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On Bounded Solutions to Convolution Equations PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 11, November 2006, Pages 3275–3286 S 0002-9939(06)08608-4 Article electronically published on May 8, 2006 ON BOUNDED SOLUTIONS TO CONVOLUTION EQUATIONS ENRICO PRIOLA AND JERZY ZABCZYK (Communicated by Jonathan M. Borwein) Abstract. Periodicity of bounded solutions for convolution equations on a separable abelian metric group G is established, and related Liouville type the- orems are obtained. A non-constant Borel and bounded harmonic function is constructed for an arbitrary convolution semigroup on any infinite-dimensional separable Hilbert space, generalizing a classical result by Goodman (1973). 1. Introduction Let µ be a probability measure defined on the σ-algebra B(G) of Borel subsets of a separable abelian metric group G, with the group operation “+”. This group might be non-locally compact. This paper is concerned with convolution equations of the type (1.1) f ∗ µ(x):= f(x − y)µ(dy)=f(x),x∈ G, G where f : G → R is a Borel and bounded function, written f ∗ µ = f for short. Our aim is to investigate bounded Borel solutions f to (1.1). These functions are also called bounded µ-harmonic functions; see [6]. Special attention will be paid to the case when G is a real separable Hilbert space. Convolution equations arise naturally in several areas of pure and applied math- ematics such as harmonic analysis (see [6], [4] and [15]), the theory of Markovian semigroups (see [9], [1] and [11]), and the renewal theory (see [10]). Define the shift µh of the measure µ by the element h ∈ G, through the formula µh(A)=µ(A − h),A∈B(G). Consider the set Mµ ⊂ G of all elements h ∈ G such that measures µh are abso- lutely continuous with respect to µ. The set Mµ has been introduced in [12] and investigated, for instance, in [24], [23] and [2]. Our main result (see Theorem 2.5) shows that each Borel and bounded solution f to (1.1) is periodic with periods in Mµ i.e., (1.2) f(x + h)=f(x),x∈ G, h ∈ Mµ. Received by the editors May 25, 2005. 2000 Mathematics Subject Classification. Primary 43A55, 68B15, 47D07, 31C05. Key words and phrases. Convolution equations on groups, bounded harmonic functions, L´evy processes. The first author was partially supported by Italian National Project MURST “Equazioni di Kolmogorov” and by Contract No ICA1-CT-2000-70024 between European Community and the Stefan Banach International Mathematical Center in Warsaw. c 2006 American Mathematical Society 3275 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3276 ENRICO PRIOLA AND JERZY ZABCZYK The theorem holds in the more general setting of measurable abelian groups; see Remark 2.11. Moreover the set Mµ can be replaced by a larger set Eµ;see(2.1). The result seems particularly useful in infinite dimensions, when G is a separable Hilbert space H and there is no reference to any Haar measure. In Section 3.1 we consider an application to α−stable measures µ on H, α ∈ (0, 2]. A related equation, (1.3) ν ∗ µ = ν, on a locally compact group G, with unknown non-negative measure ν,wasthe subject of a classical paper [3] by Choquet and Deny. Under suitable assumptions on ν (see Section 2.3), they proved that ν is periodic with periods in the subgroup generated by the support Sµ of µ (i.e., Sµ is the smallest closed set of G on which µ is concentrated). Within the class of locally compact groups, this result and related Liouville theorems have been extended in several directions; see [6], [15], [4], [21] and the references therein. Even when G is locally compact, our result does not follow from [3]; see Remark 2.8. Moreover (1.2) does not hold if Mµ is replaced by Sµ. On the other hand, as a consequence of our main result, in Proposition 2.12 we obtain a version of the Choquet-Deny theorem concerning (1.3), which holds on metric groups (replacing the subgroup generated by Sµ with a smaller subgroup which contains Mµ). We also show in Theorem 2.9 that uniformly continuous and bounded solutions to (1.1) are periodic with periods in Sµ (see also Corollary 3.3.2 in [20]). Equation (1.1) and related Liouville type theorems on a separable Hilbert space H are considered in