Remarks to the Convolution Equations of Choquet and Deny

H. Herrmann

Abstract

Let {A_{t}, t ∈ H} be a semigroup of linear operators on a topological vector space, where H is a subsemigroup of a topological group. Let the operator A_{σ} be a mixture of the operators A_{t} with respect to the probability measure σ (concentrated on H). Further we assume that the equation A_{σ}x_{0} = x_{0} holds true for a certain point x_{0}. We study the behaviour of the operators A_{t} at the point x_{0}. For example, we give sufficient conditions for A_{t}x_{0} = x_{0} to be right for all t ∈ H. Under certain restrictions we generalize the convolution equations of Choquet and Deny. Some important special cases are considered.

1. Introduction

Let G denote a locally compact abelian Hausdorff topological group, σ and α regular Borel measures on G and f a real measurable function on G. For a given σ we consider the convolution equation σ ∗ α = α (1)

and the equation in the form

∫ f(y + z) σ(dz) = f(y), ∀y ∈ G, (2)

for which we want to find the solutions α and f, respectively.

Under certain additional assumptions the measures α which satisfy (1) and the functions f ≥ 0 which satisfy (2) were determined by Choquet and Deny in 1960 (see [2] and [3]). Lau and Zeng have solved a corresponding modifyed equation (1) and the equation (2) on a semigroup in 1990 (see [6]). A systematic representation of these problems with reference to the original literature can be found in the monograph [1].

We restrict our considerations of the equations (1) and (2) to the case that σ is a probability measure, α a (signed) measure satisfying a certain boundedness condition and f is supposed to be a bounded function. Then these specialized equations we can essentially generalize in another direction. The determination of the solutions to the equations (1) and (2) is then a special case of the following problem, which is the main subject of the present paper.

Let V be a Hausdorff topological vector space, G as before a locally compact abelian Hausdorff topological group, H a (abstract) subsemigroup of G, and σ a probability measure on G with σ(H) = 1. For every t ∈ H let A_{t} : V ⟶ V such a linear operator that {A_{t}, t ∈ H} forms a semigroup. We assume that there exists a linear operator A_{σ} : V ⟶ V, which is the mixture of operators A_{t} with respect to σ. (We characterize A_{σ} by means of continuous linear functionals defined on V.) Now let x_{0} be a fixed point of the operator A_{σ}:

A_{σ}x_{0} = x_{0}. (3)

We are interested in the following question: Which conditions, concerning the probability measure σ and the semigroup of operators {A_{t}, t ∈ H}, do ensure that x_{0} is also a fixed point of every operator A_{t}, t ∈ H? The answer is given using some asymptotically uniformly distributional properties of the sequence of the convolution powers of σ (see [4]).

Section 2 contains the needed notations and the main results, which we prove in Section 3. In Section 4 we discuss two special cases of the general problem: First we assume the topological space V to be a Hilbert space and afterwards the operator A_{σ} is supposed to be the Laplace transform of a semigroup of operators. Section 5 treats a further interesting special case: A semigroup of stochastic kernels operates in a space of signed measures and in a space of measurable functions, respectively. Finally, in Section 6 we return to the equations (1) and (2) and consider in this connection a result by Klimov (see [5]).

A possible characterization of the Haar measure by means of the convolution equation (1) is given in the Appendix.

2. Notations and Results

The following assumptions contained in (A1) - (A5) are the general assumptions for the discussion of the equation (3).

(A1) Let V be a Hausdorff topological vector space over the real number field R. Let L be a set of continuous linear functionals defined on V which is separating, that is for every pair [x, y] ∈ V^{2}, x ≠ y, a x′ ∈ L exists such that < x, x′ > ≠ < y, x′ >. (The value x′(x) of the functional x′ at the point x is denoted by < x, x′ >.)

(A2) Let G be a locally compact abelian Hausdorff topological group and G the σ-algebra of its Borel sets. We assume all measures considered on G are regular. Let λ be a Haar measure on G. For a probability measure ν on G let T_{ν} denote the support of ν and G_{ν} the closed subgroup spanned by the support of ν.

(A3) Let H ∈ G be a subsemigroup of the abstract group G. For every t ∈ H let A_{t} be a linear operator A_{t} : V ⟶ V such that

A_{t}A_{s} = A_{t+s}, ∀t, s ∈ H,

that is the set of linear operators {A_{t}, t ∈ H} is a semigroup.

