Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 04, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu POISSON MEASURES ON SEMI-DIRECT PRODUCTS OF INFINITE-DIMENSIONAL HILBERT SPACES RICHARD C. PENNEY, ROMAN URBAN Abstract. Let G = X o A where X and A are Hilbert spaces considered as additive groups and the A-action on G is diagonal in some orthonormal basis. We consider a particular second order left-invariant differential operator L on G which is analogous to the Laplacian on Rn. We prove the existence of “heat kernel” for L and give a probabilistic formula for it. We then prove that X is a “Poisson boundary” in a sense of Furstenberg for L with a (not necessarily) probabilistic measure ν on X called the “Poisson measure” for the operator L. 1. Introduction In recent years, there has been considerable interest in the study of second order differential equations on infinite dimensional Hilbert spaces and their generaliza- tions. See for example Da Prato [3, 7] and the references contained therein. In this work we study a specific class of second order linear differential equations, the “Laplacians”, on a very specific set of Hilbert spaces, the meta-abelian solvmani- folds. (See (1.1) below.) The form of the differential operator that we are considering in this work, as well as the form of the algebraic structure of the space on which this operator acts has its origin in the analysis on Lie groups [18, 21]. In a sense, the context considered here can be treated as a generalisation of the (finite dimensional) setting considered in a series of papers on estimates of Poisson kernels for the second-order left-invariant differential operators on NA Lie groups, i.e. on the semi-direct products of nilpotent and Abelian Lie groups A = RD, D ≥ 1 (see [4, 5, 6, 10, 11, 12, 13, 14, 15, 16]). All these analytical problems have their source in probabilistic considerations (see e.g. [2, 8, 17]) of harmonic (with respect to a probability measure) functions on groups. Our techniques are probabilistic and are generalizations of known results in the finite dimensional Lie group case. The fact that these techniques are useful in this context is in itself an interesting result. More specifically, for p > 0, and d ∈ N ∪ {∞}, let `dp = { `p({1, . . . , d}) if d <∞, `p(N) if d =∞ 2010 Mathematics Subject Classification. 35C05, 60J25, 60J60, 60J45. Key words and phrases. Poisson measure; Gaussian measure; Hilbert space; Brownian motion; evolution kernel; diffusion processes. ©2022. This work is licensed under a CC BY 4.0 license. Submitted March 23, 2021. Published January 10, 2022. 1 2 R. C. PENNEY, R. URBAN EJDE-2022/04 with respect to the counting measure. We identify `dp with the subspace of `∞p of elements supported in {1, . . . , d}. We often omit d in our notation so that `p denotes `dp for some fixed d. We let `+p be the set of elements of `p with positive entries and ` + p the set of elements with non-negative entries. For u ∈ Rd, d ∈ N∪{∞}, we let [u] = diag u be the corresponding d×d diagonal matrix. Let A = X = `2 considered as Abelian groups. The general element of A and X are denoted respectively by a = (a1, a2, . . . ), x = (x1, x2, . . . ). We define