Multi-occupation field generates the Borel-sigma-field of loops
aa r X i v : . [ m a t h . P R ] D ec Multi-occupation field generates theBorel-sigma-field of loops.
Yinshan Chang
Abstract
In this article, we consider the space of c`adl`ag loops on a Polish space S .The loop space can be equipped with a “Skorokhod” metric. Moreover, it is Polishunder this metric. Our main result is to prove that the Borel- s -field on the space ofloops is generated by a class of loop functionals: the multi-occupation field. Thisresult generalizes the result in the discrete case, see [LJ11]. The Markovian loops have been studied by Le Jan [LJ11] and Sznitman [Szn12].Under reasonable assumptions of the state space, as an application of Blackwell’stheorem, we would like to prove that multi-occupation field generates the Borel- s -field on the space of loops, see Theorem 1. This generalizes the result in [LJ11], seethe paragraph below Proposition 10 in Chapter 2 of [LJ11]. For self-containedness,we introduce several necessary definitions and notations in the following para-graphs.Let ( S , d S ) be a Polish space with the Borel- s -field. As usual, denote by D S ([ , a ]) the Skorokhod space, i.e. the space of c`adl`ag -paths over time interval [ , a ] whichis also left-continuous at time a . We equip it with the Skorokhod metric and thecorresponding Borel- s -field. Definition 1 (Based loop).
A based loop is an element l ∈ D S ([ , t ]) for some t > l ( ) = l ( t ) . We call t the duration of the based loop and denote it by | l | . Yinshan ChangMax Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany, e-mail: [email protected]
This is part of author’s PHD work in Department of Mathematics in Universit´e Paris Sud The terminology “c`adl`ag” is short for right-continuous with left hand limits. 1 Yinshan Chang
Definition 2 (Loop).
We say two based loops are equivalent iff. they are identicalup to some circular translation. A loop is defined as an equivalence class of basedloops. For a based loop l , we denote by l o its equivalence class. Definition 3 (Multi-occupation field/time).
Define the rotation operator r j as fol-lows: r j ( z , . . . , z n ) = ( z + j , . . . , z n , z , . . . , z j ) . For any f : S n → R measurable, definethe multi-occupation field of based loop l of time duration t as h l , f i = n − (cid:229) j = Z < s < ··· < s n < t f ◦ r j ( l ( s ) , . . . , l ( s n )) ds · · · ds n . If l and l are two equivalent based loops, they correspond to the same multi-occupation field. Therefore, the multi-occupation field is well-defined for loops.For discrete S , define the multi-occupation time ˆ l x ,..., x n of a (based) loop to be h l , ( x ,..., x n ) ( · ) i where1 ( x ,..., x n ) (( y , . . . , y n )) = (cid:26) ( y , . . . , y n ) = ( x , . . . , x n ) l and l , they can be normalized to have duration 1 by lineartime scaling. Denote them l normalized1 and l normalized2 . As S is Polish, by Theorem 5.6in [EK86], the Skorokhod space ( D S ([ , ]) , d ) is also Polish under the followingmetric: d ( l , l ) def = inf l sup s < t (cid:12)(cid:12)(cid:12)(cid:12) log l ( t ) − l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) + sup u ∈ [ , ] d S ( l ( l ( u )) , l ( u )) ! (1)where the infimum is taken over all increasing bijections l : [ , ] → [ , ] . Then, itis straightforward to see that the space of based loops under the following metric D is also Polish: D ( l , l ) def = (cid:12)(cid:12)(cid:12) | l | − | l | (cid:12)(cid:12)(cid:12) + d ( l normalized1 , l normalized2 ) . Definition 4 (Distance on loops).
Define the distance D o of two loops l o and l o by D o ( l o , l o ) def = inf { D ( l , l ′ ) : l ∈ l o and l ′ ∈ l o } . Remark 1.
This is not the standard way to define a pseudo metric on quotient space.In general, the above definition does not satisfy the triangular inequality. In thisspecial situation, the distance D is in fact invariant under suitable circular translationwhich guarantees the triangular inequality.We provide the proofs of the following three propositions in Section Appendix. ulti-occupation field generates the Borel-sigma-field of loops. 3 Proposition 1.
The above distance D o is well-defined. Proposition 2.
The loop space is Polish under the metric D o . Then, we equip the loop space with the Borel- s -field. The next proposition statesthe measurability of the multi-occupation field. Proposition 3.