(A4) Let for every x ∈ V and every x′ ∈ L

(a) sup_{t∈H} | < A_{t}x, x′ > | < ∞,

(b) < A_{t}x, x′ > be a (G ∩ H)-measurable function.

(A5) Let Σ be the set of the probability measures σ defined on G with σ(H) = 1. For every σ ∈ Σ let exist a linear operator A_{σ} : V ⟶ V such that

(a)

< A_{σ}x, x′ > = ∫ < A_{t}x, x′ > σ(dt), ∀x ∈ V, ∀x′ ∈ L,

(b) A_{σ}A_{t} = A_{t}A_{σ} for every t ∈ H.

Further assumptions are partially needed and given in the following conditions (S1) - (S3). Let σ ∈ Σ.

(S1) Let λ_{σ} denote a Haar measure on the topological group G_{σ}. Let σ be finally absolutely continuous, that is for some natural number n the probability measure σ^{n∗} (the n-fold convolution of σ with itself) is nonsingular with respect to the measure λ_{σ}.

(S2) Let λ_{σ}(H ∩ U(t_{0})) > 0 for every t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0, and every neighbourhood U(t_{0}) of t_{0} (in the topological group G_{σ}).

(S3) For the fixed point x_{0} of the operator A_{σ} and every x′ ∈ L let < A_{t}x_{0}, x′ >, t ∈ H, be a continuous function for t ∈ H ∩ G_{σ}, t ≠ 0.

Under the assumptions (A1) - (A5) we prove the following statements.

Theorem 1. If for some σ ∈ Σ and some x_{0} ∈ V it holds that A_{σ}x_{0} = x_{0}, then for every x′ ∈ L the function < A_{t}x_{0}, x′ >, t ∈ H, is constant for λ_{σ}-almost all t ∈ H ∩ G_{σ}.

Theorem 2. Let A_{σ}x_{0} = x_{0}. Assume further that either condition (S1) or the both conditions (S2) and (S3) are satisfied. Then A_{t}x_{0} = x_{0} for all t ∈ H ∩ G_{σ}.

Therefore, x_{0} is fixed point of every operator A_{t}, t ∈ H, if in addition to the assumptions of Theorem 2 the inclusion G_{σ} ⊇ H is satisfied.

3. Proofs

We prove Theorem 1 and Theorem 2 in several steps.

3.1. First we note the asymptotically uniformly distributional properties of the sequence of the convolution powers of a probability measure which we will use. For t ∈ G let δ_{t} denote the degenerate probability measure on G concentrated on the set {t}. Further, we denote by ∥ ν_{1} − ν_{2} ∥ the distance in variation of the probability measures ν_{1} and ν_{2}.

Let 0 ∈ T_{σ}. Then the sequence of the convolution powers of the probability measure σ satisfies the following two basic limit relations. If (S1) holds, then

lim_{n→∞} ∥σ^{n∗} − δ_{t}∗σ^{n∗}∥ = 0, ∀t ∈ G_{σ}. (4)

If β_{1}, β_{2} are probability measures on G which are absolutely continuous with respect to λ_{σ}, then

lim_{n→∞} ∥β_{1}∗σ^{n∗} − β_{2}∗σ^{n∗}∥ = 0. (5)

(See [4], 3.1. and 3.5. We note that 0 ∈ T_{σ} implies that the closed subgroups spanned by T_{σ} and (T_{σ} − T_{σ}), respectively, coincide).

Investigating the equation (3) we may suppose 0 ∈ T_{σ} without loss of generality. Since otherwise with

τ := \frac{1}{2}δ_{0} + \frac{1}{2}σ, (A_{0}x := x, ∀x ∈ V,) A_{τ} := \frac{1}{2}A_{σ} + \frac{1}{2}A_{0}

we consider the equation A_{τ} x_{0} = x_{0} which is equivalent to (3). Further, the group spanned by the support of τ coincides with G_{σ} and τ^{n∗} is nonsingular with respect to λ_{σ} if and only if σ^{n∗} is nonsingular with respect to λ_{σ}.