an action of A on X by xa = Ada x = ead(a)(x) = (eλ1(a)x1, . . . , e λd(a)xn, . . . ), where λi : A→ R is given by λi(a) = ai. Recall that `2 ⊂ `∞, so that for a ∈ `2, both a and ea are bounded. Then for a and x in `2, the xa defined above is also in `2. We consider the corresponding semi-direct product G = X o A which is X × A with the product (x, a)(y, b) = (x+ ya, a+ b). (1.1) For g ∈ G we let x(g) = x ∈ X and a(g) = a ∈ A denote the components of g in this product so that g = (x, a). Let q, β ∈ `+2 ∩ `1 and let α ∈ `+2 . We consider the differential operators L = ∆α + Laβ , (1.2) where Laβ = ∑ j e2λj(a)βj∂ 2 xj and ∆α = ∆q α = ∑ j qj∂ 2 aj − ∑ j 2αj∂aj . We do not typically carry q in our notation as we consider it fixed. However, when necessary, we will indicate the dependence with a superscript q. Following Da Prato and Zabczyk [7], we consider our operators as densely defined operators on the respective spaces UCb(G), UCb(A), and UCb(X) of uniformly continuous, bounded functions on the given space, depending on context. We also let Cb(·) denote the space of bounded continuous functions on the given space. We require some technical assumptions on the growth rates of the coefficients. Explicitly, we assume that there is a constant c > 0 such that, for all j ≥ 1, 2αj βj > c. (1.3) We assume additionally that there is a γ ∈ `2 such that [q]γ = α. (1.4) Recall that [q] is a matrix whose diagonal entries are equal to q. For d ∈ N we let the superscript d denote the corresponding sequences obtained by considering only elements of `dp ⊂ `p. Our main results are in Theorem 5.1, which firstly proves the existence of a heat semigroup PLt for L, and secondly provides a probabilistic formula for the EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 3 semigroup. This formula is based on the existence of a diffusion kernel for Lσtβ (σt is a continuous trajectory of the process generated by ∆α) which provides the unique solution to system (4.1) below, which is our second main result. Finally, we prove that X is a “Poisson boundary” for L in the sense that there is a “Poisson measure” ν on X for L. That is to say, there exists a probability measure ν on X such that µLt ∗ ν = ν, where µLt is a semi-group of probability measures generated by L and the convolu- tion is defined by the action of A on X, i.e. if λ is a probability measure on G and ρ is a probability measure on X, then∫ X f(x)(λ ∗ ρ)(dx) = ∫ X ∫ G f(x+ ya)λ(dyda)ρ(dx). To prove existence of the “Poisson measure” ν we give its construction. The outline of the paper is as follows. In Section 2 we recall the basic properties of Gaussian measure in the Hilbert space `2 which is the basic component in our results. In Section 3 we define a vertical component, i.e. a stochastic process on A generated by ∆α. The vertical component is one of the two components of the skew-product formula (proved in Section 5) for the heat semigroup PLt . The second component, called a horizontal component, is a diffusion on X generated by a time-dependent operator Lσtβ . The horizontal component is considered in Section 4. Finally, in Section 6 we construct the Poisson measure ν on X for L. 2. Gaussian measures on `2 The