Fix any bounded Borel measurable function f on S n , the followingmap is Borel measurable functional on the loop space:l → h l , f i . Our main result is the following theorem.
Theorem 1.
The Borel- s -field on the loops is generated by the multi-occupationfield if ( S , d S ) is Polish. We will prove the main theorem in this section as an application of the followingBlackwell’s theorem.
Theorem 2 (Blackwell’s theorem, Theorem 26, Chapter III of [DM78]).
Sup-pose ( E , E ) is a Blackwell space, S , F are sub- s -field of E and S is separable.Then F ⊂ S iff every atom of F is a union of atoms of S . As a consequence, we have the following lemma.
Lemma 1.
Suppose ( E , B ( E )) is a Polish space with the Borel- s -field. Let { f i , i ∈ N } be measurable functions and denote F = s ( f i , i ∈ N ) . Then, F = B ( E ) iff forall x = y ∈ E, there exists f i such that f i ( x ) = f i ( y ) .Proof. Since E is Polish, B ( E ) is separable and ( E , B ( E )) is Blackwell space. Theatoms of B ( E ) are all the one point sets. Obviously, F ⊂ B ( E ) and F is separable.By Blackwell’s theorem, F = B ( E ) iff. the atoms of F are all the one point setswhich is equivalent to the following: for all x = y ∈ E , there exists f i such that f i ( x ) = f i ( y ) .Then, we are ready for the proof of the main theorem.From the definition of the multi-occupation field, any loop defines a finite mea-sure on S n for all n ∈ N + . Let B = ( B i , i ∈ N ) be a countable topological ba-sis of S . The s − field generated by the multi-occupation field must equal to the s − field generated by the following countable functionals {h· , B i : B ∈ ¥ S k = B k } . The countability is required by Lemma 1. Yinshan Chang
In fact, if two loop l and l are the same under these countable loop functionals {h· , B i : B ∈ ¥ S k = B k } , they must agree on all the functionals of the form h· , f i . ByLemma 1, it remains to check that two loops with the same occupation field are thesame loop.Suppose loops l o and l o have the same occupation field, i.e. h l o , f i = h l o , f i for allpositive f on some S n ( n ∈ N + ) . Recall that a loop is an equivalence class of basedloops. Take two based loops l , l in the equivalence class l o , l o respectively. Then, h l , f i = h l o , f i = h l o , f i = h l , f i . Define m ( A ) = h l , A i and m ( A ) = h l , A i for A ∈ B ( S ) . Then, we have m = m which means that the time spent in some Borelmeasurable set is the same for these two loops. In particular, the two (based) loopshave the same time duration, say t . For simplicity of the notations, we will use m instead of m and m . Now, we are ready to show that l o = l o in three steps. Letus present the sketch of the proof before providing the details. We first decomposethe space into an approximate partition which is used in [LJQ13]. Next, we replacearcs of trajectory in each part by a single point with corresponding holding times. Inthis way, we get two loops in the same discrete space. By the construction of thesediscrete loops, their multi-occupation fields coincide. It is known that Theorem 1is true for loops in discrete space. Thus, these two modified loops in the discretespace are exactly the same. Moreover, when the rough partition is small enough,these modified loops are actually good approximation of the original loops l o and l o in the sense of Skorokhod. For that reason, we conclude in the last step that l o = l o .I. For all e > U e i satisfying the follow-ing properties:– their boundaries are negligible with respect to m ,– they have positive distances from each other,– the complement of their union has mass smaller than 2 e with respect to m ,– their diameters are smaller than e .Let U e be the union of ( U e i ) i .Actually, the rough partitions ( U e i ) i are chosen in the following way. It is well-known that every finite measure on the Borel- s -field of a Polish space is regular.Therefore, for e >
0, we can find some compact set K e such that m ( K c e ) < e where K c e is the complement of K e . Let D = { x , · · · , x n , · · · } be a countabledense subset of S . Fix any x ∈ S , except for countable many r ∈ R + , the mea-sure m does not charge the boundary ¶ ( B ( x , r )) of the ball B ( x , r ) . Then, forany e >