3.2. Every operator A_{σ}, σ ∈ Σ, is characterized by (A5), (a). (For every σ ∈ Σ the integral in (A5), (a) exists because of the conditions (A4)). Now we claim

A_{τ} A_{σ} = A_{σ∗τ} , ∀σ, τ ∈ Σ. (6)

For every x ∈ V and every x′ ∈ L we obtain by virtue of (A5), (b)

< A_{τ} A_{σ}x, x′ > =
∫ < A_{t}A_{σ}x, x′ > τ(dt) = ∫ < A_{σ}A_{t}x, x′ > τ(dt) =
∫ ( ∫ < A_{s}A_{t}x, x′ > σ(ds)) τ(dt) = ∫ ∫ < A_{s+t}x, x′ > σ(ds)τ(dt) =
∫ < A_{u}x, x′ > (σ ∗ τ)(du) = < A_{σ∗τ} x, x′ >,

which gives (6). From (6) we conclude the commutative property A_{σ}A_{τ} = A_{τ} A_{σ} and, because of (3), the equations A_{σ^{n∗}}x_{0} = x_{0}, n = 1, 2, … (7)

3.3. By our assumption x_{0} is a fixed point to A_{σ}. We compare x_{0} with A_{s}x_{0} for some s ∈ H ∩ G_{σ}. Making use of (7) and (6) we have

x_{0} = A_{σ^{n∗}} x_{0},
A_{s}x_{0} = A_{s}A_{σ^{n∗}} x_{0} = A_{δ_{s}∗σ^{n∗}} x_{0}, n = 1, 2, …,

and therefore for every x′ ∈ L it holds that

| < x_{0}, x′ > − < A_{s}x_{0}, x′ > | =
| < A_{σ^{n∗}} x_{0}, x′ > − < A_{δ_{s}∗σ^{n∗}} x_{0}, x′ > | =
| ∫ < A_{t}x_{0}, x′ > σ^{n∗}(dt) − ∫ < A_{t}x_{0}, x′ > (δ_{s} ∗ σ^{n∗})(dt) | ≤

sup_{t∈H} | < A_{t}x_{0}, x′ > | ∥σ^{n∗} − δ_{s} ∗ σ^{n∗}∥, n = 1, 2, …

If σ satisfies (S1) then (4) yields

< x_{0}, x′ > = < A_{s}x_{0}, x′ >, ∀x′ ∈ L,

so that x_{0} = A_{s}x_{0}, ∀ s ∈ H ∩ G_{σ}, follows. Thus the assertion of Theorem 2 is proved provided (S1) is satisfied.

3.4. Suppose that probability measures β_{1}, β_{2} on G exist which are concentrated on H ∩ G_{σ} and absolutely continuous with respect to λ_{σ}. (This is satisfied if and only if λ_{σ}(H ∩ G_{σ}) > 0.) We form the linear operators A_{β_{1}} and A_{β_{2}} and compare A_{β_{1}}x_{0} with A_{β_{2}}x_{0}, where again x_{0} is a fixed point to A_{σ}. According to (7) and (6) it holds

A_{β_{i}}x_{0} = A_{β_{i}}A_{σ^{n∗}} x_{0} = A_{β_{i}∗σ^{n∗}} x_{0}, i = 1, 2, n = 1, 2, …

For every x′ ∈ L we obtain

| < A_{β_{1}}x_{0}, x′ > − < A_{β_{2}}x_{0}, x′ > | =
| < A_{β_{1}∗σ^{n∗}} x_{0}, x′ > − < A_{β_{2}∗σ^{n∗}} x_{0}, x′ > | =

| ∫ < A_{t}x_{0}, x′ > (β_{1} ∗ σ^{n∗})(dt) − ∫ < A_{t}x_{0}, x′ > (β_{2} ∗ σ^{n∗})(dt) | ≤

sup_{t∈H} | < A_{t}x_{0}, x′ > | ∥β_{1} ∗ σ^{n∗} − β_{2} ∗ σ^{n∗}∥, n = 1, 2, …

Now (5) yields that

< A_{β_{1}}x_{0}, x′ > = < A_{β_{2}}x_{0}, x′ >, ∀x′ ∈ L,

so that A_{β_{1}}x_{0} = A_{β_{2}}x_{0}. Therefore there exists a certain y_{0} ∈ V such that

A_{β}x_{0} = y_{0} (8)

for all probability measures β which are concentrated on H ∩G_{σ} and absolutely continuous with respect to λ_{σ}.

3.5. Now we are able to prove Theorem 1. Let (3) be satisfied. Without loss of generality let λ_{σ}(H ∩ G_{σ}) > 0, since in the case λ_{σ}(H ∩ G_{σ}) = 0 the assertion is trivial. We choose some x′_{0} ∈ L. For y_{0} defined in (8) we put z_{0} =< y_{0}, x_{0} > and define a set B ∈ G by

B := {t ∈ H ∩ G_{σ} : < A_{t}x_{0}, x′_{0} > > z_{0}}.

