QUADERNI. del. Dipartimento di Matematica. Enrico Priola & Jerzy Zabczyk ON BOUNDED SOLUTIONS TO CONVOLUTION EQUATIONS QUADERNO N.

Transcription

1 UNIVERSITÀ DI TORINO QUADERNI del Dipartimento di Matematica Enrico Priola & Jerzy Zabczyk ON BOUNDED SOLUTIONS TO CONVOLUTION EQUATIONS QUADERNO N. 11/2005 Periodicity of bounded solutions for convolution equations on a separable abelian metric group is established and related Liouville type theorems are obtained. The group is not locally compact in general. 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 oodman.

2 Stampato nel mese di Maggio 2005 presso il Centro Stampa del Dipartimento di Matematica dell Università di Torino. Address: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, Torino, ITALY Phone: ( ) Fax: ( ) E mail: dm.unito.it

3 ON BOUNDED SOLUTIONS TO CONVOLUTION EQUATIONS May 9, 2005 Enrico Priola ( 1 ) Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123, Torino, Italy. priola@dm.unito.it Jerzy Zabczyk Instytut Matematyczny, Polskiej Akademii Nauk, ul. Sniadeckich 8, , Warszawa, Poland. zabczyk@impan.gov.pl Mathematics Subject Classification (2000): 43A5, 68B15, 47D07, 31C05. Key words: Convolution equations on groups, bounded harmonic functions, Lévy processes. Abstract: Periodicity of bounded solutions for convolution equations on a separable abelian metric group is established and related Liouville type theorems 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 oodman [13]. 1 Introduction Let µ be a probability measure defined on the σ-algebra B() of Borel subsets of a separable abelian metric group, with the group operation +. This group might be non-locally compact. The paper is concerned with convolution equations of the type f µ(x) := f(x y)µ(dy) = f(x), x, (1.1) where f : R is a Borel and bounded function, written shortly f µ = f. 1 Partially supported by Italian National Project MURST Equazioni di Kolmogorov and by Contract No ICA1-CT between European Community and the Stefan Banach International Mathematical Center in Warsaw. 1

4 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 is a real separable Hilbert space. Convolution equations arise naturally in several areas of applied and pure mathematics, like harmonic analysis, see [6], [4] and [14], the renewal theory, see [9], the theory of Markovian semigroups, see [8], [1] and [10]. Define the shift µ h of the measure µ by the element h, through the formula: µ h (A) = µ(a h), A, A B(). Consider the set M µ of all elements h such that measures µ h are absolutely continuous with respect to µ. The set M µ has been introduced in [11] and investigated, for instance, in [21], [22] 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 M µ i.e., f(x + h) = f(x), x, h M µ. (1.2) This theorem seems particularly useful in infinite dimensions, when is a Hilbert space H and so there is no a reference Haar measure L. In Section 3.1 we consider an application to α stable measures µ, α (0, 2]. For a more general setting of measurable abelian groups in which our theorem continues to hold, we refer to Remark A related equation, ν µ = ν, (1.3) on a locally compact group, with unknown a non-negative measure ν, was the 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 group generated by the support S µ of µ (i.e., S µ is the smallest closed set of 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], [14], [4], [19] and references therein. Even when is locally compact, our result does not follow from [3], see Remark 2.7. Moreover (1.2) does not hold if M µ is replaced by S µ. On the other hand, as a consequence of our main result, in Proposition 2.10 we obtain a version of the Choquet- Deny theorem concerning (1.3), which holds on general metric groups (replacing the group generated by S µ with a smaller group which contains M µ ). We also show in Theorem 2.8 that uniformly continuous and bounded solutions to (1.1) are periodic with periods in S µ. In Section 3 we consider equation (1.1) and related Liouville type theorems on a separable Hilbert space H. Let (µ t ) t 0 be a convolution semigroup of probability measures of H. A function f B b (H) such that, for any t 0, f µ t (x) = f(x), x H, (1.4) is called a bounded harmonic function with respect to (µ t ). In Theorem 3.3 we prove that if the space H is infinite dimensional, then one can always construct a discontinuous nonconstant solution f to (1.4). This result generalises a well known example of oodman [13], concerning the case when (µ t ) are aussian measures. The oodman result shows clearly that (1.2) can not hold if M µ is replaced by S µ. This was one motivation for us to investigate (1.1). Theorem 3.3 also implies that Markovian convolution semigroups associated to (µ t ) are never strong Feller in infinite dimensions, cf. [7]. 2