concept of Gaussian measures is fundamental to our results. A detailed and extensive discussion of Gaussian measures in the infinite-dimensional Hilbert and Banach spaces can be found e.g. in [3, 7, 1]. We restrict to the mean 0 case since this is what we require. For λ ≥ 0, the corresponding Gaussian measure on R is then Nλ(dx) = { (2πλ)−1/2e− x2 2λ dx, λ > 0, δ0(dx), λ = 0. Let λ ∈ Rd with λj ≥ 0, where 1 ≤ d ≤ ∞. Let [λ] denote the corresponding d× d diagonal matrix. The corresponding Gaussian measure on Rd is by definition N[λ](dx) = d∏ j=1 Nλj (dxj). (2.1) The product measure exists as a measure on (Rd,B(Rd)) since each of the compo- nents is a probability measure [9, p. 157, Theorem B]. Let |A| denote the cardinality of a set A, and let Rm = ∏ {j:λj 6=0} R, where m = |{j : λj 6= 0}|, Rk = ∏ {j:λj=0} R, where k = |{j : λj = 0}|. Then we may write Rd = Rm⊕Rk. For x ∈ Rd let x = xm+xk in this decomposition. 4 R. C. PENNEY, R. URBAN EJDE-2022/04 Lemma 2.1. Let notation be as above. Then∫ Rd f(x)N[λ](dx) = ∫ Rm f(xm, 0)N[λm](dx m) = ∫ Rm f ∣∣ Rm(x)N[λm](dx m) = ∫ Rd f(x)χRm(x)N[λ](dx), where dxm = ∏m j=1 dxj. Proof. By definition N[λ](dx) = ∏ {j:λj 6=0} Nλj (dxj)× ∏ {j:λj=0} Nλj (dxj) which is equivalent to the statement in the lemma. � We say that two sequences λ1 and λ2 in Rd are disjoint if λ1iλ 2 i = 0 for all i. Equivalently, λ1 and λ2 are disjoint if the sets {i : λ1i 6= 0} and {i : λ2i 6= 0} are disjoint. The following lemma is clear. Lemma 2.2. Assume that λ ∈ Rd where λj > 0 for all j. Suppose that λ1 and λ2 are disjoint elements of Rd and λ = λ1 + λ2. Then Rd = Rd1 × Rd2 , N[λ] = N[λ1] ⊗N[λ2], where N[λi] are measures on Rdi and di = |{j : λij 6= 0}|. The statement in [7, Theorem 1.2.1] implies that if [λ] is trace class, i.e. λ ∈ `1, then N[λ](dx) restricts uniquely to a measure on `2. In our case, trace class will always be assumed, so we consider N[λ](dx) as a measure on `2. The following lemma is simple and it is left to the reader to prove it. Lemma 2.3. Let P be a bounded, symmetric, positive operator on `2 which is diagonal in the standard basis {ej} of `2 with eigenvalues pj. Then∫ `2 f(x)NP [λ](dx) = ∫ `2 f(P 1/2x)N[λ]dx. 3. Vertical component By definition, the vertical component is the diffusion on A generated by ∆α. This is the Markov process with transition kernel Pαt,s(a, db) which gives the solution to the following initial problem on A: ∂tf(t, a) = ∆αf(t, a), t > s ≥ 0, f(s, a) = fo(a). (3.1) In the α = 0 case, from [7, (3.1.9)], the solution for fo ∈ UCb(A) is f(t, x) = ∫ A fo(x+ y)N(t−s)[q](dy) = ∫ A fo(y)R(−x)N(t−s)[q](dy), P 0 t,s(x, dy) = R(−x)N(t−s)[q](dy). (3.2) EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 5 Furthermore as an operator on UCb(A), t 7→ P 0 t,0 defines a continuous semi-group. In the α 6= 0 case, by hypothesis (1.4) there is a γ ∈ `2 such that [q]γ = α. Note also that from [7, Proposition 1.2.5], the measure in the second equation in (3.3) below is finite. Corollary 3.1. In the case α ∈ `2 (arbitrary case), the solution to (3.1) for fo ∈ UCb(A) is f(x, t) = e−t(γ,α) ∫ `2 fo(x+ y)e−(γ,y)N(t−s)[q](dy), Pαs,t(x, dy) = e−t(γ,α)R(−x) ( e−(γ,y)N(t−s)[q](dy) ) . (3.3) Furthermore as an operator on