Suppose that λ_{σ}(B) > 0. Then there exists a probability measure β_{B}, which is concentrated on B and absolutely continuous with respect to λ_{σ}. But this implies

< A_{β_{B}}x_{0}, x′_{0} > = ∫ < A_{t}x_{0}, x′_{0} > β_{B}(dt) > z_{0},

contrary to (8). So we see that λ_{σ}(B) = 0. In the same way we obtain

λ_{σ} ({t ∈ H ∩ G_{σ} : < A_{t}x_{0}, x′_{0} > < z_{0}}) = 0

such that

λ_{σ} ({t ∈ H ∩ G_{σ} : < A_{t}x_{0}, x′_{0} > ≠ < y_{0}, x_{0} >}) = 0

for every x′_{0} ∈ L. Thus Theorem 1 is proved.

3.6. It remains to prove Theorem 2 in case of (S2) and (S3) are satisfied. We choose some t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0, and some x′_{0} ∈ L. Let ε > 0. From (S3) it follows that there exists a neighbourhood U(t_{0}) of t_{0} in the topological group G_{σ} such that

| < A_{t}x_{0}, x′_{0} > − < A_{t_{0}}x_{0}, x′_{0} > | < ε, ∀t ∈ U(t_{0}).

Because of (S2) there exists a pobability measures β_{0} which is concentrated on H ∩ U(t_{0}) and absolutely continuous with respect to λ_{σ}. For the linear operator A_{β_{0}} we obtain

| < A_{β_{0}}x_{0}, x′_{0} > − < A_{t_{0}}x_{0}, x′_{0} > | ≤
∫ | < A_{t}x_{0}, x′_{0} > − < A_{t_{0}}x_{0}, x′_{0} > | β_{0}(dt) < ε.

Combining this result with (8) we arrive at

< A_{t_{0}}x_{0}, x′_{0} > = < y_{0}, x′_{0} >, ∀t_{0} ∈ H ∩ G_{σ}, ∀x′_{0} ∈ L.

Therefore for every operator A_{t}, t ∈ H ∩ G_{σ}, we have A_{t}x_{0} = y_{0}. Now it follows from equation (3) that y_{0} is identical with x_{0}.

4. Two Examples

For the following two examples the assumption (A5), (b) is always satisfied, since the operators A_{t} are continuous and L is the set of all continuous linear functionals:

< A_{σ}A_{t}x, x′ > = ∫ < A_{s}A_{t}x, x′ > σ(ds) = ∫ < A_{s+t}x, x′ > σ(ds) =
∫ < A_{s}x, x′(A_{t}) > σ(ds) = < A_{σ}x, x′(A_{t}) > = < A_{t}A_{σ}x, x′ > .

4.1. We consider the special case that V is a Hilbert space V_{H}. Now for all x, y ∈ V_{H} let < x, y > be the scalar product of x and y.

L_{H} := {< · , y >, y ∈ V_{H}}

as the set of all continuous linear functionals on V_{H} is separated.

Let (A2) and (A3) be satisfied. Further, let the linear operators A_{t}, t ∈ H, be continuous, let

sup_{t∈H} ∥ A_{t} ∥ < ∞

(where ∥A_{t}∥ denotes the norm of the operators A_{t}) and let (A4), (b) be satisfied. Finally, for every σ ∈ Σ let the operator A_{σ} be defined by the bilinear form

σ(x, y) = ∫ < A_{t}x, y > σ(dt) =: < A_{σ}x, y >, ∀x, y ∈ V_{H}.

Then the assumptions for Theorem 1 are satisfied. With the additional conditions (S1), (S2), (S3) we have also Theorem 2 in this special case. (Now (S3) means: For the fixed point x_{0} of the operator A_{σ}

lim_{t→t_{0},t∈H∩G_{σ}} A_{t} x_{0} = A_{t_{0}}x_{0}, ∀ t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0,

is satisfied in the sense of the weak convergence in V_{H}.)

4.2. Now let V be a Banach space V_{B} and let L_{B} be the set of all continuous linear functionals defined on V_{B}. Let G be the group (in reference to the addition) of the real numbers and let the subsemigroup H be the half axis [0, ∞).