5 2 Convolution equations on metric groups Let be a separable abelian metric group with group operation indicated by +, see [16] and [22]. For A, B, we set A + B = {x + y, x A, y B}, A = { x, x A}. Moreover r(a) denotes the smallest subgroup containing A and A the closure of A. The indicator function of a set F will be indicated by 1 F. When is a real separable Hilbert space, we denote it by H. The inner product of H is then, and its norm. The probability measures on we consider will be always Borel probability measures on the σ-algebra B(). Let σ be a probability measure on. The support of σ is denoted by S σ ; it is the smallest closed set in which has measure 1 with respect to σ. With σ we indicate its reflection measure with respect to 0, see [20, Chapter 1], i.e. σ(a) = σ( A), for any A B(). The probability measure σ is called symmetric if σ = σ, i.e. σ(a) = σ( A), A B(). For µ and ν probability measures on, the convolution measure µ ν is defined by µ ν(a) := µ(a x)ν(dx), A B(), see for instance [16] or [22]. Note that the operation is commutative and associative. We write µ n = µ µ (n times), n N (N denotes the set of all positive integers). We also set µ 0 = δ 0, where δ 0 is the Dirac measure concentrated in 0. By B b () we denote the Banach space of all real, Borel and bounded functions f : R, endowed with the supremum norm. If g B b (), we set 2.1 Admissible shifts g µ(x) = g(x y)µ(dy), x. Let T a, a, be the translation operator, i.e., T a (x) = x + a, x, and denote by T a µ the image of a probability measure µ under T a, i.e., (T a µ)(a) = µ(a a), A B(). We also set (T a µ) = µ a. According to [11, page 449], a is called an admissible shift for µ if T a µ is absolutely continuous with respect to µ. Let us denote by M µ the set of all admissible shifts. Note that 0 M µ. Moreover it is known that M µ is always a semigroup of, see [11, page 450]. Since M µ = M µ, where µ is the reflection measure of µ, see [21], we have that M µ is a subgroup of if µ is symmetric. If is locally compact and µ is equivalent to the Haar measure of, then it is easy to show that M µ =. We introduce the set E µ = n 0 M µ n, µ n = µ µ (n times) n N. (2.1) We will need the following elementary result. 3

6 Proposition 2.1. Let µ and ν be probability measures on. Then Moreover E µ is a semigroup in. M µ + M ν M µ ν. (2.2) Proof. Take any A B() such that µ ν(a) = 0 and a M µ, b M ν. It is enough to show that (µ ν) a+b (A) = 0. We have (µ ν) a+b (A) = 1 A (x + y)p(x)q(y)µ(dx)ν(dy), where p and q are the densities of µ a and ν b respectively. For any N N, set p N = N p and q N = N q. Then 1 A (x + y)p N (x)q N (y)µ(dx)ν(dy) N 2 1 A (x + y)µ(dx)ν(dy) = 0. Taking N, we get the first assertion. This implies that E µ is an increasing union of semigroups. The second assertion follows easily. The next example shows that the inclusion in (2.2) can be strict. Example 2.2. There exists a probability measure µ on R n such that M µ = 0 and M µ 2 = R n = E µ. Define µ = k 1 p kν k, where ν k, are uniform distributions on the spheres centered in 0, of radiuses k N, and k 1 p k = 1, p k > 0, k N. Since the measures ν k ν l are absolutely continuous with respect to Lebesgue measure, see [9], and the density of ν k ν k is positive on the ball B(0, 2k), we have µ µ = p k p l ν k ν l, k,l=1 and µ µ has a positive density on R n. The assertion follows. We compare now the set E µ with the group generated by the support S µ of µ. Proposition 2.3. Let µ be a probability measure on. Then E µ r(s µ ). Proof. Fix any h M µ. It is straightforward to check that S µ + h S µ. Now take x S µ, x 0 (if S µ = {0}, then µ is the Dirac measure concentrated in 0 and M µ = {0} as well). We know that x + h S µ r(s µ ). Since also x r(s µ ), it follows that h r(s µ ). We have proved that M µ r(s µ ). For any n N, one has M µ n r(s µ n) = r(s µ ). Hence the assertion holds. In general the sets M µ and S µ are different. Example 2.4. There exists a probability measure µ on R n such that M µ and S µ are disjoint sets. Take x 0 R n, x 0 0, and u R n such that u = 1. Consider the line L = {x R n : x = λu + x 0, for λ R}. Let µ be a probability measure on R n, concentrated on L, having a positive density with respect to the one dimensional Lebesgue measure. We have that M µ = {λu} λ R and S µ = L. 4