UCb(A), t 7→ Pα0,t defines a continuous semi-group. Proof. A simple computation shows that ∆q α = e(γ,x)∆q 0e −(γ,x) + (α, γ). This suggests that as operators on UCb(A) Pαs,t = Pα,qs,t = e(t−s)(α,γ)e(γ,x)P 0,q s,t e −(γ,x) which is equivalent with the stated identity. A rigorous proof can be constructed either using finite rank approximations as in [7] or using the stated formula for ∆q α. We omit the details. � 4. Horizontal component Let σ ∈ C([0,∞)), X) and fo ∈ UCb(X). For k ∈ N, let UCkb (X) denote the set of elements of UCb(X), all of whose derivatives up to order k belong to UCb(X). Consider the initial value problem on R+ ×X, ∂tf(t, b) = Lσtβ f(t, b), t > s, f(s, b) = fo(b). (4.1) For s < t, let Aσ(s, t) = ∫ t s e2σ(u)du := (∫ t s e2σ(u)1du, . . . , ∫ t s e2σ(u)ddu )T ∈ Rd, that is we consider Aσ(s, t) as a column vector, and Aσj (s, t) = ∫ t s e2σ(u)jdu. Note that for a ∈ A, (AdaA σ(s, t))j = eajAσj (s, t) = ∫ t s e2σ(u)j+ajdu = Aσ+a/2(s, t)j . (4.2) Theorem 4.1 below is one of our main results. We prove it in a series of propo- sitions. Assume first that d <∞. Under the Fourier transform in x, (4.1) implies ∂tf̂(t, ξ) = ( − d∑ j=1 e2λj(σt)βjξ 2 j ) f̂(ξ, t), f̂(0, ξ) = f̂o(ξ). 6 R. C. PENNEY, R. URBAN EJDE-2022/04 This ODE is easily solved, f̂(t, ξ) = exp ( − d∑ j=1 Aσj (s, t)βjξ 2 j ) f̂o(ξ) = e− 1 2 (2[A σ(s,t)][β]ξ,ξ)f̂o(ξ). If d < ∞, the next follows immediately from [7, Theorem 1.2.1]. The general case is Proposition 4.3 below. Theorem 4.1. Assume that [β] is trace class. Let Pσ,βs,t (dy) = N2[β]Aσ(s,t)(dy). (4.3) Then for fo ∈ Cb(X) (UCb(X), resp.) the unique solution to (4.1) in Cb(X) (UCb(X), resp.) is Uσ(s, t)fo(x) = ∫ X fo(x+ y)Pσ,βs,t (dy) = ∫ X fo(x+ [Aσ(s, t)]1/2y)N2[β](dy). (4.4) In the finite dimensional case, the following properties follow from Theorem 4.1 and are well known (see [10, 19, 20]). The process corresponding to the transition kernel (4.3) is called the horizontal component. Corollary 4.2. Assume d <∞. Then (i) Uσ(s, r)Uσ(r, t) = Uσ(s, t), 0 ≤ s ≤ r ≤ t, (ii) ∂tU σ(s, t)f = Uσ(t, s)Lσ(t)f , (iii) ∂sU σ(s, t)f = −Lσ(s)Uσ(s, t)f , (iv) Uσ(s, t) : C2 b (Rd)→ C2 b (Rd), s ≤ t. Assume now that d =∞. For m ∈ N and β as in (1.2) let βm = (β1, . . . , βm, 0, 0, . . . ) ∈ R∞, Pσ,ms,t (dy) = N2[βm]Aσ,m(s,t)(dy), where Aσ,m(s, t) = (Aσ1(s, t), . . . , Aσm(s, t), 0, 0, . . . )T . Then m 7→ Pσ,ms,t (dy) is referred to as the finite rank approximation to Pσs,t(dy). We apply Lemma 2.2 with λ = 2β, λ1 = 2βm, λ2 = 2β − 2βm. Then in this lemma d1 = m and d2 =∞. Restricting to `2, this lemma implies that `2 = Rm × `2, N2[β]Aσ(s,t) = N2[βm]Aσ(s,t) ⊗N[λ2]Aσ(s,t). For x ∈ `2, let xm and x∞ be the components of x with respect to the above decomposition. Let uσm(s, t) = ∫ `2 fo(x+ y)χRm(y)N2[β]Aσ(s,t)(dy) = ∫ Rm×`2 fo(x m + ym, x− xm + y∞)χRm(ym, y∞) ×N2[βm]A σ(s, t)(dym)N[λ2]Aσ(s,t)(dy ∞) = ∫ Rm×`2 fo(x m + ym, x− xm + 0)N2[βm]Aσ(s,t)(dy m) EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 7 = ∫ Rm×`2 fo(x+ ym)N[βm]Aσ(s,t)2(dym) = ∫ Rm fo(x+ ym)Pσ,ms,t (dym). Recall that for a set B, χB is the indicator function of B, i.e. χB(x) = 1 if x ∈ B and χB(x) = 0 if x 6∈ B. From the dominated convergence theorem it follows that lim m→∞ Pσ,ms,t (fo) = Pσ,βs,t (fo). In particular from Corollary 4.2, for 0 ≤ s ≤ r ≤ t, Pσ,βs,r P σ,β r,t = Pσ,βs,t . (4.5) Theorem 4.1 (for d =∞) now follows from the next proposition. Proposition 4.3. For fo ∈ UC2 b (X), u(s, t) = Uσ(s, t)fo is the unique solution to the initial value problem (4.1). Proof. We use the second formula in (4.4). Let B = 2[β], C = [Aσ(s, t)]1/2, S = C2(D2fo)(x). From the mean value theorem for integrals applied in each variable C2 = [( ∫ t s e2σ(u)1du, ∫ t s e2σ(u)2du, . . . )T ] = (t− s)[(e2σ(u1), e2σ(u2), . . . )T ] = (t− s)e2σ(u), u ∈ Rd, (4.6) where ui ∈ (s, t) for all i. From Taylor’s Theorem applied to the function t 7→ fo(x+ ty) at t = 1, fo(x+ y) = fo(x) + (Dfo(x), y) + 1 2 ( D2fo(x+ ξy)y, y ) where differentiation is with respect to x and ξ = ξ(x, y) ∈ [0, 1]. Then fo(x+ Cy) = fo(x) + (Dfo(x), Cy) + 1 2 (C2D2fo(x+ ξCy)y, y). (4.7) From [7, Proposition 1.2.4], ∫ (fo(x), Cy)NB(dy) = 0,∫ SijyiyjNB(dy) = 2Sijβij ,∫ (Sy, y)NB(dy) = ∑ ij 2Sijβij = 2(t− s) Tr([β](e2σ(u))D2)(fo) = 2(t− s) ∑ i βie 2σ(ui)∂2i fo(x). We integrate (4.7) with ξ = 0 against N2[β](dy) using (4.6) and rearrange to find that Uσ(s, t)(x)− fo(x)− (t− s) ∑ i βie 2σ(ui)∂2i fo(x) 8 R. C. PENNEY, R. URBAN EJDE-2022/04 = t− s 2 ∫ [eσ(ui)] ( (D2fo(x+ (t− s)1/2[e2σ(ui)]ξy)−D2fo(x))y, y ) N2[β](dy). Hence, as t→ s+, Uσ(s, t)(x)− fo(x) t− s − ∑ i βie 2σ(ui)∂2i fo(x)→ 0, that is, ∂tU σ(s, t) ∣∣ t=s+ = Lσsβ fo(x). This proposition now follows as in the proof of [7, Theorem 3.2.3] using Corollary 4.2 (i). The uniqueness follows from the uniqueness in the finite dimensional case. � 4.1. Ad-invariance. For f a C∞ function on either G or X and a ∈ A let ada f = f ◦ ada . For X in the Lie algebra G of G we have ad−a ◦X ◦ ada = Ada(X ). Consequently, as an operator on X, Lσtβ = ∞∑ j=1 e2λj(σt)βj∂ 2 xj = Adσt ( ∞∑ j=1 βjX 2 j ) , Lσt+aβ = Ada(Lσtβ ) = Ad−a ◦(Lσtβ ) ◦Ada . Corollary 4.4. As operators on Cb(X), for a ∈ A, Uσ+a(s, t) = Ad−a ◦Uσ(s, t) ◦Ada . Proof. From Corollary 4.2, integration against the quantity on the right solves the initial value problem (4.1) with σ replaced by σ + a. � Corollary 4.5. For fo ∈ Cb(X), Uσ+a(s, t)fo(x) = ∫ X N2[β]Aσ+a(s,t)fo(x+ y)dy. Proof. From (4.4) and (2.3), Uσ(s, t)(Ada(fo))(x) = ∫ X (fo ◦Ada)(y)Pσs,t(x, dy) = ∫ X N2[β]Aσ(s,t)fo(Ada(x+ y))(dy) Hence, Ad−a ◦Uσ(s, t) ◦Ada(fo)(x) = ∫ X N2[β]Aσ(s,t)fo(x+ Ada y)dy = ∫ X N2[β] Ad2 a A σ(s,t)fo(x+ y)dy = ∫ X N2[β]Aσ+a(s,t)fo(x+ y)dy. � EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 9 5. A skew-product formula Here we present our second main result. Let Pσs,t be the measure defined in (4.3). We refer to the formula given by the next theorem as to a skew-product formula. It gives both the existence of and a disintegration of the heat semigroup TLt of L into a “horizontal” diffusion τ(t) defined on X = `2 and a “vertical” one on A = `2 generated by ∆α. In finite dimensions, this is a special case of [6, Theorem 3.1], see also [12]. Theorem 5.1. For f ∈ UC2 b (G) and t ≥ 0 we have TLt f(x, a) = Ea ∫ f(x+ y, σt)N2[β]Aσ(0,t)(dy) ≡ v(t, x, a), (5.1) where the expectation is taken with respect to the distribution of the process σ in `2 generated by ∆α, and starting from a, i.e., σ0 = a. Proof. We