0, there exists a collection of open ball B ( x i , r i ) with radius smallerthan e such that their boundaries are negligible with respect to m . Then, theycover the compact set K e as D is dense in S . Therefore, we can extract a finiteopen covering { B ( y , r ) , · · · , B ( y k , r k ) } . These open balls cut the whole space S into a finite partition of the space S \ S i ¶ ( B ( y i , r i )) : P , · · · , P q open set with ulti-occupation field generates the Borel-sigma-field of loops. 5 P = ( S i B ( y i , r i )) c . Let U e i , d = { y ∈ S : d S ( y , P ci ) > d } which is contained in P i . Infact, one can always choose some d small enough and good enough such thatthe boundary sets { y ∈ S : d S ( y , P ci ) = d } of these open sets are negligible under m . Moreover, m ( S \ ( S i U e i , d ∩ K e )) < e . Set U e i = U e i , d , i = , · · · , q . Then, theysatisfy the desired properties stated above.II. From the based loop l j ( j = , ) , we will construct two piecewise-constant basedloops l e j ( j = , ) with finitely many jumps such that l e and l e are the same inthe sense of loop and that they are quite close to the trace of l and l on U e respectively.To be more precise, define A e j , u = u R { l j ( s ) ∈ U e } ds with the convention that l j ( s + kt ) = l j ( s ) for s ∈ [ , t ] and k ∈ Z where t is the time duration of the based loops.Then, ( A e j , u , u ∈ R + ) is right-continuous and increasing for j = ,
2. Let ( s e j , s , s ∈ R + ) be the right-continuous inverse of ( A e j , u , u ∈ R + ) for j = , s e j , s = inf { s ∈ R + : A e j , u > s } . Let t e = A e , t = A e , t = m ( U e ) to be the total occupation time of the loops within U e . Then, A e j , u + kt = kt e + A e j , u and A e j , u ≤ u for u ∈ R + . Thus, s e j , s + kt e = s e j , s + kt for k ∈ N , s ∈ [ , t e [ and s e j , s ≥ s for s ∈ R + . Moreover, as e ↓ ( s e j , s , s ∈ R + ) de-creases to ( s , s ∈ R + ) uniformly on any compact of R + . We know that l j ( s e j , s ) ∈ S i U e i . We choose in each U e i a point y i and define l e j ( s ) = y i iff. l j ( s e j , s ) ∈ U e i for j = ,
2. Then, as the diameters of ( U e i ) i are less than e , sup s d S ( l e j ( s ) , l j ( s e j , s )) ≤ e for j = ,
2. Moreover, s → l e j ( s ) is c`adl`ag for j = ,
2. Since the based loops l and l are c`adl`ag and all the U e i have a positive distance from each other, s → l e j ( s ) has finitely many jumps in any finite time interval for j = ,
2. Then, ( l e j ( s ) , s ∈ [ , t e ]) o is a loop on the same finite state space for j = , U e i in negligible with respect to m , by Lebesgue’s changeof measure formula, (( l e j ( s ) , s ∈ [ , t e ]) o ) y i , ··· , y in = h l j , U e i · · · U e in i for j = , . Therefore, ( l e ( s ) , s ∈ [ , t e ]) o and ( l e ( s ) , s ∈ [ , t e ]) o have the same multi-occupationfield. It is known that Theorem 1 is true for loops in finite discrete space, see theparagraph below Proposition 10 in Chapter 2 of [LJ11]. Thus, ( l e ( s ) , s ∈ [ , t e ]) o = ( l e ( s ) , s ∈ [ , t e ]) o . Consequently, there exists some T ( e ) ∈ [ , t e [ such that l e ( s + T ) = l e ( s ) for s ≥ l = l up to circular translation. Yinshan Chang
We can find a sequence ( e k ) k with lim k → ¥ e k = T ( e k ) converges to T ∈ [ , t ] as k → ¥ . Then, lim k → ¥ s e k , s + T = s + T for fixed s ≥
0. Accordingly,lim k → ¥ min { d S ( l ( s e k , s + T ) , l ( s + T )) , d S ( l ( s e k , s + T ) , l (( s + T ) − )) } = . (2)On the other hand, we have lim k → ¥ s e k , s = s and s e k , s ≥ s for all k ∈ N . Therefore, bythe right continuity of l ,lim k → ¥ d S ( l ( s e k , s ) , l ( s )) = d S ( l ( s +) , l ( s )) = . (3)From the constructions of l e and l e and the argument in part II, we see thatsup s d S ( l ( s e k , s + T ) , l ( s e k , s )) ≤ sup s d S ( l e ( s + T ) , l ( s e k , s + T ))+ sup s d S ( l e ( s ) , l ( s e k , s )) ≤ e k . (4)As a result, by (2) + (3) + (4), for any s ≥
0, either d S ( l ( s + T ) , l ( s )) = d S ( l (( s + T ) − ) , l ( s )) =
0. Finally, by right-continuity of the paths l and l , l ( s + T ) = l ( s ) and the proof is complete. Appendix
As promised, we give the proofs for Proposition 1, 2 and 3 in this section. For thatreason, we prepare several notations and lemmas in the following.