7 2.2 Characterization theorems Theorem 2.5. Let be a separable abelian metric group. Let µ be a probability measure on and let E µ be defined in (2.1). Let f B b () be a µ-harmonic function. Then one has: f(x + a) = f(x), x, a r(e µ ). (2.3) If, in addition, f is continuous on and r(e µ ) is dense in, then f is constant. Proof. We first define, similarly to [3], suitable auxiliary functions and then obtain the required characterization arguing by contradiction. Both steps are accomplished differently from [3]. In particular, instead of the Ascoli-Arzela theorem, we use arguments based on L -weak compactness. Let us introduce f, f(x) = f( x), x. In this way the equation f µ = f is equivalent to f µ = f, where µ is the reflection measure of µ. Now we consider some auxiliary functions as in [3]. Take a M µ and introduce the function g(x) = f(x) f(x + a). It is clear that g B b () and g µ = g on. Let 2c = sup g(x) x and (x n ) such that g(x n ) 2c as n. Consider the functions g n : R, g n (x) = g(x + x n ), x. Each g n B b () and solves the convolution equation (1.1). Now we set L = L (, µ) and use weak L -convergence. Now the proof is divided into some steps. Step I. The sequence (g n ) is L -weak compact. It is enough to remark that (g n ) is bounded in L. Hence, possibly passing to a subsequence, still denoted by (g n ), we know that there exists g 0 L such that, for any f L 1 (, µ), g n (y)f(y)µ(dy) g 0 (y)f(y)µ(dy), as n. Step II. The limit function g 0 = 2c, µ-a.s.. Note that, for x, g n (x) = g n (x y) µ(dy) = g n (x + y)µ(dy), n N. (2.4) Set x = 0 in (2.4). Passing to the limit as n, using the L -weak convergence, we have: 2c = lim g n(0) = g 0 (y)µ(dy). (2.5) n Now we prove that g 0 (x) 2c, µ a.s. If this does not hold, then there exists ɛ > 0 such that B = {x, g 0 (x) 2c + ɛ} verifies µ(b) > 0. 5