claim first that for f ∈ UC2 b (G), v(t, x, a) defined in(5.1) is a solution of the integral equation v(t, x, a) = Eaf(x, σt) + Ea ∫ t 0 Lσt−sβ ∣∣ x v(s, x, σt−s)d s (5.2) We claim first that the only question is the second term. Let f ∈ UC2 b (G). Then from (4.1) for 0 < s < t,∫ t 0 EaLσt−sβ v(s, x, σt−s) = Ea ∫ t 0 Lσt−sβ Uσ(0, s)f(x, σt−s) = ∫ t 0 Ea ∫ `2 ∞∑ j=1 e2λj(σt−s)βj∂ 2 xjv(x+ y)N2[β]Aσ(s,0)(dy) ds. Since ∣∣Ea ∫ t 0 e2λj(σt−s) ds ∣∣ = ∣∣Ea ∫ t 0 e2λj(σu) du ∣∣ = ∣∣Ea ∫ t 0 e2(σu)j du ∣∣ = ∣∣Ea ∫ t 0 e2(bu)j−αjt du ∣∣ ≤ ∣∣∣∣Ea ∫ t 0 e2(bu)j du ∣∣∣∣ = Ct we obtain (since β ∈ `1) that |Laβv(s, x, a)| ≤ ∣∣∣‖v‖2∞ ∞∑ j=1 βjEa ∫ t 0 e2λj(σt−s) ds ∣∣∣ ≤ ‖v‖2∞Ct ∣∣ ∞∑ j=1 βj ∣∣ ≤ Ct‖v‖∞‖β‖`1 . Let dWa(b) be a “Wiener measure”, i.e. probability measure one C([0,∞), A) such that, for every a ∈ A, Wa(b : b(0) = a) = 1. 10 R. C. PENNEY, R. URBAN EJDE-2022/04 We calculate EaLbt−sβ v(s, x, bt−s) = ∫ Lbt−sβ v(s, x, bt−s) dWa(b) = ∫ Lbt−sβ ∫ Uσ(0, s)f(x, σs)dWbt−s(σ)dWa(b) = ∫ ∫ Lbt−sβ Uσ(0, s)f(x, σs)dWbt−s(σ)dWa(b) = ∫ Lbt−sβ Uσ(t− s, t)f(x, bt)dWa(b), (5.3) where in the last equality we have used the Markov property of the process. We apply Fubini’s theorem to obtain∫ t 0 EaLbt−sβ u(s, x, bt−s)d s = ∫ ∫ t 0 Lbt−sβ U b(t− s, t)f(x, bt)d sdWa(b), but ∫ t 0 Lbt−sβ U bt−s(t− s, t)f(x, bt)d s = U b(0, t)f(x, bt)− f(x, bt). Indeed by property (iii) of U b we obtain d d s U b(t− s, t)f(x, bt) = − d d s U b(·, t)f(x, bt) ∣∣∣∣ t−s = − ( −Lbt−sβ U b(t− s, t)f(x, bt) ) = Lbt−sβ U b(t− s, t)f(x, bt). Therefore, ∫ t 0 EaLbt−sβ u(s, x, bt−s)d s = ∫ U b(0, t)f(x, bt)dWa(b)− ∫ f(x, bt)dWa(b) = u(t, x, a)−Eaf(x, bt). Thus TLt f(x, a) = Ea (∫ f(x+ y, σt)N2[β]Aσ(0,t)(dy) ) (5.4) as claimed. � 6. Construction of the Poisson measure Let operators be defined as in (1.2) and TLt be defined by (5.1). For b ∈ A we let TL,bt be defined by replacing σt with σt + b in the initial value problem (4.1). For a measure µ on G we define µ̌(ϕ) = ∫ G ϕ(g−1)µ(dg). We fix t > 0 and consider a Markov chain Rn with the starting point R0 = e = (0, 0) and the transition kernel p(·, ·) = p̌t(·, ·). We have Rn = (0, 0)(X1, A1) . . . (Xn, An) EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 11 = (X1 +XA1 2 + · · ·+XA1+···+An−1 n , A1 + · · ·+An). The distribution of Rn is equal to p(n)(e, dg) - the n-folded composition of kernel p. For i = 1, 2, let πi : X oA = `2 o `2 → `2 be the projection on the i-th variable, i.e. π1(x, a) = x, π2(x, a) = a. Then π1(Rn) = X1 +XA1 2 + · · ·+XA1+···+An−1 n . For i = 1, 2, . . . , we denote Xi = (Xi,1, Xi,2, . . . ) ∈ `2, Ai = (Ai,1, Ai,2, . . . ) ∈ `2. Theorem 6.1. The limit lim n→∞ π1(Rn) = Z (6.1) exists a.e., and the distribution ν(0, dy) of the random variable Z is the Poisson kernel for L. Proof. The proof is based on ideas from [17, 8]. First we prove the existence of (6.1) and then we show the claim about the Poisson measure. To show the existence of (6.1) it is sufficient to show that lim sup n→∞ ‖XA1+···+An−1 n ‖1/n`2 < 1 a.e. We have, ‖XA1+···+An−1 n ‖`2 = ‖(eλ1(A1···+An−1)Xn,1, e λ2(A1···+An−1)Xn,2, . . . )‖`2 = ‖(eA1,1···+An−1,1Xn,1, e A1,2···+An−1,2Xn,2, . . . )‖`2 = ‖AdA1+···+An−1 Xn‖`2 ≤ ‖AdA1+···+An−1 ‖`2→`2‖Xn‖`2 . Thus it suffices to show