required by Lemma 1.

References

  1. C. Dellacherie, P-A. Meyer, Probabilities and Potential, vol. 29 of North-Holland Mathematics Studies (North-Holland Publishing, Amsterdam, 1978)

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  2. S.N. Ethier, T.G. Kurtz, Markov Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics (Wiley, New York, 1986). Characterization and convergence

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  3. Y. Le Jan, Markov Paths, Loops and Fields, vol. 2026 of Lecture Notes in Mathematics (Springer, Heidelberg, 2011) Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour [Saint-Flour Probability Summer School]

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  4. Y. Le Jan, Z. Qian, Stratonovich’s signatures of Brownian motion determine Brownian sample paths. Probab. Theory Relat. Fields 157(1–2), 209–223 (2013)

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  5. A-S. Sznitman, Topics in Occupation Times and Gaussian Free Fields. Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2012)

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Acknowledgements

The author is grateful to Professor Y. Le Jan for inspiring discussions, valuable suggestions and great help in preparation of the manuscript. The author also thanks Titus Lupu for inspiring discussions.

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Correspondence to Yinshan Chang .

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Appendix

Appendix

As promised, we give the proofs for Propositions 12 and 3 in this section. For that reason, we prepare several notations and lemmas in the following.

Definition 5

Suppose λ: [0, 1] → [0, 1] is a increasing bijection. For t ∈ [0, 1[, define

$$\displaystyle{\theta _{t}\lambda (s) = \left \{\begin{array}{ll} \lambda (t + s) -\lambda (t) &\text{ for }s \in [0,1 - t]\\ 1 -\lambda (t) +\lambda (t + s - 1) &\text{ for } s \in [1 - t, 1]. \end{array} \right.}$$

In fact, we cut the graph of λ at the time t, exchange the first part of the graph with the second part and then glue them together to get an increasing bijection over [0, 1].

Lemma 2

$$\displaystyle{\sup \limits _{s<t}\left \vert \log \frac{\theta _{r}\lambda (t) -\theta _{r}\lambda (s)} {t - s} \right \vert =\sup \limits _{s<t}\left \vert \log \frac{\lambda (t) -\lambda (s)} {t - s} \right \vert.}$$

Proof

Denote by ϕ(λ, s, t) the quantity \(\vert \log \frac{\lambda (t)-\lambda (s)} {t-s} \vert \). We see that for a < b < c,

$$\displaystyle{\max (\phi (\lambda,a,b),\phi (\lambda,b,c)) \geq \phi (\lambda,a,c).}$$

Thus, \(\sup \limits _{s<t}\phi (\lambda,s,t) =\sup \limits _{s<t,t-s\text{ is small}}\phi (\lambda,s,t)\). As a result, \(\sup \limits _{s<t}\vert \log \frac{\lambda (t)-\lambda (s)} {t-s} \vert \) is a function of λ which is invariant under θ t .

Definition 6

For a based loop l of time duration t and r ∈ [0, t[, denote by Θ r the circular translation of l:

$$\displaystyle{\varTheta _{r}(l)(u) = \left \{\begin{array}{ll} l(u + r) &\text{ for }u \in [0,t - r]\\ l(u + r - t) &\text{ for } u \in [t - r, t]. \end{array} \right.}$$

Then, we can extend Θ r for all \(r \in \mathbb{R}\) by periodical extension.

Notice that Θ r (l) is a based loop iff the periodical extension of l is continuous at time r. Nevertheless, we define the distance D(Θ r l, l) in the same way. The next lemma shows the continuity of r → Θ r l at time r when the based loop l is continuous at r.

Lemma 3

Suppose l is a based loop. Then, \(\lim \limits _{h\rightarrow 0}D(\varTheta _{h}l,l) = 0\) .

Proof

Without loss of generality, we can assume l has time duration 1. By definition, we have that

$$\displaystyle\begin{array}{rcl} D(\varTheta _{h}(l),l)& =& d(\varTheta _{h}(l),l) {}\\ & =& \inf \Big\{\sup \limits _{s<t}\left \vert \log \frac{\lambda (t) -\lambda (s)} {t - s} \right \vert +\sup \limits _{u\in [0,1]}d_{S}\left (l(\lambda (u)),\varTheta _{h}(l)(u)\right ): {}\\ & & \lambda \text{ increasing bijection on }[0,1]\Big\}. {}\\ \end{array}$$

Fix 0 < a < b < 1, take λ(0) = 0, λ(a) = a + h, λ(b) = b + h, λ(1) = 1 and linearly interpolate λ elsewhere. Then,

$$\displaystyle\begin{array}{rcl} D(\varTheta _{h}(l),l)& \leq & \max \left (\left \vert \log \frac{a + h} {a} \right \vert,\left \vert \log \frac{1 - b - h} {1 - b} \right \vert \right ) {}\\ & +& 2\sup \limits _{u,v\in [0,a+\vert h\vert ]\cup [b-\vert h\vert,1]}\vert l(u) - l(v)\vert. {}\\ \end{array}$$

