Let ( T,τ ) be a nonvoid sigma-compact connected topological space. A nonempty class A of compact connected subsets of T is called an indexing collection if it satisfies the following: (1)

∅∈A . In addition, there is an increasing sequence ( Bn ) of sets in A such that T=⋃n=1∞Bn∘ .

(a) Note that any collection of sets closed under intersections is a semilattice with respect to the partial order of the inclusion.

(b) Definition 0 implies that a space T cannot be discrete and that A is at least a continuum.

(a) The classical example is T=ℜ+2 and A=A(ℜ+2)={[0,x]:x∈ℜ+2} (This example can be extended to T=ℜ+d and A(ℜ+d)={[0,x]:x∈ℜ+d} , which will give rise to a sort of 2d -sides process).

(b) The example (a) may be generalized as follows. Let T=ℜ+2 and take A (or A(Ls)) to be the class of compact lower sets, i.e. the class of compact subsets A of T satisfying t∈A implies [0,t]⊆A .

Let (Ω,F,P) be a complete probability space equipped with an A-indexed filtration {FA:A∈A} that satisfies the following conditions: (i)

for all A∈A , we have FA⊆F , and FA contains the P-null sets.

A set-indexed stochastic process X={XA:A∈A} is additive if it has an (almost sure) additive extension to C: X∅=0 , and if C,C1,C2∈C with C=C1∪C2 and C1∩C2=∅ , then almost surely XC1+XC2=XC . In particular, if C∈C and C=A∖⋃i=1nAi , A,A1,…,An∈A , then almost surely XC=XA−∑i=1nXA∩Ai+∑i<jXA∩Ai∩Aj−⋯+(−1)nXA∩⋂i=1nAi .

Let (G,·) be a group. The group G will be called a permutation group on [a,b] if G={π:[a,b]→[a,b]∣π is a one-to-one and onto function}, and we denote this group by S[a,b] (i.e., S[a,b] is the class of all the bijection functions from [a,b] to [a,b] ).

A positive measure σ on ( T,B ) is called strictly monotone on A if σ∅′=0 and σA<σB for all A,B∈A such that A⊊B . The collection of these measures is denoted by M(A) .

Let σ∈M(A) , and let (G,·) be a group. A group action ∗ of (G,·) on A is defined by g∗(A∪B)=g∗A∪g∗B , g∗(A∖B)=g∗A∖g∗B for all A,B∈A and g∈G , and there exists η:G→ℜ+ such that σ(g∗A)=η(g)σ(A) for all A∈A and g∈G .

Let I⊆ℜ , and let A={Aα}α∈I be increasing sequence in A(u) . (a)

The sequence A is called “strictly increasing” if Aα⊊Aβ for all α,β∈I such that α<β .

Given a set-indexed stochastic process X and increasing sequence {Aα}α∈[a,b] in A(u) , we define a process Y indexed by [a,b] as follows: Ys=XAs=XsA for all s∈[a,b] .

A set-indexed stochastic process X is called outer-continuous if X is finitely additive on C and for any decreasing sequence {An}∈A , X⋂nAn=limnXAn and is called inner-continuous if for any increasing sequence {An}∈A such that ⋃nAn‾=A∈A , XA=limnXAn .

Let A′={∅′=A0,…,Ak} be any finite subsemilattice of A equipped with a numbering consistent with the strong past. Then there exists a continuous (strictly) flow f:[0,k]→A(u) such that the following are satisfied: (i)

f(0)=∅′ , f(k)=⋃j=0kAj .

Let A′={∅′=A0,…,Ak} be any finite subsemilattice of A equipped with a numbering consistent with the strong past. (a)

Then there exists a strictly increasing and continuous sequence B(k)={Bα(k)}α∈[0,k] in A(u) such that the following are satisfied:

(a) It is clear from Lemma 1 by setting Bi(k)=f(i) , 1≤i≤k .

(b) Notice that for each k, B(k)=B(k+1) on [0,k] . Then we can define the sequence B={Bα}α∈[0,∞) in A(u) by Bα=Bα(α+1) for all α.  □

Similarly to the construction performed in Lemma 2, we can prove that for all increasing sequences {Bn}n=1∞∈A(u) , there exists a strictly increasing and continuous sequence {Aα}α∈[0,∞) in A(u) such that An=Bn .

Let σ∈M(A) . We say that an A-indexed process X is a Brownian motion with variance σ if X can be extended to a finitely additive process on C(u) and if for disjoint sets C1,…,Cn∈C , XC1,…,XCn are independent zero-mean Gaussian random variables with variances σC1,…,σCn , respectively.

For any σ∈M(A) , there exists a set-indexed Brownian motion with variance σ [6].

(a) Let X={Xt:t≥0} be a stochastic process, and let ∗ be a group action of (G,·) on ℜ+ . The process X is called a G-time-changed Brownian motion if there exists g∈G such that Xg={Xg∗t:t≥0} is a Brownian motion.