that lim n→∞ ‖AdA1+···+An ‖ 1/n `2→`2 < 1 a.e. (6.2) and lim sup n→∞ ‖Xn+1‖1/n`2 ≤ 1 a.e. (6.3) First we prove (6.2). Clearly, ‖Ada x‖`2 ≤ sup j≥1 eλj(a)‖x‖`2 . Therefore ‖Ada ‖`2→`2 ≤ sup j≥1 eλj(a) = sup j≥1 eaj and consequently ‖AdA1+···+An ‖ 1/n `2→`2 ≤ sup j≥1 e 1 n (A1,j+···+An,j). Thus to prove (6.2) we have to show that 0 < sup j≥1 lim n→∞ e 1 nλj(A1+···+An) = sup j≥1 lim n→∞ e 1 n (A1,j+···+An,j) < 1, a.e. 12 R. C. PENNEY, R. URBAN EJDE-2022/04 Equivalently we have to show that −∞ < sup j≥1 lim n→∞ 1 n (A1,j + · · ·+An,j) < 0. Let π2,j(x, a) = aj . Then π2,jµt is a gaussian semigroup of measures with the generator qj∂ 2 aj − 2αj∂aj = qj ( ∂2aj − 2αj qj ∂aj ) ; that is, π2,jµt(daj) = 1√ 4πqjt exp ( − (aj − 2αjqjt/qj) 2 4qjt ) daj . Thus by the assumption (1.3), there is a constant c > 0 such that∫ `2 ∫ `2 π2,j(x, a)µ(dx da) = ∫ R ajπ2,jµt(da) = −2αj qj < −c. Notice that for every j ≥ 1, A1,j , A2,j , . . . is a sequence of i.i.d. random variables with values in R and with gaussian distribution π2,jµt(daj). By the strong law of large numbers lim n→∞ 1 n (A1,j + · · ·+An,j) = ∫ R ajπ2,jµt(daj) = −2αj qj < −c, a.e. Hence we obtain (6.2). To prove (6.3) we proceed as follows. Let f(x, a) = log(1 + ‖x‖). We will prove that TLt f is finite. By the skew-product formula (5.1), TLt f(x, a) = E0 (∫ log(1 + ‖x+ y‖)NAσ+a(0,t)(dy) ) , (σ is involved in the Gaussian measure NAσ(0,t)(dy)). Clearly there is C > 0 such that E0 ∫ log(1 + ‖x+ y‖)NAσ+a(0,t)(dy) ≤ E0 ∫ (log(1 + ‖x‖) + log(1 + ‖y‖))NAσ+a(0,t)(dy) ≤ E0 log(1 + ‖x‖) + CE0 ∫ (1 + ‖y‖2)NAσ+a(0,t)(dy) < +∞. Hence from [7, Proposition 1.2.4],∫ `2o`2 log(1 + ‖π1(g)‖`2)p(e, dg) = ∫ `2o`2 log(1 + ‖x‖`2)p(e, dg) < +∞, g = (x, a). (6.4) Notice that the real random variable Y is integrable if and only if ∞∑ n=1 P(|Y | ≥ n) < +∞. EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 13 It follows from (6.4), that for every r > 0 the random variable 1 r ‖π1(g)‖`2 = 1 r ‖Xn‖`2 is integrable. Therefore, ∞∑ n=1 p (e, {g ∈ G : log(1 + ‖Xn‖`2) ≥ nr}) = ∞∑ n=1 p(e, {g ∈ G : log(1 + ‖X1‖`2) ≥ nr}) < +∞ ∀r > 0. Let An = {log(1 + ‖Xn‖`2) ≥ nr}. By Borel-Cantelli lemma lim sup n≥1 An := ∩∞n=1 ( ∪∞m=n Am ) = ∅ a.e. Thus, lim sup n≥1 (1 + ‖Xn‖`2)1/n ≤ er a.e. and consequently lim sup n≥1 ‖Xn‖1/n`2 ≤ er, a.e. Now taking r → 0 inequality (6.3) is proved. Now we prove that the object we have defined is the Poisson kernel. The argu- ment is standard and is taken from [17, 4, 8]. The operator L generates semigroup of probability measures µt. Consider probability measure νn(dy) = νn(0, dy). That is νn is the distribution of Rn with R0 = (0, 0). Recall that the transition kernel for Rn was defined as µt for some fixed t. Therefore, for the random walk Rn we have µ ∗ νn = νn+1. Obviously for every bounded continuous function f defined on X we have lim n→∞ (f, νn) = (f, ν). Consequently µ ∗ ν = ν and the proof is complete. � Remark 6.2. Using methods of [4, 5] it is possible to prove that in fact νn(x, dy) and ν(x, dy) do not depend on x. Now we recall a version of Doob’s theorem [5, p. 17]. Theorem 6.3. Let Xt be an almost surely continuous stochastic process. Suppose that for every sequence 0 < tn →∞ such that lim n→∞ tn n = 1 