8 But then we have, using that g n (x) 2c, x, 2cµ(B) g n (y)µ(dy), n N. Passing to the limit, as n, we infer a contradiction: 2cµ(B) g 0 (y)µ(dy) (2c + ɛ)µ(b). Hence, using also (2.5), we get the claim. B B Step III. There exists a subsequence of (g n ), still denoted by (g n ), which converges pointwise to 2c, µ a.s. It is enough to show that (g n ) converges to 2c in probability (with respect to µ). To this purpose, we write, using that g n 2c, n 1, ( ) µ x : g n (x) 2c > ɛ 1 (2c g n (y))µ(dy) 0, as n. ɛ Step VI. For any x M µ, lim g n(x) = 2c. (2.6) n By (2.4) we have, for any x M µ, g n (x) = g n (y)(t x µ)(dy) = g n (y)f x (y)µ(dy), x M µ, n N, (2.7) where T x is the translation operator and F x denotes the density of (T x µ) with respect to µ. Passing to the limit, as n, in (2.7) we infer lim g n(x) = 2c (T x µ)(dy) = 2c, x M µ. n Final Step. Recall that M µ is a semigroup in, see [11, page 450]. This fact and (2.6) imply that g 0 (ka) = 2c, k N. Now we complete the proof similarly to Choquet-Deny [3]. For any integer m, there exists ˆn such that gˆn (ka) = f(xˆn + ka) f(xˆn + (k 1)a) > c, (2.8) for k = 1,..., m. Summing (4.3) m-times, we get f(xˆn + ma) f(xˆn ) > mc. Letting m, we find that c 0, since f is bounded. This means that g(x) 0, x, i.e., f(x) f(x + a), x, Repeating the previous argument with f instead of f, one has: f(x) = f(x + a), x. Thus (2.3) holds, for any a M µ. Equation (1.1) implies that, for any n N, f µ n (x) = f(x), x. Repeating the previous argument, we get that (2.3) holds even for any a E µ. The assertion follows remarking that the set of all periods of a given real function on is a subgroup of. The proof is complete. 6

9 Remark 2.6. If µ is symmetric then r(e µ ) = E µ and so the formulation of Theorem 2.5 simplifies. Indeed if µ is symmetric, then M µ is a subgroup of. By Proposition 2.1 we know that E µ is an increasing union of subgroups. Hence E µ is a group. Remark 2.7. Theorem 2.5 does not hold if we replace E µ with the group generated by the support S µ of µ. Let = R d and take Q d = {q n } be the set of all points in R d having rational coordinates. Let (p n ) R + be such that n 1 p n = 1. Define µ = n 1 p n δ qn, where δ qn are Dirac measures concentrated in q n. Take f = 1 Q d be the indicator function of Q d. It is straightforward to check that µ f(x) = f(x), x R n. Moreover the support S µ = R d and M µ = Q d. Note that if h Q d, then f(x + h) = f(x) only if x A h, where A h = h + Q d has Lebesgue measure 0. If we restrict our attention to bounded µ-harmonic functions which are also uniformly continuous on, then we can prove periodicity with respect to r(s µ ). In the terminology of [4, page 396] the next result shows that any Polish abelian group has the Liouville property. We denote by UC b () the space of all uniformly continuous and bounded functions from into R. Theorem 2.8. Let be a Polish abelian group. Let µ be a probability measure on. Let f UC b () be a solution to f µ = f. Then, f(x + a) = f(x), for any x, a r(s µ ). (2.9) We provide the proof in Appendix. It uses the same arguments given in [3] but the next lemma is needed. Lemma 2.9. Let be an abelian Polish group. There exists a subgroup S 0, which is a countable union of compact sets and has the property that µ(s 0 ) = 1. Proof. Chose first compact sets n such that 0 n, n n+1 and µ(\ n ) < 1/n. Define new compacts F n = ( n ) n, n 1 and finally set K 1 = F 1, K 2 = F 2 + F 2,..., K n = F n F n (n times),.... It is easy to check that S 0 = n 1 K n, (2.10) has all the required properties. 2.3 Connections with the Choquet-Deny theorem It is interesting to compare our result with the remarkable theorem due to Choquet and Deny, mentioned in Introduction, valid in locally compact groups, see [3]. Their theorem is concerned with the equation ν µ = ν, (2.11) 7