Definition 5.
Suppose l : [ , ] → [ , ] is a increasing bijection. For t ∈ [ , [ , define q t l ( s ) = (cid:26) l ( t + s ) − l ( t ) for s ∈ [ , − t ] − l ( t ) + l ( t + s − ) for s ∈ [ − t , ] . In fact, we cut the graph of l at the time t , exchange the first part of the graph withthe second part and then glue them together to get an increasing bijection over [ , ] . Lemma 2. sup s < t (cid:12)(cid:12)(cid:12)(cid:12) log q r l ( t ) − q r l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) = sup s < t (cid:12)(cid:12)(cid:12)(cid:12) log l ( t ) − l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) . ulti-occupation field generates the Borel-sigma-field of loops. 7 Proof.
Denote by f ( l , s , t ) the quantity | log l ( t ) − l ( s ) t − s | . We see thatmax ( f ( l , a , b ) , f ( l , b , c )) ≥ f ( l , a , c ) . Thus, sup s < t f ( l , s , t ) = sup s < t , t − s is small f ( l , s , t ) . As a result, sup s < t | log l ( t ) − l ( s ) t − s | is a func-tion of l which is invariant under q t . Definition 6.
For a based loop l of time duration t and r ∈ [ , t [ , denote by Q r thecircular translation of l : Q r ( l )( u ) = (cid:26) l ( u + r ) for u ∈ [ , t − r ] l ( u + r − t ) for u ∈ [ t − r , t ] . Then, we can extend Q r for all r ∈ R by periodical extension.Notice that Q r ( l ) is a based loop iff. the periodical extension of l is continuousat time r . Nevertheless, we define the distance D ( Q r l , l ) in the same way. The nextlemma shows the continuity of r → Q r l at time r when the based loop l is continuousat r . Lemma 3.
Suppose l is a based loop. Then, lim h → D ( Q h l , l ) = .Proof. Without loss of generality, we can assume l has time duration 1. By defini-tion, we have that D ( Q h ( l ) , l ) = d ( Q h ( l ) , l )= inf n sup s < t (cid:12)(cid:12)(cid:12)(cid:12) log l ( t ) − l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) + sup u ∈ [ , ] d S ( l ( l ( u )) , Q h ( l )) : l increasing bijection on [ , ] o . Fix 0 < a < b <
1, take l ( ) = , l ( a ) = a + h , l ( b ) = b + h , l ( ) = l elsewhere. Then, D ( Q h ( l ) , l ) ≤ max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) log a + ha (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) log 1 − b − h − b (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + u , v ∈ [ , a + | h | ] ∪ [ b −| h | , ] | l ( u ) − l ( v ) | . Thus, for any 0 < a < b < h → D ( Q h ( l ) , l ) ≤ u , v ∈ [ , a ] ∪ [ b , ] | l ( u ) − l ( v ) | . Since l is a based loop, inf a , b ( sup u , v ∈ [ , a ] ∪ [ b , ] | l ( u ) − l ( v ) | ) =
0. Therefore,lim h → D ( Q h l , l ) = . Yinshan Chang
Lemma 4.
Suppose l is a based loop with time duration t and l is continuous attime r ∈ [ , t [ . Then, inf { D ( l , l ) : l ∈ l o } = inf { D ( Q r ( l ) , l ) : l ∈ l o } . Proof.
