Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝ^d

Definition 2.1

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

1. 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(I_k) : k ∈ K} < ∞, here and in what follows K is an arbitrary countable index set;

2. 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.

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:

1. π is a Markov partition i.e., for each k ∈ K,

   φ(Ik)=⋃{Ij:m(φ(Ik)∩Ij)>0};

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

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 [5, 6].

Formula (2.1) is equivalent to the following one:

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_τf ≥ 0 and ∥P_τf∥ = ∥f∥.

The last equality follows immediately from the second formula equivalent to (2.1) if one puts A = I. 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) = τ_j ∘ … ∘ τ₁, τ_(j,…,i+1) = τ_j ∘ … ∘ τ_i+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:

1. Y is a Polish metric space and the map I × Y ∋ (x, y) → φ_y(x) ∈ I is Σ (I × Y)/ Σ(I)-measurable;

2. 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

then by condition (2.M2), φ_y(j)k(j) is one-to-one mapping of Ik(j)y(j−1) onto

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_k is one-to-one mapping of I_k onto Jky=φyk(Ik).

We have to adjust the indecomposable condition (2.M4) to the new case when a single Markov map φ is replaced by a family {φ_y}_y∈Y of Markov maps. We propose the following condition (see note at the end of Rem. 2.4):

1. for each (j, k) ∈ K² there exist an integer s > 0, and a subset Y͠^s ⊆ Y^s with p^s(Y͠^s) > 0 such that I_k ⊆ φ_y(s)(I_j) for all y(s) ∈ Y͠^s. Here p is a probability measure on Σ(Y), and ps=p×⋯×p⏟s−times.

From the properties of φ_y(r) it follows that the formula

defines an absolutely continuous measure which is 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 m_y(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, …)

next,

and finally

Then the F-P operator P_y(r) of the Markov map φ_y(r)k(r) can be written in the following form

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

where Ay(r)k(r)=φy(r)k(r)−1(A), and dm_y(r)k(r) is given by (2.5). 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, instead of (2.M_y6), the following less restrictive condition:

1. there is a Markov map φ_ỹ such that for each y ∈ Y:

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

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

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.

We close this section with the following criterion of the convergence in L¹ of the iterates Pⁿ of Markov operator. It is used in the proof of Th. 3.3.

Theorem 2.5

Let there exist h ∈ L¹, h ≥ 0 with ∥h∥ > 0, and a dense subset G₀ ⊆ G such that limj→∞∥(Pjg−h)−∥ = 0, for g ∈ G₀, where (P^j g − h)⁻ = max{0, −(P^j g − h)}. Then there exists exactly one P-fixed point g₀ ∈ G such that

Proof

We refer to [6], Theorem 3.□

Proposition 3.1

If Prob(x0∈B)=df∫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 of F-P type defined by (3.1).

Proof

We refer to [7], Prop. 3.1.□

The convergence of the sequence {P^j} of the iterates of the Markov operator P of F-P type associated with {x_j} is established under two general conditions. We are going now to formulate the first of them.

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)=df{x:g(x)>0}.

Now we give the following:

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. A_y(j)k(g) ≤ C_y(j)(g)a_y(j)k(g) a.e. [p^j], j ≥ j₁(g), and

2. lim supj→∞∫ln⁡Cy(j)(g)dpj<C∗.

Having defined the set G(C^*) we are in a position to formulate the first condition:

1. (Distortion Inequality for the family P_y(j), y(j) ∈ Y^j) There exists a constant 0 < C^* < ∞ such that the set G(C^*) defined by Def. 3.2 contains a subset dense in G.

   To formulate the second condition we define first the following auxiliary function:

   uw(2r)(x)=dfinf{gw(2r)k(r)(x):k(r)∈Kr,andIk(r)w(r−1)≠∅}, (3.4)

   where

   gw(2r)k(r)=df∑k~(r)σ~w~(r)k~(r)∫Ik~(r)w~(r−1)σ~w(r)k(r)dm, (3.5)

   w(2r) = (w̃(r), w(r)) ∈ Y^r × Y^r, (in (3.5) we put w̃(r) = (w̃_r, …, w̃₁), and k̃(r) = (k̃₀, …, k̃_r−1)); further

   σ~w(r)k(r)=dfσw(r)k(r)m(Ik(r)w(r−1)), (3.6)

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

   The second condition reads as follows:

   1. There exists r̃ ≥ 1 such that 0 < ∥ ∫ u_w(2r̃) dp^2r̃ ∥ < ∞.

   The theorem below states that the semi-dynamical system given by (3.3) evolves to a stationary distribution under the above two conditions.

Theorem 3.3

(Convergence Theorem) Assume that a family {φ_y}_y∈Y of Markov maps satisfies (3.H1) and (3.H2). Then there exists exactly one P-fixed point g₀ ∈ G, that is Pg₀ = g₀, such that

Proof

The point is to show that for each r ≥ 1, the function

is a function for P which, under condition (3.H1), plays the role of h from Theorem 2.5. That is, it satisfies the relation

To this end note that by condition (3.H1) there exists a subset G͠ ⊆ G(C^*) dense in G. Thus for any g ∈ G͠ there exists, by Def. 3.2 (a), j₁ = j₁(g) such that the following inequalities hold:

for each j ≥ j₁, all y(j) ∈ Y^j, any I_k ∈ π and m × m a.e. (x, y) ∈ I_k × I_k.