Thus, for any 0 < a < b < 1,

$$\displaystyle{\limsup \limits _{h\rightarrow 0}D(\varTheta _{h}(l),l) \leq 2\sup \limits _{u,v\in [0,a]\cup [b,1]}\vert l(u) - l(v)\vert.}$$

Since l is a based loop, \(\inf \limits _{a,b}(\sup \limits _{u,v\in [0,a]\cup [b,1]}\vert l(u) - l(v)\vert ) = 0\). Therefore,

$$\displaystyle{\lim \limits _{h\rightarrow 0}D(\varTheta _{h}l,l) = 0.}$$

Lemma 4

Suppose l 1 is a based loop with time duration t and l 1 is continuous at time r ∈ [0,t[. Then,

$$\displaystyle{\inf \{D(l_{1},l): l \in l_{2}^{o}\} =\inf \{ D(\varTheta _{ r}(l_{1}),l): l \in l_{2}^{o}\}.}$$

Proof

Recall that \(D(l_{1},l) =\Big \vert \vert l\vert -\vert l_{1}\vert \Big\vert + d(l_{1}^{\text{normalized}},l^{\text{normalized}})\) where

$$\displaystyle\begin{array}{rcl} & & d(l_{1}^{\text{normalized}},l^{\text{normalized}}) =\inf \Big\{\sup \limits _{ u\in [0,1]}d_{S}(l_{1}^{\text{normalized}}(u),l^{\text{normalized}}(\lambda (u))) {}\\ & & \qquad \qquad \qquad +\sup \limits _{s<t}\left \vert \log \frac{\lambda (t) -\lambda (s)} {t - s} \right \vert:\lambda \text{ increasing bijection over }[0,1]\Big\}. {}\\ \end{array}$$

Then, for ε > 0, there exists \(l \in l_{2}^{o}\) and λ such that

$$\displaystyle\begin{array}{rcl} & & \sup \limits _{s<t}\left \vert \log \frac{\lambda (t) -\lambda (s)} {t - s} \right \vert +\sup \limits _{u\in [0,1]}d_{S}(l_{1}^{\text{normalized}}(u),l^{\text{normalized}}(\lambda (u))) \\ & & \qquad \qquad \qquad \qquad <\inf \{ D(l_{1},l): l \in l_{2}^{o}\} +\epsilon. {}\end{array}$$
(5)

Since the paths are càdlàg, for fixed l 1 and l, the following set is at most countable:

$$\displaystyle{\{a: l_{1}\text{ jumps at time }a\text{ or }l\text{ jumps at }\vert l\vert \lambda (a/\vert l_{1}\vert )\}.}$$

Thus, we can find a sequence (r n ) n such that

  • \(r_{n} \downarrow r\) as n → ,

  • \(\varTheta _{r_{n}}(l_{1})\) and \(\varTheta _{\vert l\vert \lambda (r_{n}/\vert l_{1}\vert )}(l)\) are both based loops.

By Lemma 2, we have that

$$\displaystyle{ \sup \limits _{s<t}\left \vert \log \frac{\lambda (t) -\lambda (s)} {t - s} \right \vert =\sup \limits _{s<t}\left \vert \log \frac{\theta _{r_{n}/\vert l_{1}\vert }\lambda (t) -\theta _{r_{n}/\vert l_{1}\vert }\lambda (s)} {t - s} \right \vert. }$$
(6)

Meanwhile, we have that

$$\displaystyle\begin{array}{rcl} & & \sup \limits _{u\in [0,1]}d_{S}(l_{1}^{\text{normalized}}(u),l^{\text{normalized}}(\lambda (u))) \\ & & \qquad =\sup \limits _{u\in [0,1]}d_{S}\left ((\varTheta _{r_{n}}l_{1})^{\text{normalized}}(u),(\varTheta _{\vert l\vert \lambda (r_{n}/\vert l_{1}\vert )}l)^{\text{normalized}}(\theta _{r_{n}/\vert l_{1}\vert }\lambda (u))\right ).{}\end{array}$$
(7)

Notice that \(\varTheta _{\vert l\vert \lambda (r_{n}/\vert l_{1}\vert )}l \in l_{2}^{o}\). Thus, by (5)+(6)+(7), for any ε > 0, there exists \((r_{n})_{n}\) with decreasing limit r such that

$$\displaystyle{ \inf \{D(\varTheta _{r_{n}}l_{1},l): l \in l_{2}^{o}\} <\inf \{ D(l_{ 1},l): l \in l_{2}^{o}\} +\epsilon. }$$
(8)