10 where µ is a given probability measure on and unknown ν is a σ finite Borel measure on such that, for any compact set K, the Borel non-negative function: R +, x ν(x K) = 1 K ν(x) is finite and bounded on. (2.12) It turns out that ν is periodic with periods in the group generated by the support S µ of µ, i.e. ν(a + h) = ν(a), h r(s µ ), A B(). (2.13) This result can be applied to the study of equation (1.1). Let us interpret function f as a density of a measure ν with respect to the Haar measure L of : f = dν dl. Then (2.13) implies that, for any h S µ, f(x + h) = f(x), L a.s., where the set of L-measure 0 depends, in general, on h, see Remark 2.7. The following corollary of Theorem 2.5 can be regarded as a version of [3, Theorem 1] in the non-locally compact case. Proposition Let µ be a given probability measure on a Polish abelian group. Let ν be a σ finite Borel measure on satisfying the condition (2.12) for any compact set K. If ν µ = ν, (2.14) then ν is periodic with periods in E µ, see (2.13). Proof. Take first a compact set K and consider the indicator function of K, i.e. 1 K. We have: (1 K ν) µ = 1 K ν. Indeed, by the Fubini theorem, 1 K (ν µ)(x) = 1 K (x y z)ν(dy)µ(dz) = µ(dz) 1 K (x y z)ν(dy) 2 = (1 K ν) µ(x), x. By Theorem 2.5 we get that 1 K ν(x+h) = 1 K ν(x), x, h E µ. Hence, by taking x = 0, we get ν(h K) = ν(k), and so ν(h + K) = ν(k), for any compact set K, h E µ. (2.15) By the inner regularity of ν, for any Borel set A, with ν(a) <, there exists an increasing sequence of compact sets (K n ), such that K n A, n N, and lim n ν(k n ) = ν(a). It follows that ν(h + A) = ν(a), for any h E µ. The proof is complete. We do not know if the previous result holds when is a Polish abelian group and E µ is replaced by the group generated by the support of µ. Remark Theorems 2.5 and Proposition 2.10 hold in a more general setting, with the same proof. Theorem 2.5 holds more generally if is a measurable abelian group, see [22, page 63]. A measurable space (, A) which is also a group (with additive notation) is said to be a measurable group if the group operations: (x, y) x + y and x x are both measurable (on one considers the product σ-algebra A A). In measurable groups 8

11 the convolution of finite measures on A is naturally defined. Note that a separable metric group and a locally compact group, with A being the Borel σ-algebra, are both examples of measurable groups. Proposition 2.10 holds more generally if is a topological abelian group and we assume that the measures µ and ν in (2.14) are both Radon measures (a non-negative Borel σ-finite measure γ on a Hausdorff topological space X is called Radon, if for each Borel set B X, with γ(b) <, for any ɛ > 0, there exists a compact set K B such that γ(b \ K) < ɛ, see for instance [22]). 3 Convolution equations on Hilbert spaces 3.1 The case of stable measures Let = H be a real separable Hilbert space, Q a non-negative trace class operator on H and α (0, 2]. A probability measure µ on H is said to be (α, Q)-stable, α (0, 2], centered at x H, if its characteristic function is ˆµ(h) := H ( e i h,y µ(dy) = exp(i x, h ) exp ( Qh, h 2 ) α/2 ), h H, (3.1) see [16], [15], [20]. Such measures will be denoted by N α (x, Q). Measures N 2 (x, Q) are aussian. In this case we also write N(x, Q). The following result extends Theorem in [7]. Proposition 3.1. Let µ = N α (x, Q), α (0, 2]. If f B b (H) solves equation (1.1) then f(y + Q 1/2 a) = f(y), y H, a H. (3.2) If, in addition, f is continuous and Q positive definite, then f is constant on H. Proof. First we show the result for α = 2. Lemma 3.2. Let µ = N(x, Q) and ν = N(y, S) be aussian measures on H. Then, In particular E µ = M µ = Q 1/2 H. M µ ν = M µ + M ν = Q 1/2 H + S 1/2 H. (3.3) Proof. It is well known that M µ = Q 1/2 H, see [7]. Moreover µ ν = N(x + y, Q + S). Define the linear operator T : H H H, T (x, y) = Q 1/2 x + S 1/2 y (where as usual (x, y), (x, y ) := x, x + y, y ). We check easily that (Q + S) 1/2 h 2 = T h 2, h H, (3.4) where T denotes the adjoint of T. By a classical duality argument, we have that (Q + S) 1/2 H = Q 1/2 H + S 1/2 H. The proof is complete. Continuing the proof of the proposition note that by Lemma 3.2, Q 1/2 H = E N2. Thus Theorem 2.5 gives the first claim. The second one follows from the density of Q 1/2 H in H when Q is non-degenerate. 9