Let {A_{t}, t ≥ 0} be a semigroup of continuous linear operators A_{t} : V_{B} ⟶ V_{B}, let

sup_{t≥0} ∥A_{t}∥ < ∞

and for every x ∈ V_{B} let A_{t}x be a continuous function of 0 ≤ t < ∞.

Then there exists for every probability measure σ ∈ Σ (that is T_{σ} ⊆ [0, ∞) ) and every x ∈ V_{B} the Riemann integral

∫_{0}^{∞} A_{t}x σ(dt).

We define the operator A_{σ}, σ ∈ Σ, by

A_{σ}x = ∫_{0}^{∞} A_{t}x σ(dt), ∀ x ∈ V_{B}.

(It is obvious that then

< A_{σ}x, x′ > = ∫_{0}^{∞} < A_{t}x, x′ > σ(dt), ∀ x ∈ V_{B}, ∀ x′ ∈ L_{B}.)

Thus in this special case the assumptions (A1) - (A5) are satisfied and from Theorem 2 we may conclude the following:

Let σ be finally absolutely continuous and let G_{σ} ⊇ [0, ∞). Then the equation A_{σ}x_{0} = x_{0} is equivalent to the equations A_{t}x_{0} = x_{0} for all t ≥ 0.

We choose in this statement for σ the exponential distribution E_{λ} with a parameter λ > 0. Then the operator A_{E_{λ}} is the Laplace transform L_{λ} (with the parameter λ) of the semigroup of operators {A_{t}, t ≥ 0} and we obtain:

An element x_{0} ∈ V_{B} is a fixed point of the Laplace transform L_{λ} if and only if x_{0} is a fixed point of all operators A_{t}, t ≥ 0.

5. Convolution Semigroups of Stochastic Kernels

Let S be a locally compact Hausdorff topological space and S the σ-algebra of its Borel sets. We assume that all signed measures on S, which we consider, are regular.

Let (A2) be satisfied. For every t ∈ H, defined in (A3), let P^{t} be a stochastic kernel on S × S, which means P^{t}(y, ·) is a probability measure on S for every y ∈ S and P^{t}( · , B) is a S-measurable function for every B ∈ S ). Let the following conditions be satisfied.

(a) P^{t_{2}}P^{t_{1}} = P^{t_{1}+t_{2}}, ∀ t_{1}, t_{2} ∈ H,

where P^{t_{2}}P^{t_{1}} denotes the convolution of the kernels P^{t_{1}} and P^{t_{2}}:

(P^{t_{2}}P^{t_{1}})(y, B) = (P^{t_{2}} ∗ P^{t_{1}})(y, B) = ∫ P^{t_{2}}(z, B)P^{t_{1}}(y, dz), ∀y ∈ S, ∀B ∈ S.

(b) P^{t}(y, B) is for every fixed B ∈ S a ((G ∩ H) × S)- measurable function of [t, y].

We consider signed measures α on S. |α| denotes the variation measure of α. Let V_{M} be the set of all signed measures α on S, for which with every compact set K ∈ S it holds that

(a) |α|(K) < ∞, (b) sup_{t∈H} ∫ P^{t}(y, K) |α|(dy) < ∞. (9)

In this vector space V_{M} we use the weak topology.

Let C_{0}(S) be the set of all continuous real functions defined on S with compact support. Let L_{M} be the set of the continuous linear functionals defined on V_{M} by

x′_{f}(α) = ∫ f(x) α(dx), f ∈ C_{0}(S).

L_{M} separate the points of the Hausdorff topological vector space V_{M}.

For every t ∈ H, by the usual convolution of the measure α with the stochastic kernel P^{t}

P^{t}α = ∫ P^{t}(y, ·) α(dy)

we get a linear operator P^{t} : V_{M} ⟶ V_{M} and {P^{t}, t ∈ H} represents a smigroup of operators which satisfies the conditions formulated in (A4).

Now for every σ ∈ Σ defined in (A5) let P^{σ} be the stochastic kernel

P^{σ}(y, B) = ∫ P^{t}(y, B) σ(dt), ∀y ∈ S, ∀B ∈ S, (10)

and for α ∈ V_{M} let P^{σ}α be the convolution of α with P^{σ}

∫∫_{y∈S,t∈H} P^{t}(y, ·) σ(dt) α(dy).

Thus also the assumption (A5) is satisfied.