By triangular inequality of D,

$$\displaystyle{D(\varTheta _{r}l_{1},l) \leq D(\varTheta _{r_{n}}l_{1},\varTheta _{r}l_{1}) + D(\varTheta _{r_{n}}l_{1},l).}$$

We take the infimum on both sides, then

$$\displaystyle{ \inf \{D(\varTheta _{r}l_{1},l): l \in l_{2}^{o}\} \leq D(\varTheta _{ r_{n}}l_{1},\varTheta _{r}l_{1}) +\inf \{ D(\varTheta _{r_{n}}l_{1},l): l \in l_{2}^{o}\}. }$$

By (8),

$$\displaystyle{ \inf \{D(\varTheta _{r}l_{1},l): l \in l_{2}^{o}\} \leq D(\varTheta _{ r_{n}}l_{1},\varTheta _{r}l_{1}) +\inf \{ D(l_{1},l): l \in l_{2}^{o}\} +\epsilon. }$$
(9)

By Lemma 3, for the based loop l 1, \(\lim \limits _{n\rightarrow \infty }D(\varTheta _{r_{n}}l_{1},\varTheta _{r}l_{1}) = 0\). By taking n →  in (9), we see that

$$\displaystyle{\inf \{D(\varTheta _{r}l_{1},l): l \in l_{2}^{o}\} \leq \inf \{ D(l_{ 1},l): l \in l_{2}^{o}\} +\epsilon \text{ for all }\epsilon > 0.}$$

Therefore,

$$\displaystyle{\inf \{D(\varTheta _{r}l_{1},l): l \in l_{2}^{o}\} \leq \inf \{ D(l_{ 1},l): l \in l_{2}^{o}\}.}$$

If we replace r by | l 1 | − r and l 1 by \(\varTheta _{r}l_{1}\), we have the inequality in opposite direction:

$$\displaystyle{\inf \{D(\varTheta _{r}l_{1},l): l \in l_{2}^{o}\} \geq \inf \{ D(l_{ 1},l): l \in l_{2}^{o}\}.}$$

Then, we turn to prove Propositions 12 and 3.

Proof (Proof of Proposition 1)

 

  • Reflexivity: straightforward from the definition.

  • Triangular inequality: directly from Lemma 4.

  • \(D^{o}(l_{1}^{o},l_{2}^{o}) = 0\Longrightarrow l_{1}^{o} = l_{2}^{o}\): by Lemma 4, it is enough to show that

    $$\displaystyle{\inf \{D(l_{1},l): l \in l_{2}^{o}\} = 0\Longrightarrow l_{ 1} \in l_{2}^{o}.}$$

    Suppose \(\inf \{D(l_{1},l): l \in l_{2}^{o}\} = 0\). Then, we can find a sequence \((r_{n})_{n}\) with limit r such that \(\lim \limits _{n\rightarrow \infty }D(\varTheta _{r_{n}}l_{2},l_{1}) = 0\). Since \(l_{1}(\vert l_{1}\vert -) = l_{1}(0)\), l 2 must be continuous at r and \(\lim \limits _{n\rightarrow \infty }\varTheta _{r_{n}}l_{2} =\varTheta _{r}l_{2}\) by Lemma 3. Thus, \(l_{1} =\varTheta _{r}l_{2}\).

Proof (Proof of Proposition 2)

 

  • Completeness: given a Cauchy sequence \((l_{n}^{o})_{n}\), one can always extract a sub-sequence \((l_{n_{k}}^{o})_{k}\) such that \(D^{o}(l_{n_{k}}^{o},l_{n_{k+1}}^{o}) < 2^{-k}\). By Lemma 4, one can find in each equivalence class \(l_{n_{k}}^{o}\) a based loop L k such that \(D(L_{k},L_{k+1}) < 2^{-k}\). By the completeness of D, there exists a based loop L such that \(\lim \limits _{k\rightarrow \infty }L_{k} = L\). Thus, \(\lim \limits _{k\rightarrow \infty }l_{n_{k}}^{o} = L^{o}\). So it is the same for \((l_{n}^{o})_{n}\).

  • Separability: the based loop space is separable. Then, as a continuous image, the loop space is separable.

Proof (Proof of Proposition 3)

For any bounded continuous function \(f: S^{n} \rightarrow \mathbb{R}\), \(l \rightarrow \langle l,f\rangle\) is continuous in l. In particular, it is measurable. By monotone class theorem for functions, \(l \rightarrow \langle l,f\rangle\) is measurable for all bounded measurable \(f: S^{n} \rightarrow \mathbb{R}\).

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Chang, Y. (2014). Multi-Occupation Field Generates the Borel-Sigma-Field of Loops. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLVI. Lecture Notes in Mathematics(), vol 2123. Springer, Cham. https://doi.org/10.1007/978-3-319-11970-0_17

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