the L´evy Transformation of Brownian Mo-
tion
In this self-contained section we construct an internal transformation τ on a hyperfinite dimensional Euclidean space and show that τ is ergodic if and only if L is ergodic, where L denotes the L´evy transformation of Brownian motion. As, in opposition to L, the mapping τ is constructively given, this approach allows to see the open problem whetherLis ergodic from a different angle.
LetbC[0,1] be the Borelσ-algebra onC[0,1] and let W be Wiener-measure on
bC[0,1]. Furthermore let ψ be the canonical Brownian motion on the classical Wiener-space (C[0,1],bC[0,1], W), i.e.
ψ :C[0,1]×[0,1]3(x, t)7→x(t) .
Define a measure-preserving transformationL by
C[0,1]3ω 7→L(ω) := Z ·
0
sgn(ψ)d ψ(ω)∈C[0,1],
where the sgn-mapping is defined as usual by
sgn : R3x7→ 1 if x >0 0 if x= 0 −1 if x <0 .
The map L is referred to as theL´evy transformation of Brownian mo- tion. It is an open problem whether L is ergodic, i.e. whether for each
A∈bC[0,1] the implication
A isL-invariant =⇒ W(A)∈ {0,1}
is valid, where A is called L-invariant ifW(A4L−1(A)) = 0. This question is mentioned for example in [3], [10], [22], [28], [30] and [34]. In [11], Du- bins, ´Emery and Yor have introduced a condition in terms of time-changing a Brownian motion and show that this condition is equivalent toLbeing er- godic; in [12], Dubins and Smorodinsky have proven that a discrete analogue of L is ergodic.
There is an infinitesimal approach to the Malliavin calculus (on the classical Wiener space) due to Cutland, Ng (see [8]) and Osswald (see [20]) that
replacesC[0,1] by a hyperfinite dimensional Euclidean space. Our aim is to search for a transformation in this internal setting which corresponds toL. We want the internal counterpart of the mapping sgn, denoted by Sgn, to take only values in {−1,1} and therefore define
Sgn : ∗R3x7→
1 if x≥0
−1 if x <0 .
LetH ∈∗Nbeunlimited, i.e. H > nfor everyn ∈N. DefineT :={1, .., H} and Ω := ∗RH. For k ∈T let
πk : Ω3X = (X1, ..., XH)7→Xk
be the projection onto the k-th component. Let bΩ be the internal Borel
∗σ-algebra on Ω. There is exactly one internal probability measure such that
the projections πk are independent and normal distributed with mean 0 and
variance H1. Denote this measure by Γ. This construction of Cutland and Ng gives rise to an internal probability space (Ω,bΩ,Γ). Define τ : Ω→Ω by
τ(X)t := Xt if t= 1 Sgn(Pt−1 i=1Xi)Xt if t∈ {2, ..., H} .
14.1
Lemma
The mapping τ : (Ω,bΩ,Γ)→(Ω,bΩ,Γ) is bijective and measure preserving.
Proof. Fix X,Y ∈ Ω with X 6= Y. Set k := min{l ∈ T | Xl 6= Yl}.
Obviously, τ(X)k 6= τ(Y)k and therefore τ(X) 6= τ(Y), thus τ is injective.
Now letY ∈Ω. DefineX1 :=Y1, Xn+1 := Sgn( Pn
i=1Xi)Yn+1 and verify that
τ(X) = Y. Hence τ is onto. Since the projections πt and the function Sgn
are measurable,τ is measurable. Finally, prove that (πt◦τ)t≤H is a sequence
of independent variables and that eachπt◦τ is normal distributed with mean
0 and variance H1, which implies that Γ = Γ◦τ−1.
14.2
Lemma
The transformation τ : (Ω,bΩ,Γ)→(Ω,bΩ,Γ) is not ergodic.
Proof. Ergodicity fails, since the set{π1 <0}isτ-invariant but has measure 1/2.
Let (Ω, LΓ(bΩ),Γ) be the Loeb space of (Ωb ,bΩ,Γ). Set NbΓ := n A∈LΓ(bΩ)|bΓ(A) = 0 o .
The next lemma is an immediate consequence of Lemma 14.1 and of the definition of Loeb spaces.