Summarizing we see that we have a special case of the general structure described in (A1) - (A5). By specialization of Theorem 1 we obtain

Theorem 3. If P^{σ}α_{0} = α_{0} for some σ ∈ Σ and some α_{0} ∈ V_{M}, then for every f ∈ C_{0}(S) the function

∫∫_{x∈S,y∈S} f(x) P^{t}(y, dx) α_{0}(dy), t ∈ H,

is constant for λ_{σ}-almost all t ∈ H ∩ G_{σ}.

Before transfering Theorem 2 we discuss the condition (S3), which now means that for the fixed point α_{0} of the operator P^{σ} and for every x′_{f} ∈ L_{M}

< P^{t}α_{0}, x′_{f} > = ∫∫_{x∈S,y∈S} f(x) P^{t}(y, dx) α_{0}(dy), t ∈ H,

is continuous with respect to t ∈ H ∩ G_{σ}, t ≠ 0. For this both of the following two conditions are sufficient.

(S3,1) For every y ∈ S and every t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0, we have the weak convergence

P^{t}(y, ·) ⟹ P^{t_{0}}(y, ·), as t → t_{0}, t ∈ H ∩ G_{σ},

and - with respect to y - this convergence is uniform in every compact set K ∈ S (that is

lim_{t→t_{0},t∈H∩G_{σ}} sup_{y∈K} | ∫ f(x)(P^{t}(y, dx) − P^{t_{0}}(y, dx)) | = 0, ∀f ∈ C_{0}(S) ).

(S3,2) For every compact set K ∈ S and every t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0, there exists a neighbourhood U(t_{0}) of t_{0} in the topological group G_{σ} such that the family of functions

{ P^{t}( · , K), t ∈ U(t_{0}) ∩ H }

is uniformly integrable with respect to α_{0}.

Now we get from Theorem 2 the following statement.

Theorem 4. Let P^{σ}α_{0} = α_{0}. Assume, further, that either conditions (S1) or the conditions (S2), (S3,1) and (S3,2) are satisfied. Then P^{t}α_{0} = α_{0} for all t ∈ H ∩ G_{σ}.

Now the convolution semigroup {P^{t}, t ∈ H} will operate in a space of measurable functions.

Let V_{F} be the set of the real functions which are defined on S, S-measurable and bounded. In this vector space V_{F} we use the topology induced by the norm

∥f∥ = sup_{y∈S} |f(y)|, f ∈ V_{F} .

For every y ∈ S let y′ be the continuous linear functional defined on V_{F} by y′(f) = f(y). L_{F} := {y′ : y ∈ V_{F} } separate the points of the Hausdorff topological vector space V_{F} .

By the usual convolution of the function f with the stochastic kernel P^{t}

P^{t}f = ∫ f(x) P^{t}( · , dx)

we get for every t ∈ H a linear operator P^{t} : V_{F} ⟶ V_{F} and {P^{t}, t ∈ H} is again a semigroup of operators satisfying the conditions formulated in (A4).

For every σ ∈ Σ let P^{σ} be the stochastic kernel defined in (10) and for every f ∈ V_{F} let P^{σ}f be the convolution of f with P^{σ}

P^{σ}f = ∫∫_{x∈S,t∈H} f(x) P^{t}( · , dx) σ(dt).

The assumptions (A1) - (A5) are satisfied and Theorem 1 yields our next result.

Theorem 5. If P^{σ}f_{0} = f_{0} for some σ ∈ Σ and some f_{0} ∈ V_{F}, then for every y ∈ S the function

∫ f_{0}(x) P^{t}(y, dx), t ∈ H,

is constant for λ_{σ}-almost all t ∈ H ∩ G_{σ}.

In the present case condition (S3) means: For the fixed point f_{0} of the operator P^{σ} and for every y′ ∈ L_{F}

< P^{t}f_{0}, y′ > = ∫ f_{0}(x) P^{t}(y, dx), t ∈ H,

is continuous with respect to t ∈ H ∩ G_{σ}, t ≠ 0. For this the continuity of the kernels P^{t} with respect to the convergence in variation is sufficient:

(S3,3) For every y ∈ S and every t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0, we have

lim_{t→t_{0},t∈H∩G_{σ}} ∥ P^{t}(y, ·) − P^{t_{0}}(y, ·) ∥ = 0.

If the fixed point f_{0} is continuous, then the continuity of the kernels P^{t} with respect to the weak convergence is sufficient:

(S3,4) For every y ∈ S and every t_{0} ∈ H ∩ G_{σ}, t_{0} ≠ 0, we have the weak convergence

P^{t}(y, ·) ⟹ P^{t_{0}}(y, ·), as t → t_{0}, t ∈ H ∩ G_{σ}.