(b) Let X={XA:A∈A} be a set-indexed process, {Aα}α∈[0,∞) be an increasing sequence in A(u) , and ∗ be a group action of (S[0,∞),∘) on ℜ+ . The process XA (see Definition 6) is called a G-time-changed Brownian motion if there exists π∈S[0,∞) such that Xπ,A={Xπ∗Aα:α∈[0,∞)}={XAπ(α):α∈[0,∞)} is a Brownian motion.

Let X={XA:A∈A} be a square-integrable set-indexed stochastic process. Suppose that there exists a group action ∗ of (S[0,∞),∘) on A. Let σ∈M(A) . Then X is a set-indexed Brownian motion with variance σ if and only if the process XA={XAα:α∈[0,∞)} is an S[0,∞) -time-changed Brownian motion for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) . In other words (by Definition 8), for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) , there exists π∈S[0,∞) such that Xπ,A={Xπ∗Aα:α∈[0,∞)}={XAπ(α):α∈[0,∞)} is a Brownian motion.

(if) Suppose that X is a set-indexed Brownian motion with variance σ. Define θ:ℜ+→ℜ+ by θ(α)=σ(Aα) , α∈[0,∞) . The function θ is strictly increasing and continuous because A is strictly increasing and continuous. Since σ∈M(A) , θ is invertible. Let π(α)=θ−1(α) ; π is continuous, and σ(Aπ(α))=α . Then π∈S[0,∞) , and Xπ,A={Xπ∗Aα:α∈[0,∞)}={XAπ(α):α∈[0,∞)} is a Brownian motion.

(only if) Suppose that for all strictly continuous sequences {Aα}α∈[0,∞) in A(u) , there exists π∈S[0,∞) such that Xπ,A is a Brownian motion. It must be shown that if {C1,…,Ck}∈C are disjoint, then XC1,…,XCk are independent normal random with variances σ(C1),…,σ(Ck) , respectively. Without loss of generality, we may assume that the sets {C1,…,Ck} are the left neighborhoods of the subsemilattice A′ of A equipped with a numbering consistent with the strong past. By Lemma 2 there exists a strictly increasing and continuous sequence {Aα}α∈[0,∞) in A(u) such that each left neighborhood generated by A′ is of the form Ci=Ai∖Ai−1 , 1≤i≤k . Thus, XA is an S[0,∞] -time-changed Brownian motion such that XCi=XAi−XAi−1 and σ(Ci)=σ(Ai)−σ(Ai−1) ; therefore, XC1,…,XCk are independent normal random with variances σ(C1),…,σ(Ck) , respectively.  □

Let X={XA:A∈A} be a square-integrable set-indexed stochastic process with X∅′=0 that is inner- and outer-continuous. Let σ∈M(A) . Then X is a set-indexed Brownian motion with variance σ if and only if for all strictly continuous sequences {Aα}α∈[0,∞) in A(u) , the process XA has independent increments and there exists π∈S[0,∞) such that Xπ,A={XAπ(α):α∈[0,∞)} has stationary increments. (The definition and more details about independent increments and stationary increments can be found in [6].)

(if) Obvious.

(only if) Suppose that for all strictly continuous sequences {Aα}α∈[0,∞) in A(u) , the process XA has independent increments and there exists π∈S[0,∞) such that Xπ,A={XAπ(α):α∈[0,∞)} has stationary increments. Since X is inner- and outer-continuous, XA is continuous (see [6]). The process XA has independent increments, and there exists π∈S[0,∞) such that Xπ,A has stationary increments; therefore, XA is an S[0,∞) -time-changed Brownian motion for all strictly continuous sequences {Aα}α∈[0,∞) in A(u) . Thus, from Theorem 1 we conclude that X is a set-indexed Brownian motion with variance σ.  □

Let X={XA:A∈A} be an integrable additive set-indexed stochastic process adapted with respect to filtration F={FA:A∈A} . The process X is said to be: 1.

A C-strong martingale (or in short notation, a strong martingale) if for all C∈C , we have E[XC|GC∗]=0 ;

Under some hypotheses, we can define ⟨X⟩ to be the compensator associated with the submartingale X2 . The definition and more details regarding ⟨X⟩ can be found in [6, 2].

Let X={XA:A∈A} be a square-integrable set-indexed martingale with X∅′=0 that is inner- and outer-continuous. Let σ∈M(A) . Then X is a set-indexed Brownian motion with variance σ if and only if ⟨XA⟩ is deterministic for all strictly increasing and continuous sequence {Aα}α∈[0,∞) in A(u) .

(if) Suppose that X is a set-indexed Brownian motion with variance σ. By Theorem 1 the process XA is an S[0,∞) -time-changed Brownian motion for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) . Then from the Lévy characterization we conclude that ⟨XA⟩ is deterministic for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) .

(only if) Suppose that ⟨XA⟩ is deterministic for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) . Since X is inner- and outer-continuous, XA is continuous (see [6]). Since X is a set-indexed martingale, XA is a martingale. But if the process XA is a martingale and ⟨XA⟩ is deterministic, then based on the Lévy characterization we get that XA is an S[0,∞) -time-changed Brownian motion for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) . Thus, from Theorem 1 we conclude that X is a set-indexed Brownian motion with variance σ.  □

Let X={XA:A∈A(ℜ+2)} be a set-indexed Brownian motion with variance σ (Lebesgue measure). Then P[La⪯A]=2−2Φ(aσA)for allA∈A(ℜ+2) (Φ is the standard Gaussian distribution function).

