Distortion inequality for a Markov operator generated by a randomly perturbed family of Markov Maps in ℝ^d

Definition 2.1

A nonsingular transformation φ from I into itself is said to be a piecewise invertible iff

(2.M1) one can find a finite or countable partition π = {I_k : k ∈ K} of I,which consists of measurable subsets (of I) such that m(I_k) > 0 for each k ∈ K, and sup{m(Ik):k∈K}<∞;

(2.M2) for each I_k ∈ π, the mapping φ_k = φ_|Ik is one-to-one of I_k onto J_k = φ_k(I_k) and its inverse φk−1 is measurable.

Several various important classes of piecewise invertible transformations e.g., Anosov diffeomorphisms [15], some expanding mapping [24, 25], or unimodal mapping [16] admit partitions with so-called Markov property.

In this paper we study random families of such piecewise invertible transformations. First however we give a definition of a single Markov map:

Definition 2.2

A piecewise invertible transformation φ is said to be a Markov map iff its corresponding partition π satisfies the following two conditions:

(2.M3) π is a Markov partition i.e., for each k ∈ K,

(2.M4) φ is indecomposable (irreducible) with respect to π i.e., for each (j, k) ∈ K² there exists an integer n > 0 such that Ik⊆φn(Ij).

In what follows we denote by ‖ · ‖ the norm in L¹ = L¹(I, Σ, m) and by G = G(m) the set of all (probabilistic) densities i.e.,

Let τ : I → I be a nonsingular transformation. Then the formula

where dm_f = f dm, and ddm denotes the Radon-Nikodym derivative, defines a linear operator from L¹ into itself. It is called the Frobenius-Perron operator (F-P operator, in short) associated with τ [20, 26].

From the definition of P_τ it follows that it is a Markov operator, i.e., P_τ is a linear operator and for any f ∈ L¹(m) with f ≥ 0, P_τ f ≥ 0 and ∥Pτf∥=∥f∥. Further, P_τG ⊆ G, and P_τ is a contraction, i.e., ‖P_τ‖ ≤ 1.

Let τ₁, . . . , τ_j, (j ≥ 2), be some nonsingular transformations. Denote by P(j,…,1),P(j,…,i+1) and P_(i,...,1), 1 ≤ i < j, the F-P operators associated with the transformations τ(j,…,1)=τj∘…∘τ1,τ(j,…,i+1)=τj∘…∘τi+1 and τ_(i,...,1) = τ_i ◦ · · · ◦ τ₁, respectively. Then

in particular P_(j,...,1) = P_j . . . P₁.

Definition 2.3

In what follows we consider a family {φ_y}_y∈Y of Markov maps such that:

(2.My5) Y is a complete separable metric space and the map I × Y (x, y) → φ_y(x) ∈ I is Σ(I × Y)/Σ(I)-measurable;

(2.My6) there exists a partition π of I such that π_y = π for each y ∈ Y, where π_y is a Markov partition associated with φ_y.

For j ≥ 1, and y₁, . . . , y_j ∈ Y, we denote y(j) = (y_j , . . . , y₁) and then we set

Clearly, φ_y(j) : I → I is a Markov map. Its Markov partition is given by

It consists of the sets of the form:

where k(j) = (k₀, k₁, . . . , k_j−1) ∈ K^j.

Let φy(j)k(j):=(φy(j))|Ik(j)y(j−1); then by condition (2.M2), φ_y(j)k(j) is one-to-one mapping of Ik(j)y(j−1) onto Jk(j)y(j):= φy(j)k(j)Ik(j)y(j−1)=φyjIkj−1. It is nonsingular, and φy(j)k(j)−1, the mapping inverse to φ_y(j)k(j), is measurable. By Def. 2.3, π_y(1) = π_y1 = π and therefore Ik(1)y(0)=Ik0; consequently φ_y(1)k(1) = (φ_y1 )_|Ik0 = φ_y1k0 and, according to (2.M2), φ_yk is one-to-one mapping of I_k onto Jky=φyk(Ik).

In the case of a family {φ_y}_y∈Y of Markov maps we use the following analogue of the indecomposable condition (2.M4):

(2.My4) for each (j, k) ∈ K² there exist an integer s > 0, and a subset Y˜s⊆Ys with ps(Y˜s)>0 such that Ik⊆φy(s)(Ij) for all y(s)∈Y˜s. Here p is a probability measure on Σ(Y), and ps=p×…×p⏟s−times.

Note that in the particular case: {φ_y}_y∈Y , where Y = {y}, p({y}) = 1, the above conditions yields condition (2.M4) (see also note at the end of Rem. 2.4).