12 Let us consider now α (0, 2) and set N α = N α (x, Q). We show that Q 1/2 H M Nα, α (0, 2]. For this we use subordination. Let ν α by an α stable distributions ν α on [0, + ), with the Laplace transform given by It is easy to check that 0 N α (B) := e λs ν α (ds) = e (λ)α/2, λ > 0. 0 N(x, sq)(b) ν α (ds), B B(H). (3.5) Take A B(H) such that N α (A) = 0, then, N(x, sq)(a) = 0, for any s S ν = R +. Let g = Q 1/2 h, for some h H. By the absolute continuity of the aussian measures (T g N α )(A) = N α (A + g) = Hence, Q 1/2 (H) M Nα. By Theorem 2.5 we get the claim. 0 N(x g, sq)(a)ν α (ds) = 0. For informations about the set of all admissible shifts for general α-stable measures we refer to [2] and [21]. 3.2 Liouville type theorems on Hilbert spaces Let µ t, t 0, be a convolution semigroup of measures on a real separable Hilbert space H. This means that µ t µ s = µ t+s, t, s 0, µ 0 is the Dirac measure concentrated in 0 and µ t is weakly continuous at t = 0. Let P t be the Markovian semigroup determined by µ t, t 0, P t f(x) = f(x y)µ t (dy) = f µ t (x), x H, t 0, f B b (H), (3.6) H see [16], [22], [12] and [20] for more information on convolution semigroups and Lévy processes. A function h B b (H) is said to be a bounded harmonic function for P t, briefly a BHF for P t, see [1], [8], [17] and [18] if (tλ) k k! P t h = h, t 0. (3.7) Note that in the case when ) P t is a compound convolution semigroup, i.e., P t f(x) = (f (ν) k (x), where λ > 0 and ν is a given probability measure on H, e λt k 0 one can check that h is a BHF for P t if and only if h is a bounded ν-harmonic function. We present now a Liouville type theorem about BHFs for convolution semigroups. The first part is a consequence of Theorem 2.5. The second one is a generalization of a surprising result obtained by oodman [13]. It states that in infinite dimension there exists non-constant BHF for the heat semigroup (see also [7], Section 4.3.1). Theorem ) Let P t be the Markovian semigroup (3.6) on a separable Hilbert space H and let ( Γ = r M µt ). (3.8) t 0 10

13 Then each BHF h for P t is periodic with periods in Γ. If h is continuous and Γ = H, then h is constant. 2) For arbitrary semigroups (3.6) there exists a non-constant BHF φ if H is infinite dimensional. Proof. ( 1) It follows from Theorem 2.5. We only note that, by Proposition 2.1, one has: ) r t 0 E µ t = Γ. 2) We use a probabilistic representation of convolution semigroups, see [12]. There exists a Lévy process (Z t ) on a stochastic basis (Ω, F, (F t ) t 0, P), with values in H, such that the law of each Z t is µ t, t 0. The process (Z t ) can be represented as Z t = at + η t + ξ t, t 0, (3.9) where a R n, (η t ) is a square integrable martingale and (ξ t ) is a compound Poisson process. Moreover the processes (η t ) and (ξ t ) are independent and so in particular P t f(x) = Ef(x ξ t η t at) = f ν t r t (x), (3.10) where ν t is the law of at + η t and r t the law of ξ t. Thus it is enough to construct a non-constant function φ such that φ ν t = φ and φ r t = φ, t 0. (3.11) Let us first consider η t + at with law ν t. Remark that there exists a non-negative trace class operator Q : H H, such that this holds: Qh, k = 1 t E( η t, h η t, k ), t > 0, h, k H. Let us choose an orthonormal basis (e k ) in H, such that Qe k = λ k e k. We have that k 1 λ k <. Let (α k ) be a sequence of positive numbers, diverging to +, such that a 2 k α k <, a k = a, e k, k N, (3.12) k 1 λ k α k + k 1 and define the linear subspace K, K = {x H : g(x) < }, g(x) = k 1 x 2 k α k, x H, x k = x, e k. (3.13) Since (α k ) diverges, one has that K is strictly contained in H. Moreover a K by construction. It turns out that the law of η t + at is concentrated on K, for any t 0. Indeed one has: Eg(η t + at) = α k η t, e k 2 + t a 2 k α k < k 1 k 1 and so η t + at is almost surely in K, for any t 0. Note that (I K ) ν t (x) = I K (x y)ν t (dy) = ν t (x K) = I K (x), x H, t 0. H Indeed if x K, then x k K, for any k K. This gives that I K is a non-constant BHF for the convolution semigroup determined by (η t + at). 11