Using Theorem 2 we obtain in this special case the following theorem.

Theorem 6. Let P^{σ}f_{0} = f_{0}. Then in each of the cases (a) (S1) is satisfied,

(b) (S2) and (S3,3) are satisfied,

(c) f_{0} is continuous and (S3,4) is satisfied,

it holds that P^{t}f_{0} = f_{0} for all t ∈ H ∩ G_{σ}.

6. Shift Groups

We are going to consider two special cases of the structures which we have studied already in Section 5. We identify the space S with the group G defined in (A2) and the semigroup H defined in (A3) also with the group G. Now for every t ∈ G let P^{t} be the stochastic kernel on G × G

P^{t}(y, B) = δ_{t}(B − y), ∀y ∈ G, ∀B ∈ G.

Condition (9) passes into

sup_{t∈G} |α|(K − t) < ∞

to be valid for every compact set K ∈ G. We denote the space of these shift bounded signed measures α with V_{M_{0}}.

{P^{t}, t ∈ G} is a shift group of operators: The convolution of a measure α ∈ V_{M_{0}} and a function f ∈ V_{F} with a kernel P^{t} leads to the shift of α and f, respectively, by t, it is

(P^{t}α)(B) = α(B − t), ∀B ∈ G, (P^{t}f)(y) = f(y + t), ∀y ∈ G.

For every probability measure σ on G now P^{σ}(y, B) is the convolution kernel σ(B − y), y ∈ G, B ∈ G. Therefore, if we search for solutions in V_{M_{0}} and V_{F}, respectively, then the equations (1) and (2) are identical with the equations

P^{σ}α_{0} = α_{0}, P^{σ}f_{0} = f_{0}.

From the conditions (S1) - (S3) the condition (S2) is always fulfilled (because H = G). In the case that the shift group operates in V_{M_{0}} also the condition (S3) is always satisfied, because for every α ∈ V_{M_{0}} and for every x′_{f} ∈ L_{M} (that is for every f ∈ C_{0}(G))

< P^{t}α, x′_{f} > = ∫ f(y + t) α(dy)

is a continuous function of t ∈ G. In the case that the shift group operates in V_{F} the condition (S3) is satisfied if and only if the fixed point f_{0} is a continuous function (because we have

< P^{t}f_{0}, y′ > = f_{0}(y + t), ∀y′ ∈ L_{F}).

From the Theorems 2, 3 and 4 we obtain the following propositions.

Proposition 1. If σ ∗ α = α for some probability measure σ on G and some shift bounded signed measure α on G, then for every compact set K ∈ G and every t ∈ G_{σ}

α(K) = α(K − t).

In the case G_{σ} = G it is α = cλ with a constant c ∈ R.

Remark to Proposition 1. If we investigate the equation (3) for a fixed probability measure σ_{0} ∈ Σ, then we use only operators A_{t}, t ∈ H ∩G_{σ_{0}}. Therefore, we can put H ∩ G_{σ_{0}} instead of H in the assumptions (A1) - (A5). Using this fact, for example, in the assumptions of Proposition 1 ”α is shift bounded” can be replaced by the restricted shift boundedness of α

sup_{t∈G_{σ}} |α| (K − t) < ∞ for every compact set K ∈ G.

Proposition 2. If

∫ f(y + t) σ(dt) = f(y), ∀y ∈ G,

for some probability measure σ on G and some bounded measurable function f on G, then for every y ∈ G the function

f_{y}(t) := f(y + t), t ∈ G,

is constant for λ_{σ}-almost all t ∈ G_{σ}. (In the case G_{σ} = G the function f is constant λ-almost everywhwere.)

Proposition 3. If

∫ f(y + t) σ(dt) = f(y), ∀y ∈ G,

and if in addition the probability measure σ satisfies (S1) or the bounded measurable function f is continuous, then for every y ∈ G it holds that

f(y) = f(y + t), ∀t ∈ G_{σ}.

In the case G_{σ} = G the function f is constant.