These inequalities imply the following estimate:

for every r ≥ 1, j ≥ j₁ and all w(r) ∈ Y^r, y(j) ∈ Y^j; where C_y(j)(g) are constants involved in Def. 3.2, and F_rw(r) is defined by the following formula:

In the last formula σ̃_w(r)k(r) and Ik(r)w(r−1) are defined by (3.6) and (2.4), respectively.

To see it note that from (3.9) 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

(Py(j)g)¯w(r)k(r)(x)=df(Py(j)g)∘φw(r)k(r)−1(x), or 0, according as x∈Jk(r)w(r), or x∈I∖Jk(r)w(r).

Integrating the above inequalities with respect to x on Jk(r)w(r) and multiplying by σ̃_w(r)k(r)(y), then summing the resulting inequalities with respect to all k(r) and finally using equality (2.8) one gets the desired double inequality (3.10).

Let w(2r) = (w̃(r), w(r)) ∈ Y^r × Y^r, then iterating the first of the double inequality (3.10), by using the equalities

and the formula (3.11), one gets for every r ≥ 1, j ≥ j₁(g), and all w̃(r), w(r) ∈ Y^r, and y(j) ∈ Y^j:

where z(r + j) = (w(r), y(j)) and u_w(2r) is defined by formulas (3.4), and (3.5).

Integrating the above inequalities with respect to w(2r) = (w̃(r),w(r)), and y(j), using Jensen’s inequality and condition (b) of Def. 3.2, and applying formulas (2.2) together with (3.2), give:

where u_2r is defined by (3.7).

The last inequality implies that (3.8) holds. This is so because G͠ ⊆ G is dense, and P is a contraction.

Thus we have proved that for each r ≥ 1, u_2r indeed plays the role of h from Theorem 2.5 for P; possibly the trivial one, if ∥ ∫ u_w(2r) dp^2r∥ = 0. To exclude the trivial possibility we have to assume the existence of a nontrivial function u_2r for P, for some r ≥ 1, that is condition (3.H2). Then by Theorem 2.5 we have limj→∞ P^j g = g₀, for all g ∈ G. From this and the inequality

it follows that Pg₀ = g₀, i.e. the density g₀ is P-invariant. This finishes the proof of the theorem.□

Definition 4.1

We denote by G_α, 0 < α ≤ 1, the class of all densities g ∈ G satisfying the following three conditions:

1. spt(g)=df{x∈I:g(x)>0} is a sum of a number of I_k ∈ π;

2. for each I_k ∈ π, g_∣Ik ∈ C^0+α(I_k), and

   |g(x)−g(y)|≤C(g)g(y)|x−y|αfor allx,y∈spt(g)∩Ik;

   where C(g) is a constant depending on g.

The following theorem is a consequence of Th. 3.3:

Theorem 4.2

Let a family {φ_y}_y∈Y of C^1+α-smooth Markov maps satisfy conditions (4.H_y4[a, b]), (4.H_y7), and (3.H). Then there exists exactly one P-fixed point g₀ ∈ G such that

Proof

We show that

here in (4.1), for g ∈ G_α, we define

where C~0α=sup{(diam(Ik))α:k∈K}.

Note that by conditions (4.H_y4[b]) and (4.H_y7) we have

that is condition (b) of Def. 3.2 holds.

It remains to show the first condition of that definition holds. Let g ∈ G_α, then for any y(j) ∈ Y^j, k(j) ∈ K^j, j = 1, 2, …, and for any x, z ∈ I_r the following inequality holds:

Next, by condition (4.H_y4[a]), we have the following inequality (for any y(j) ∈ Y^j, k(j) ∈ K^j, j = 1, 2, …, and for any x, z ∈ I_r):

where C~0α=sup{(diam(Ik))α:k∈K}.

Therefore for any y(j) ∈ Y^j, j = 1, 2, …, I_k ∈ π, and for any x, z ∈ I_k we have

where C_y(j) are defined by (4.2). Hence condition (a) of Def. 3.2 holds for g ∈ G_α too.

The last inequality, and relations (4.2), (4.3) show that (4.1) holds. This implies condition (3.H1) because G_α is dense in G.□

Remark 4.3

(Final Remark) We present two cases of particular nature of the system (3.3) (for more details see [7], Examples (5.1), and (5.3)).

Example 4.4

The first particular case is the following x_j(x, ω) = φ^ξ_j(ω)(x_j−1), for j = 1, 2, …. The stochastic perturbation of the system arises from not knowing the precise number of iterations. That kind of stochastic perturbation has no influence on the statistical behaviour of the deterministic system x_j = φ^j(x_j−1), for j = 1, 2, ….

Example 4.5

The second case of stochastic perturbation is the following x_j(x, ω) = ζ_j(ω)φ(x_j−1), for j = 1, 2, …. In that case stochastic perturbation appears in a multiplicative way (it is the so called parametric noise). Such a perturbation changes essentially the statistical behaviour of the system. It illustrates the example: Let φ_y(x) = y tan(x), y ∈ Y = {b, 1}; and p₁(ζ_j = b) = 1 − a, and p₁(ζ_j = 1) = a, for j = 1, 2, …, where b > 1, and 0 < a < 1.

Here φ₁(x) = tan(x) is (a Markov map) without any invariant density [11]. However, the considered random system has P-invariant density, but its deterministic counterpart, i.e. when Y = {1} with p₁(ζ_j = 1) = 1, not.