14 Let us consider the remainder compound Poisson process (ξ t ), see (3.9). Denote by λ > 0 its intensity, by ν its Lévy measure and by S t the associated convolution semigroup. One has: S t f(x) = Ef(x ξ t ) = e λt k 0 (tλ) k k! ( f (ν) k) (x), (3.14) where E(e i ξt,h ) = e tψ(h), h H, t 0, and ψ(h) = H (ei x,h 1)ν(dx). As already noted, h B b (H) is a BHF for S t if and only if h ν = h. Let us first construct a non-constant bounded ν-harmonic function h. Let us introduce λ k = E(e U U, e k 2 ), where U : Ω H is a random variable with law ν. It is clear that λ k = E(e U U 2 ) <. k 1 Take a diverging sequence of positive real numbers (α k ), such that k 1 λ k α k < and define the linear subspace K, K = {x H : k 1 x2 k α k < }. One has that h is a non-costant BHF for S t. To finish the proof, define α k = min(α k, α k ), and introduce the new subspace K = {x H, k 1 x2 k α k < }. Repeating the previous arguments, we find that φ = I K is non-constant and verifies (3.11). This completes the proof. Corollary 3.4. Convolution semigroups P t, given by (3.6) on an infinite dimensional Hilbert space H, are never strong Feller. Acknowledgments The authors thank the Institute of Mathematics of the Polish Academy of Sciences in Warsaw and Scuola Normale Superiore di Pisa, where parts of this paper were prepared, for good working conditions. References [1] Blumenthal R. M., etoor R. K., Markov processes and potential theory, Pure and Applied Mathematics, Academic Press, New York-London, [2] Cambanis S., Marques M., Admissible and singular translates of stable processes, Probability theory on vector spaces IV, , Lecture Notes in Math., 1391, [3] Choquet., Deny J., Sur l equation de convolution µ = µ σ, C. R. Acad. Sci. Paris 250 (1960), [4] Chu, Cho-Ho, Leung, Chi-Wai, The convolution equation of Choquet and Deny on [IN]- groups, Integral Equations Operator Theory 40 (2001), no. 4, [5] Chu, Cho-Ho, Leung, Chi-Wai, A matrix-valued choquet deny theorem, Proc AMS 129, 2000, [6] Chu, Cho-Ho, Lau, Anthony To-Ming, Harmonic functions on groups and Fourier algebras, Lecture Notes in Mathematics, 1782, Springer-Verlag, Berlin, [7] Da Prato., Zabczyk J., Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series 293, Cambridge University Press,