The following was proved by Klimov [5]: Let G = G_{σ} = R^{d} and let w : R^{d} ⟶ R be bounded and regular. If w(y) = ∫ w(y − t) σ(dt), ∀y ∈ R^{d}, (11)

then the function w is constant. (A measurable function w is called regular if the set of points of discontinuity is expressible as ∪A_{n} with each A_{n} nowhere dense and if w(t) ≠ const. ∀t ∈ R^{d} implies the existence of points of continuity a and b such that w(a) ≠ w(b).)

This result can be obtained from Proposition 2. The equation (11) has the form of the equation (2) with the measure σ^{−}, σ^{−}(B) = σ(−B) for all Borel sets B, instead of σ. Since G_{σ^{−}} = G_{σ} = R^{d} Proposition 2 tells us that w is λ-almost everywhere constant. But if a regular function is λ-almost everywhere constant then this function is constant. (Suppose that w is not constant. Then there exist continuity points a, b with w(a) < w(b). Therefore there exist also neighbourhoods U(a) of a and U(b) of b such that w(x) < w(y) for all x ∈ U(a) and all y ∈ U(b). This contradicts the fact that w is λ-almost everywhere constant.)

Appendix: Characterization of the Haar Measure

Proposition 1 contains a characterization of the Haar measure on a locally compact abelian Hausdorff topological group G. We can not omit the shift boundedness of α from the corresponding statement. But we can replace this assumption by additional restrictions to the probability measure σ and the topological group G. Now let α be a Radon measure (that is 0 ≤ α(K) < ∞ for every compact Borel set K).

Proposition 4. Let G be a locally compact abelian topological group which is metrizable and separable, let G denote the σ-algebra of their Borel sets. Let σ be a probability measure on G, which is

(a) non-arithmetic (that is G_{σ} = G) and

(b) symmetrical (that is σ(B) = σ(−B), ∀B ∈ G).

If for a Radon measure α on G, which is not identically zero, the convolution equation σ ∗ α = α is true then α is a Haar measure.

P r o o f. We can see in [3] or [6] that certain exponential measures generate the set of the solutions α of the equation σ ∗ α = α. These measures are absolutely continuous with respect to λ and their densities are nonnegative exponential functions g with the property

∫ g(−y) σ(dy) = 1. (12)

(A function g : G ⟶ R which is continuous and satisfies g(x + y) = g(x)g(y), ∀x, y ∈ G, is said to be an exponential function.)

Suppose that (12) is satisfied by an exponential function g_{0}, which is not identical one. Let M := {y ∈ G : g_{0}(y) = 1}. Then M is a closed subgroup of G, which is not identical with G. Since G = G_{σ}, it follows σ(M) < 1. Further, since

g_{0}(−y) = \frac{1}{g_{0}(y)}, ∀y ∈ G, we have g_{0}(y) + g_{0}(−y) \{ \begin{matrix}= 2 if & y ∈ M, \\ > 2 if & y ∉ M. \end{matrix}

Therefore, we obtain

I(g_{0}) := ∫ (g_{0}(y) + g_{0}(−y)) σ(dy) > 2.

On the other hand from (12) and the symmetry of σ it follows that I(g_{0}) = 2. This contradiction shows that (12) has only the trivial solution g_{0} ≡ 1. The proof is complete.

For special groups G we can find also other conditions (b), which guarantee together with (a) that (12) has only the trivial solution g ≡ 1. We finish this paper with such a special obvious assertion.

Remark to Proposition 4. If G is the group of all real numbers or the group of all integers, then condition (b) can be replaced by the assumption that σ is concentrated on a half axis.

References

[1] R. Balasubrahmanyan, K. S. Lau, Functional Equations in Probability Theory, Academic Press (1991)

[2] G. Choquet, J. Deny, Sur l’équation de convolution μ = μ ∗ σ, C. R. Acad. Sci. Paris 250 (1960), 799 - 801

[3] J. Deny, Sur l’équation de convolution μ ∗ σ = μ, Semin. Théor. Potent. M. Brelat, Fac. Sci. Paris, 4 ann. (1960)

[4] J. Kerstan, K. Matthes, Gleichverteilungseigenschaften von Faltungen von Verteilungsgesetzen auf lokalkompakten abelschen Gruppen I, Math. Nachr. 37 (1968), 267 - 312

[5] G. P. Klimov, A Solution of the Equation w = w ∗ F where F is a Probability Measure on R^{d}, Appl. Math. Lett. 5(5) (1992), 41 - 44

[6] K. S. Lau, W. B. Zeng, The Convolution Equation of Choquet and Deny on Semigroups, Studia Math. 97(2) (1990), 115 -135