Since φy(r)k(r):=(φy(r))|Ik(r)y(r−1) is a nonsingular mapping of Ik(r)y(r−1) onto Jk(r)y(r), then the formula

defines an absolutely continuous measure whichis concentrated on Jk(r)y(r) (i.e.,my(r)k(r)(A)=my(r)k(r)(A∩Jk(r)y(r))), and whose Radon-Nikodym derivative satisfies dmy(r)k(r)dm>0 a.e. on Jk(r)y(r).

To see the latter property of the measure my(r)k(r), note first that if dmy(r)k(r)dm=0 on A⊆Jk(r)y(r), then φy(r)k(r)−1(A)⊆I∖Ik(r)y(r−1) a.e., because m(φy(r)k(r)−1(A)∩Ik(r)y(r−1))=∫A∩Jk(r)y(r)dmy(r)k(r)dmdm=0. Therefore, A=∅ a.e.

We put (r = 1,2, . . .)

and next,

Then the F-P operator P _y(r) = P_yr . . . P_y1 corresponding to the Markov map φ_y(r), given by (2.3), can be written in the following form

Indeed, from (2.1), Defs. 2.2, 2.3 and (2.5) it follows that for any f ∈ L¹, f ≥ 0 the following equalities hold:

where Ak(r)y(r−1)=φy(r)k(r)−1(A). Hence, (2.8) follows.

Remark 2.4

The studies of this paper can be extended to the family {φ_y}_y∈Y which satisfies condition (2.M_y5) of Def. 2.3 and the following one:

(2.M˜y 6) there is a Markov map φy˜ such that for each y ∈ Y:

1. πy≺πy˜. i.e., for each V ∈ π_y , there exists U∈πy˜ which contains V, and

2. for each V ∈ π_y , φ_y(V) is a union of a number of U∈πy˜.

In this situation each φ_y(j) = φ_yj ◦ φ_yj−1 ◦ · · · ◦ φ₁ is defined on the interval of the form:

The following family can serve as a simple example: {φi}i=1∞ where φ_i = φⁱ, and φ is a Markov map. Note that conditions (2.M4) and (2.M_y4) are equivalent in this case, if p({i}) > 0 for i ≥ 1.

Let P : L¹(m) → L¹(m) be a Markov operator. The following criterion of the convergence of {Pjg}j=1∞ is used in this paper (see [14], Th. 2, and Rem. 1):

Theorem 2.5

Let there exist h ∈ L¹, h ≥ 0 with ‖h‖ > 0, and a dense subset G₀ ⊆ G such that

Then there is a unique g₀ ∈ G such that limj→∞Pjg=g0 for all g∈G.

Proposition 3.1

If Prob(x0∈B):=∫Bgdm for all B∈Σ(I), where g ∈ L¹(m) and g ≥ 0, ‖g‖ = 1, then

for all B ∈ Σ(I), where P^j is the j-th iterate of the Markov operator P defined by (3.1).

Proof. We refer to [7], Prop. 3.1.

In [6] has been established an inequality for the Frobenius-Perron operator P_φ associated with a single Markov map φ. That inequality (called Distortion Inequality) was intended as an operator-theoretic analogue of the Rényi’s Condition [21, 22]. Below we formulate a probabilistic analogue of that inequality for a family {P_y}_y∈Y of the Frobenius-Perron operators corresponding to a given family {φ_y}_y∈Y of Markov maps.

Let φ_y(j) be a Markov map given by (2.3) and let P_y(j) be its F-P operator given by (2.8). We put

where spt(g) := {x : g(x) > 0}.

Now we give the following definition:

Definition 3.2

A density g ∈ G belongs to G(C^*), 0 < C^* < ∞, iff there exist constants C_y(j)(g) ≥ 1, y(j) ∈ Y^j, j ≥ j₁(g), such that the following two conditions are satisfied:

1. Ay(j)k(g)≤Cy(j)(g)ay(j)k(g) a.e. [pj],j≥j1(g), and

2. lim supj→∞∫lnCy(j)(g)dpj<C∗.

Note that the above definition is coherent with the definition of the set G(C^*) given in [6] by formula (3.0) in the case φ_y = φ for each y ∈ Y, where φ is a fixed Markov map. Therefore the two discussed there questions connected with the set G(C^*), in the case of single F-P operator, are valid in the case of a family of F-P operators too. Thus in the first place we must assume the following property of G(C^*):

(3.H1) (Distortion Inequality for the family {P_y}_y∈Y )

There exists a constant 0 < C^* < ∞such that the set G(C^*) defined by Def. 3.2 contains a subset dense in G.