14.3
Lemma
The transformation τ : (Ω, LΓ(bΩ),bΓ) → (Ω, LΓ(bΩ),Γ)b is bijective and
measure-preserving.
After introducing a sub-σ-field W of LΓ(bΩ) that corresponds to bC[0,1], we will show that Lis ergodic if and only if
τ : (Ω,W,bΓ)→(Ω,W,bΓ)
is ergodic. This approach allows a new view on the L´evy transformation, since τ is bijective and defined pointwise, whereas
L(x) =L(−x) for W-almost all x∈C[0,1].
We will use the following lemma about stochastic integrals.
14.4
Lemma
Fix two probability spaces(Λ,C, µ),(Λ0,C0, µ0)and a measure-preserving map- ping
κ: (Λ,C, µ)→(Λ0,C0, µ0).
Let (Ct)t∈[0,1] and (Ct0)t∈[0,1] be filtrations in C, C0 respectively such that
Ct =κ−1(Ct0)∨ Nµ
for each t ∈[0,1]. Assume further that
X : Λ0×[0,1]→R
is a (Ct0)-Brownian motion and that
is in L2(Λ0×[0,1]) and (Ct0)-predictable. Then the process (Xt◦κ) is a (Ct)-
Brownian motion, the process (Yt◦κ) is(Ct)-predictable and
Z Ytd Xt ◦κ = Z (Yt◦κ) d (Xt◦κ).
Fork ∈T ∪ {0} setBk:=Pkl=1πl. We define a sub-σ-algebra of LΓ(B) by
W := σ{◦Bk |k ∈T} ∨ NbΓ,
i.e. W is the smallest σ-field on Ω that includes NbΓ and such that the functions◦Bk, k∈T are measurable. Define further forX ∈Ω,t∈[0,1] and
k∈T with k H ≈t
b(X, t) := ◦Bk(X) and set κ(X) :=b(X, · ) .
A proof of the next statement can be found in [20].
14.5
Lemma
The process b is bΓ-almost surely well defined and a Brownian motion. Fur-
thermore, the mapping
κ: (Ω,W,bΓ)→(C[0,1],bC[0,1], W)
is measure-preserving and
W =κ−1(bC[0,1])∨ NbΓ. (16)
Let (Bt)t∈[0,1] be the filtration in bC[0,1] which is generated by ψ and set for
t∈[0,1]
Wt:=κ−1(Bt)∨ NbΓ.
14.6
Lemma
For each t∈[0,1] and k∈T with Hk ≈t we have Γb-a.s. Z t 0 sgn (bs)d bs ≈ k X l=1 Sgn(Bl−1)·πl.
Proof. This follows immediately from the definition of the stochastic integral in [20], since
sgn (bt) ≈ Sgn (Bk)
for bΓ-almost all X ∈Ω.
The main proposition of this section states the equivalence of τ and L con- cerning ergodicity.
14.7
Proposition
The transformation
L: (C[0,1],bC[0,1], W)→(C[0,1],bC[0,1], W)
is ergodic if and only if the transformation
τ : (Ω,W,bΓ)→(Ω,W,bΓ)
is ergodic.
Proof. The assertion follows from Equation (16) and from the fact that the diagram
C[0,1] −→L C[0,1]
↑κ ↑κ
Ω −→τ Ω
commutes bΓ-a.s., which implies that A ∈ bC[0,1] is L-invariant if and only if
κ−1(A) is τ-invariant. Note that the fact that the diagram commutes also implies that
τ−1(W)⊂ W.
In order to prove that κ◦τ =L◦κ bΓ-a.s. we fix at∈[0,1] and show that
ψt(L◦κ) =ψt(κ◦τ) Γ-a.s.b
Fix a k ∈ T with Hk ≈ t and note that because of Lemma 14.4 and Lemma 14.6 ψt(L◦κ)(X) = Z t 0 sgn (ψs)d ψs(κ(X)) = Z t 0 sgn (bs)d bs(X)≈ k X l=1 Sgn (Bl−1)(X)·Xl = Bk(τ(X))≈bt(τ(X)) =ψt(κ◦τ)(X)