Let A∈A(ℜ+2) . Then P[XA≥a]=P[XA≥a|A≺La]P[A≺La]+P[XA≥a|A⪰La]P[A⪰La] . From the definition of La we conclude that if A≺La , then XA<a , and thus P[XA≥a|A≺La]=0 . It is clear that there exist a strictly increasing and continuous sequence {Bα}α∈[0,∞) in A(u) and α0≥0 such that Bα0= A. The sequence B={Bα}α∈[0,∞) is strictly continuous; therefore, from Theorem 1 we conclude that XB is a G-time-changed Brownian motion. (In other words, there exists π∈S[0,∞) such that Xπ,B is a Brownian motion.) By the symmetry of Xπ,B it is clear that P[XA≥a|A⪰La]=P[Xα0π,B≥a|A⪰La]=P[XBπ(α0)≥a|A⪰La]=12 , and thus P[La⪯A]=2P[XA≥a]=2−2Φ(aσA) .  □

Let X={XA:A∈A(ℜ+2)} be a set-indexed Brownian motion with variance σ (Lebesgue measure). Then, for A∈A , WA=XA,A≺La2a−XA,A⪰La is a set-indexed Brownian motion with variance σ.

We must show that if {C1,…,Ck} are disjoint, then WC1,…,WCk are independent normal random variables with variances σC1,…,σCk , respectively. Similarly to the construction done in Theorem 1, we can get that there exists a strictly increasing and continuous sequence {Aα}α∈[0,∞) in A(u) such that Ci=Ai∖Ai−1 . The sequence {Aα}α∈[0,∞) is strictly continuous; therefore, from Theorem 1 we conclude that there exists π∈S[0,∞) such that Xπ,A is a Brownian motion. Clearly, there exists αa≥0 such that Aπ(αa)∈La . We recall that if X={Xt:t≥0} is be a classical Brownian motion and Ta=inf{t≥0:Xt=a} , then Zt=Xt,t<Ta2a−Xt,t≥Ta(t≥0) is a Brownian motion [4]. Thus, if we define Wα=Xαπ,A,α<αa2a−Xαπ,A,α≥αa, then Wα turns out to be a Brownian motion, and so WC1,…,WCk are independent normal random variables with variances σC1,…,σCk , respectively.  □

Let X={XA:A∈A(ℜ+2)} be a set-indexed Brownian motion with variance σ (Lebesgue measure). If T(a,b)=⋂A∈D(a,b)A , then P[XT(a,b)=b]=|a|b+|a| .

It is clear that T(a,b)∈A(ℜ+2) . It is easy to see that there exists a strictly increasing and continuous sequence {Aα}α∈[0,∞) in A(u) and there exists 0≤β such that Aβ= T(a,b) . The sequence {Aα}α∈[0,∞) is strictly increasing and continuous; therefore, from Theorem 1 we conclude that XA is an S[0,∞) -time-changed Brownian motion. (In other words, there exists a π∈S[0,∞) such that Xπ,A={XAπ(α):α∈[0,∞)} is a Brownian motion, and there exists 0≤α(a,b) such that Aβ=Aπ(α(a,b))=T(a,b) ). We recall that if X={Xt:t≥0} is a Brownian motion and t(a,b):=inft≥0{Xt∉(a,b)} for a<0<b , then P[Xt(a,b)=b]=|a|b+|a| . Thus, P[XT(a,b)=b]=P[Xα(a,b)π,A=b]=|a|b+|a| .  □

Let σ∈M(A) , and let X={XA:A∈A} be a set-indexed Brownian motion with variance σ. Then supA∈AXA(ω)=+∞ and infA∈AXA(ω)=−∞ for almost all ω.

Clearly, we have supα≥0XAα(ω)≤supA∈AXA(ω) ( infα≥0XAα(ω)≥infA∈AXA(ω)) for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) . By Theorem 1 the process XA is an S[0,∞) -time-changed Brownian motion for all strictly increasing and continuous sequences {Aα}α∈[0,∞) in A(u) ; therefore, +∞=supα≥0XAπ(α)(ω)≤supA∈AXA(ω) for almost all ω ( −∞infα≥0XAπ(α)(ω)≥infA∈AXA(ω) for almost all ω).  □

Let σ∈M(A) , and let X={XA:A∈A} be a set-indexed Brownian motion with variance σ. Then (a)

lim_A↑TXAσA=0 for almost all ω, and for all sequences An↑T , {An}∈A .

Let An↑T . By Theorem 1 and Remark 1 there exists a strictly increasing and continuous sequence {Bα}α∈[0,∞)∈A(u) such that XB is an S[0,∞) -time-changed Brownian motion when An=Bn . (In other words, there exist a strictly increasing and continuous sequence {Bα}α∈[0,∞)∈A(u) and {αn}∈[0,∞) such that XB,π is a Brownian motion when An=Bn=Bπ(αn) and π∈S[0,∞) ). Then (a)