Next it follows from the second question discussed in the cited paper [6] that condition (3.H1) alone does not yet ensure the existence of P-invariant density. In [6] have been used two complementary conditions. Each of those two conditions completes condition (3.H1) there in an optimal way. In this paper we make use of their probabilistic analogues. They complete condition (3.H1) here in an optimal way too (see Rem. 3.6).

To formulate a probabilistic analogue of the first of the two mentioned conditions we introduce the following auxiliary function:

where

w(2r)=(w˜(r),w(r))∈Yr×Yr, and

and Ik(r)w(r−1), and σw(r)k(r) are defined, respectively by (2.4) and (2.6).

The first complementary condition reads as follows:

(3.H2) There exists r˜≥1 such that 0<∥∫uw(2r˜)dp2r˜∥<∞.

The theorem below has already been proved in [9]. It states that the semi-dynamical system given by (3.4) evolves to a stationary distribution under the above two conditions.

Theorem 3.3

(First Convergence Theorem). Assume that a family {φ_y}_y∈Y of Markov maps satisfy conditions (3.H1) and (3.H2).

Then there exists exactly one P-invariant density g₀ such that

Proof. (main idea) By condition (3.H2) there exists A⊆Y2r˜×I such that ∫A|lnuw(2r˜)|dp2r˜×m<∞. We assume for the simplicity that A=Y2r˜×B for some B ∈ Σ(I). Then the idea of the proof consists in showing that the function

whereCˆ=exp(−2C∗), satisfies the relation:

Then the proof of the theorem is completed by an appeal to Th. 2.5.

The theorem above states only that the semi-dynamical system given by (3.4) evolves to a unique P-invariant density g₀. It however gives no further information about the limit density itself. Below we prove some properties of the limit density g₀ under somewhat more restrictive condition than condition (3.H1). Namely, we assume that the constants C_y(j)(g) in Def. 3.2 depend on j and g but not on y(j) ∈ Y_j. In that case the set G(C^*) in condition (3.H1) satisfies the following equality: G(C∗)={g∈G:lim supj→∞Aj(g)<C∗}, where

Ay(j)k(g), and ay(j)k(g) are defined by (3.5).

Accordingly, the new version of condition (3.H1) can be formulated as follows:

(3.H˜1) (Uniformly Distortion Inequality for the family {P_y}_y∈Y )

There exists a constant C* > 0 such that the set

contains a subset dense in G.

We are going now to prove the following theorem:

Theorem 3.4

(Second Convergence Theorem). Let a family {φ_y}_y∈Y ofMarkov maps satisfy conditions (3.H˜1) and (3.H2).

Then:

1. There exists exactly one P-invariant density g₀ such that

   limj→∞Pjg=g0 for all g∈G.

2. There exists a density of the form

with r˜ as in (3.H2), such that

where C* is the constant in condition (3.H˜1).

(b₁) Additionally, the unique P-invariant density g₀ is estimated as follows:

where uw(2r˜) is defined by (3.6), and

The upper estimation holds provided ∫Uw(2r˜)dp2r˜<∞.

Proof. (a) Clearly, that assertion follows from Th. 3.3. Nevertheless we prove it directly (in a way close to the proof of Th. 3.3) because the proof contains a result which is exploited in the course of the proof of the two remainder assertions.

The proof is based on the fact that the function

where C* is the constant in condition (3.H˜1), satisfies relation (3.10).

By condition (3.H˜1) there exists a subset G∗⊆G˜(C∗) dense in G. Then for any g ∈ G* there exists j₁ = j₁(g) such that: for any j ≥ j₁ and all I_k ∈ π one has

From the above inequalities follows the following basic double inequality (see [7], Lemma 2.3):

for g ∈ G*, all w(r) ∈ Y^r, y(j) ∈ Y^j, r ≥ 1, and all j ≥ j₁(g); where F _rw(r) is defined by

In the last formula σ˜w(r)k(r) and Ik(r)w(r−1) are defined by (3.8) and (2.4), respectively.

Indeed, from (3.12) we obtain

for each Jk(r)w(r)=φw(r)k(r)(Ik(r)w(r−1)), all x,y∈Jk(r)w(r), and j≥j1(g);

where

Integrating the above inequalities with respect to x on Jk(r)w(r) and multiplying by σ˜w(r)k(r)(y) , then summing the resulting inequalities with respect to all k(r) and finally using the equality (2.8) one gets the desired double inequality (3.13).