15 [8] Dynkin E. B., Markov processes, Vol. I-II, Springer-Verlag, Berlin-öttingen-Heidelberg, [9] Feller, W., An introduction to probability theory and its applications, Vol. II. Second edition, John Wiley & Sons, Inc., New York-London-Sydney (1971). [10] Foguel S. R., The ergodic theory of Markov processes, Van Nostrand Mathematical Studies, No. 21, New York-Toronto, Ont.-London, [11] I.I. ihman, A.V. Skorohod, The Theory of Stochastic Processes, Vol. I, Springer-Verlag, Berlin, [12] I.I. ihman, A.V. Skorohod, The Theory of Stochastic Processes, Vol. II, Springer-Verlag, Berlin, [13] oodman V. A Liouville theorem for abstract Wiener spaces, Amer. J. Math. 95 (1973), [14] Johnson B. E., Harmonic functions on nilpotent groups, Integr. Equ. oper. theory 40 (2001), [15] Linde W. Probability in Banach spaces, stable and infinitely divisible distributions. Second edition, A Wiley-Interscience Publication, John Wiley Sons, Ltd., Chichester, [16] Parthasarathy K. R. Probability measures on metric spaces, Academic Press, New York and London, [17] Pinsky R.., Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics 45, [18] Priola E., Zabczyk J., Liouvile Theorems for non-local operators, J. Funct. Anal. 216 (2004), [19] Raugi, A., A general Choquet-Deny theorem for nilpotent groups, Ann. Inst. H. Poincar Probab. Statist. 40 (2004), no. 6, [20] Sato Keni-Iti, Lévy processes and infinite divisible distributions, Cambridge University Press, [21] Zinn, J., Admissible translates of stable measures, Studia Math. 54 (1976), [22] Vakhania, N. N.; Tarieladze, V. I.; Chobanyan, S. A., Probability distributions on Banach spaces, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht 14, Appendix: Proof of Theorem 2.8 Fix a and consider the function g : R, Now g UC b () and g µ = g on. Let g(x) = f(x) f(x a), x. 2c = sup g(x) x and (x n ) such that g(x n ) 2c as n. We introduce also the functions g n : R, g n (x) = g(x + x n ), x. 13

16 Again each g n solves (1.1). Moreover, since g UC b (), (g n ) is a relatively compact sequence in C(K) for each compact set K. Now we consider a subgroup S 0, countable union of compact sets K m, m N, and having the property that µ(s 0 ) = 1, see Lemma 2.9. Since (g n ) is relatively compact on each C(K m ), for any m 1, we get, by a diagonal procedure, that there exists a subsequence of (g n ), still denoted by (g n ), which converges to a continuous and bounded function g 0 : S 0 R, uniformly on each K m. In particular, lim n g n(x) = g 0 (x), x S 0. Passing to the limit in g n (x) = g n (x y)µ(dy), x S 0, S 0 as n, by the dominated convergence theorem, we infer: g 0 (x) = g 0 (x y)µ(dy), x S 0. (4.1) S 0 Setting x = 0, we get 2c = g 0 (0) = g 0 ( y)µ(dy) = S 0 g 0 (y) µ(dy), S 0 where µ is the reflection of µ with respect to 0. From the previous identity, using that g 0 is continuous on S 0 and that g 0 2c, we obtain g 0 (x) = 2c, x S 1 = (S 0 S µ ), where S µ = S µ denotes the support of µ. Note that µ(s 1 ) = 1. Thanks to (4.1), we get 2c = g 0 (x) = g 0 (x + y) µ(dy), x S 1. S 1 It follows that g 0 (x + y) = 2c, x, y S 1. Using an induction argument, we find that g 0 (z) = 2c, z S 2 = S 1 (S 1 + S 1 )... (S S 1 )..., (4.2) where S 2 is the semigroup generated by S 1. Note that S 2 S 0 (in fact S 0 is a group). Now we choose the initial point a in S 1. By (4.2), we deduce that g 0 (ka) = 2c, k 1. Hence for any integer m, there exists ˆn such that gˆn (ka) = f(xˆn + ka) f(xˆn + (k 1)a) > c, (4.3) for k = 1,..., m. Summing (4.3) m-times, we get f(xˆn + ma) f(xˆn ) > mc. 14

17 Letting m, we find that c 0, since f is bounded. This means that g(x) 0, x, i.e. f(x) f(x a), x, a (S 0 S µ ). Repeating the previous argument with f instead of f, one has: f(x) = f(x a), x, a (S 0 S µ ). Since µ(s 0 S µ ) = µ(s 0 S µ ) = 1, we have that (S 0 S µ ) is dense in S µ (indeed by definition S µ is the smallest closed set on which µ is concentrated). Using the continuity of f one gets f(x) = f(x a), x, a S µ. To finish the proof, one remarks that the set of all periods of a given real function on is a subgroup of. 15