Recall that D ( l , l ) = (cid:12)(cid:12)(cid:12) | l | − | l | (cid:12)(cid:12)(cid:12) + d ( l normalized1 , l normalized ) where d ( l normalized1 , l normalized ) = inf n sup u ∈ [ , ] d S ( l normalized1 ( u ) , l normalized ( l ( u )))+ sup s < t (cid:12)(cid:12)(cid:12)(cid:12) log l ( t ) − l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) : l increasing bijection over [ , ] o . Then, for e >
0, there exists l ∈ l o and l such thatsup s < t (cid:12)(cid:12)(cid:12)(cid:12) log l ( t ) − l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) + sup u ∈ [ , ] d S ( l normalized1 ( u ) , l normalized ( l ( u ))) < inf { D ( l , l ) : l ∈ l o } + e . (5)Since the paths are c`adl`ag, the following set is at most countable: { a : l jumps at time a or l jumps at | l | l ( a / | l | ) } . Thus, we can find a sequence ( r n ) n such that • r n ↓ r as n → ¥ , • Q r n ( l ) and Q | l | l ( r n / | l | ) ( l ) are both based loops.By Lemma 2, we have thatsup s < t (cid:12)(cid:12)(cid:12)(cid:12) log l ( t ) − l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) = sup s < t (cid:12)(cid:12)(cid:12)(cid:12) log q r n / | l | l ( t ) − q r n / | l | l ( s ) t − s (cid:12)(cid:12)(cid:12)(cid:12) . (6)Meanwhile, we have thatsup u ∈ [ , ] d S ( l normalized1 ( u ) , l normalized ( l ( u )))= sup u ∈ [ , ] d S (cid:16) ( Q r n l ) normalized ( u ) , ( Q | l | l ( r n / | l | ) l ) normalized ( q r n / | l | l ( u )) (cid:17) . (7)Notice that Q | l | l ( r n / | l | ) l ∈ l o . Thus, by (5)+(6)+(7), for any e >
0, there exists ( r n ) n with decreasing limit r such thatinf { D ( Q r n l , l ) : l ∈ l o } < inf { D ( l , l ) : l ∈ l o } + e . (8)By triangular inequality of D , D ( Q r l , l ) ≤ D ( Q r n l , Q r l ) + D ( Q r n l , l ) . ulti-occupation field generates the Borel-sigma-field of loops. 9 We take the infimum on both sides, theninf { D ( Q r l , l ) : l ∈ l o } ≤ D ( Q r n l , Q r l ) + inf { D ( Q r n l , l ) : l ∈ l o } . By (8), inf { D ( Q r l , l ) : l ∈ l o } ≤ D ( Q r n l , Q r l ) + inf { D ( l , l ) : l ∈ l o } + e . (9)By Lemma 3, for based loop l , lim n → ¥ D ( Q r n l , Q r l ) =
0. By taking n → ¥ in (9), wesee that inf { D ( Q r l , l ) : l ∈ l o } ≤ inf { D ( l , l ) : l ∈ l o } + e for all e > . Therefore, inf { D ( Q r l , l ) : l ∈ l o } ≤ inf { D ( l , l ) : l ∈ l o } . If we replace l by Q r l and r by | l |− r , we have the inequality in opposite direction:inf { D ( Q r l , l ) : l ∈ l o } ≥ inf { D ( l , l ) : l ∈ l o } . Then, we turn to prove Proposition 1, 2 and 3.
Proof (Proof of Proposition 1). • Reflexivity: straightforward from the definition. • Triangular inequality: directly from Lemma 4. • D o ( l o , l o ) = = ⇒ l o = l o : by Lemma 4, it is enough to show thatinf { D ( l , l ) : l ∈ l o } = = ⇒ l ∈ l o . Suppose inf { D ( l , l ) : l ∈ l o } =
0. Then, we can find a sequence ( r n ) n with limit r such that lim n → ¥ D ( Q r n l , l ) =
0. Since l ( | l |− ) = l ( ) , l must be continuousat r and lim n → ¥ Q r n l = Q r l by Lemma 3. Thus, l = Q r l . Proof (Proof of Proposition 2). • Completeness: given a Cauchy sequence ( l on ) n , one can always extract a sub-sequence ( l on k ) k such that D o ( l on k , l on k + ) < − k . By Lemma 4, one can find ineach equivalence class l on k a based loop L k such that D ( L k , L k + ) < − k . Bythe completeness of D , there exists a based loop L such that lim k → ¥ L k = L . Thus,lim k → ¥ l on k = L o . So is the same for ( l on ) n . • Separability: the based loop space is separable. Then, as the continuous image,the loop space is separable.
Proof (Proof of Proposition 3).
For bounded continuous function f : S n → R , l → h l , f i is continuous in l . In particular, it is measurable. By p − l theorem forfunctions, l → h l , f i is measurable for all bounded measurable f : S n → R . References
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