Let w(2r)=(w˜(r),w(r))∈Yr×Yr, then iterating the double inequality (3.13), by using the equalities Pw(2r)=Pw˜(r)Pw(r),

and the definition (3.14), one gets for every r ≥ 1, j ≥ j₁(g), and all w˜(r), w(r) ∈ Y^r, and y(j) ∈ Y^j:

Integrating the above two inequalities with respect to w(2r)=(w˜(r),w(r)), and y(j), and applying formulas (2.2), (3.2), and (3.3) give

for r ≥ 1, and j ≥ j₁(g).

It follows from the first part of the just derived double inequality (3.17), the two facts: G* is dense in G, and P is a contraction, that for each r ≥ 1, the function u˜2r, given by (3.11), satisfies relation (3.10).

Since however it may happen that ∥u˜2r∥=0 for each r ≥ 1 (see [4, 5] and recently [8]), one has to assume condition (3.H2) that excludes such a possibility. The proof of assertion (a) is completed by an appeal to Th. 2.5.

(b) Integrating (3.13) with respect to w(r), and y(j), and applying formulas (2.2), (3.2), and (3.3) give

for r ≥ 1, and j ≥ j₁(g).

Therefore the double inequality of the assertion under consideration follows from the last relation, and assertion (a).

Hence for each B∈Σ,μ0(B)=0⇔μ˜0(B)=0 where dμ˜0=g˜0dm, that is A=spt(g0)=spt(g˜0), where spt(g) = {x ∈ I : g(x) > 0}. To prove that g˜0>0, note first that A≠∅ consists exclusively of I_k’s. It follows from the form of g˜0 given in Th. 3.4 (b). We claim that ∥1Ikg0∥=∥1Ik∩Ag0∥>0, i.e., I_k ⊂ A for any I_k ∈ π.

Take any Ik˜⊆A, and for an arbitrary I_k let s be such that Ik⊆φy(s)(Ik˜)⊆φy(s)(A) for all y(s)∈Y˜s, where Y˜s⊆Ys with ps(Y˜s)>0 (see condition (2.M_y4)).

Then we have

Therefore

It shows that spt(g˜0)=I.

(b₁) The estimations under consideration follows from (3.17) and assertion (a) of the theorem.

We conclude this section with a result on the convergence of {j−1∑i=0j−1Pi}j=1∞. Namely, the convergence of that sequence is proved under condition (3.H˜1) and a probabilistic analogue of condition (3.H3) in [6].

Let {Vn}n=1∞ be a sequence of subsets of I such that each V_n consists of a finite number of I_k’s, V_n ⊂ V_n+1, ⋃n=1∞Vn=I˜ and m(I∖I˜)=0. Then we define

The probabilistic analogue of that condition reads as follows:

(3.H3) There exists r˜≥1 such that limn→∞dr˜n=0.

The following theorem holds:

Theorem 3.5

(Third Convergence Theorem). Let a family {φ_y}_y∈Y of Markov maps satisfy conditions (3.H˜1) and (3.H3).

Then:

(a) There exists exactly one P-invariant density g₀ such that limj→∞Sjg=g0 in L¹ for all g ∈ G, where

(b) The two assertions (b) and (b₁) of the previous theorem hold.

Proof. (a) Let V_n consists of a finite number of I_k’s. Integrating first the second part of the double inequality (3.13), with respect to y and over I_k ⊆ V_n, one gets m(Ik)Py(j)g(x)≤C∗∫IkPy(j)gdm. Then integrating the

obtained inequality with respect to y(j) ∈ Y^j one gets

where m˜(Vn)=min{m(Ik):Ik⊆Vn}.

Next, it follows from the second part of the double inequality (3.13) and the definition (3.18) that

Then the last two inequalities and condition (3.H3) imply weak compactness of {P^jg} for g ∈ G* ([10], Th. IV.8.9).

Finally, assertion (a) follows from this, the Yosida-Kakutani Ergodic Theorem ([10], Th. VIII.5.1), and the denseness of G* in G.

(b) Follows from assertion (a) of this theorem similarly as assertions (b) and (b₁) of the previous theorem follow from assertion (a) of that theorem.

Remark 3.6

It follows from (3.6), (3.7), (3.16) (a), and the first part of the double inequality (3.17) that condition (3.H2) is optimal. Likewise from (3.18), the second part of the inequality (3.13), and (3.20) follow that condition (3.H3) is optimal too.

Note also that the first complementary condition is essentially less restrictive than the second one, but the second condition is readily verifiable in practice (especially in the case m(I) < ∞, see Fact 4.1).