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
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- 1.
The terminology “càdlàg” is short for right-continuous with left hand limits.
- 2.
The countability is required by Lemma 1.
References
C. Dellacherie, P-A. Meyer, Probabilities and Potential, vol. 29 of North-Holland Mathematics Studies (North-Holland Publishing, Amsterdam, 1978)
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
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]
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)
A-S. Sznitman, Topics in Occupation Times and Gaussian Free Fields. Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2012)
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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Appendix
Appendix
As promised, we give the proofs for Propositions 1, 2 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
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
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,
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:
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
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,
Thus, for any 0 < a < b < 1,
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,
Lemma 4
Suppose l 1 is a based loop with time duration t and l 1 is continuous at time r ∈ [0,t[. Then,
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
Then, for ε > 0, there exists \(l \in l_{2}^{o}\) and λ such that
Since the paths are càdlàg, for fixed l 1 and l, the following set is at most countable:
Thus, we can find a sequence (r n ) n such that
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\(r_{n} \downarrow r\) 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
Meanwhile, we have that
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
By triangular inequality of D,
We take the infimum on both sides, then
By (8),
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
Therefore,
If we replace r by | l 1 | − r and l 1 by \(\varTheta _{r}l_{1}\), we have the inequality in opposite direction:
Then, we turn to prove Propositions 1, 2 and 3.
Proof (Proof of Proposition 1)
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Reflexivity: straightforward from the definition.
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Triangular inequality: directly from Lemma 4.
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\(D^{o}(l_{1}^{o},l_{2}^{o}) = 0\Longrightarrow l_{1}^{o} = l_{2}^{o}\): 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)
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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 \((l_{n}^{o})_{n}\).
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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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