Section 3. Let (µt)t≥0 be a convolution semigroup of probability measures on H. A function f ∈ Bb(H) such that, for any t ≥ 0, (1.4) f ∗ µt(x)=f(x),x∈ H, is called a bounded harmonic function with respect to (µt). In Theorem 3.3 we prove that if the space H is infinite dimensional, then one can always construct a discontinuous non-constant solution f to (1.4). This result generalises a well-known example of Goodman [14], concerning the case when (µt) are Gaussian measures. The Goodman result shows clearly that (1.2) cannot hold if Mµ is replaced by Sµ. This was one motivation for us to investigate (1.1). Theorem 3.3 also implies that Markovian convolution semigroups associated to (µt) are never strong Feller in infinite dimensions; cf. [7]. 2. Convolution equations on metric groups Let G be a separable abelian metric group with group operation indicated by +; see [16] and [23]. For A, B ⊂ G,weset A + B = {x + y, x ∈ A, y ∈ B}, −A = {−x, x ∈ A}. Moreover Gr(A) denotes the smallest subgroup containing A,andA denotes the closure of A. The indicator function of a set F ⊂ G will be indicated by 1F . When G is complete, we call it an abelian Polish group. When G is a real separable Hilbert space, we denote it by H. The inner product of H is then ·, · and its norm is |·|. The probability measures on G we consider will always be Borel probability measures on the σ-algebra B(G). Let σ be a probability measure on G.The License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ON BOUNDED SOLUTIONS TO CONVOLUTION EQUATIONS 3277 support of σ is denoted by Sσ; it is the smallest closed set in G which has measure 1 with respect to σ. Withσ ˜ we indicate its reflection measure with respect to 0 (see [22, Chapter 1]), i.e.σ ˜(A)=σ(−A) for any A ∈B(G). The probability measure σ is called symmetric if σ =˜σ, i.e. σ(A)=σ(−A),A∈B(G). For µ and ν probability measures on G,theconvolution measure µ ∗ ν is defined ∗ − ∈B by µ ν(A):= G µ(A x)ν(dx), A (G); see for instance [16] or [23]. Note that the operation ∗ is commutative and associative. We write µn = µ ∗···∗µ (n−times), n ∈ N (N denotes the set of all positive 0 integers). We also set µ = δ0,whereδ0 is the Dirac measure concentrated in 0. By Bb(G) we denote the Banach space of all real, Borel and bounded functions f : G → R, endowed with the supremum norm ·∞.Ifg ∈ Bb(G), we set g ∗ µ(x)= g(x − y)µ(dy),x∈ G. G 2.1. Admissible shifts. Let Ta, a ∈ G, be the translation operator, i.e. Ta(x)= x + a, x ∈ G, and denote by Ta ◦ µ the image of a probability measure µ under Ta, i.e. (Ta ◦ µ)(A)=µ(A − a), A ∈B(G). We also set (Ta ◦ µ)=µa. According to [12, page 449], a ∈ G is called an admissible shift for µ if Ta ◦ µ is absolutely continuous with respect to µ. Let us denote by Mµ the set of all admissible shifts. Note that 0 ∈ Mµ. Moreover it is known that Mµ is always a semigroup of G; see [12, page 450]. Since Mµ˜ = −Mµ,where˜µ is the reflection measure of µ (see [24]), we have that Mµ is a subgroup of G if µ is symmetric. If G is locally compact and µ is equivalent to the Haar measure of G,thenitis easy to show that Mµ = G. We introduce the set n (2.1) Eµ = Mµn ,µ= µ ∗···∗µ (n-times) n ∈ N. n≥0 We will need the following elementary result. Proposition 2.1. Let µ and ν be probability measures on G.Then (2.2) Mµ + Mν ⊂ Mµ∗ν . Moreover Eµ is a semigroup of G. Proof. Take any A ∈B(G) such that µ∗ν(A)=0anda ∈ Mµ, b ∈ Mν . It is enough to show that (µ ∗ ν)a+b(A)=0.Wehave (µ ∗ ν)a+b(A)= 1A(x + y)p(x)q(y)µ(dx)ν(dy), G G where p and q are the densities of µa and νb, respectively. For any N ∈ N,set p = N ∧ p and q = N ∧ q.Then N N 2 1A(x + y)pN (x)qN (y)µ(dx)ν(dy) ≤ N 1A(x + y)µ(dx)ν(dy)=0. G G G G Taking N →∞, we get the first assertion. This implies that Eµ is an increasing union of semigroups. The second assertion follows easily. The next example shows that the inclusion in (2.2) can be strict. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3278 ENRICO PRIOLA AND JERZY ZABCZYK n Example 2.2. There exists a probability measure µ on R such that Mµ =0and Rn Mµ2 = = Eµ. Define µ = k≥1 pkνk,whereνk are uniform distributions on the spheres cen- ∈ N ∈ N tered in 0, of radiuses k ,and k≥1 pk =1,pk > 0, k .
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