there exists a limit lim tn→∞ Xtn = Z a.e. which does not depend on {tn}. Then lim t→∞ Xt = Z a.e. 14 R. C. PENNEY, R. URBAN EJDE-2022/04 By Doob’s theorem we obtain the following corollary from Theorem 6.1. Corollary 6.4. Let µt = PLt (dy) be the semigroup of probability measures on G = X oA = `2 o `2 generated by L. Then for f ∈ Cb(X), lim t→∞ (π1(µ̌t), f) = (ν(·), f), where (µ̌, h) = (µ, ȟ), ȟ(g) = h(g−1). References [1] V. I. Bogachev; Gaussian measures, Mathematical Surveys and Monographs, 62, American Mathematical Society, Providence, RI, 1998. [2] Ph. Bougerol, L. Élie; Existence of positive harmonic functions on groups and on covering manifolds, Ann. Inst. H. Poincaré Probab. Statist. 31 (1995), no 1, 59–80. [3] Yu. L. Dalecky, S. V. Fomin; Measures and differential equations in infinite-dimensional space, translated from the Russian, with additional material by V. R. Steblovskaya, Yu. V. Bogdansky, and N. Yu. Goncharuk, Mathematics and its Applications (Soviet Series), 76, Kluwer Academic Publishers Group, 1991. [4] E. Damek; Left-invariant degenerate elliptic operators on semidirect extensions of homoge- neous groups, Studia Math. 89 (1998), 169–196. [5] E. Damek. A. Hulanicki; Boundaries for left-invariant subelliptic operators on semidirect products of nilpotent and abelian groups, J. Reine Angew. Math. 411 (1990), 1–38. [6] E. Damek, A. Hulanicki, R. Urban; Martin boundary for homogeneous Riemannian manifolds of negative curvature at the bottom of the spectrum, Rev. Mat. Iberoamericana 17(2001), no 2, 257–293. [7] G. Da Prato, J. Zabczyk; Second order partial differential equations in Hilbert spaces, London Mathematical Society Lecture Note Series, 293, Cambridge University Press, Cambridge, 2002. [8] L. Élie; Comportement asymptotique du noyau potentiel sur les groupes de Lie, Ann. Sci. École Norm. Sup. (4) 15 (1982) no 2, 257–364. [9] P. R. Halmos; Measure theory, Graduate Text in Mathematics 18, Springer, 2013. [10] R. Penney, R. Urban; Estimates for the Poisson kernel on higher rank NA groups, Colloq. Math., 118 (2010) no 1, 259–281. [11] R. Penney, R. Urban; An upper bound for the Poisson kernel on higher rank NA groups, Potential Analysis, 35 (2011) no 4, 373–386. [12] R. Penney, R. Urban; Estimates for the Poisson kernel and the evolution kernel on the Heisenberg group, J. Evol. Equ., 12 (2012), no 2, 327–351. [13] R. Penney, R. Urban; The evolution and Poisson kernels on nilpotent meta-abelian groups, Studia Math., 219 (2013) no 1, 69–96. [14] R. Penney, R. Urban; Gaussian type upper bound for the evolution kernels on nilpotent meta-abelian groups, Positivity, 20 (2016), no 2, 257-281. [15] R. Penney, Urban; Poisson kernels on nilpotent, 3-meta-abelian groups, Ann. Mat. Pura Appl.(4) 195 (2016), no 2, 293–307. [16] R. Penney, R. Urban; Poisson kernels on semi-direct products of abelian groups, Math. Slo- vaca, 66 (2016), no 6, 1375–1386. [17] A. Raugi; Fonctions harmoniques sur les groupes localement compacts à base dénombrable. Bull. Soc. Math. France Mém. 54 (1977), 5–118. [18] D. W. Robinson; Elliptic operators and Lie groups, Oxford Mathematical Monographs, Ox- ford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. [19] D. W. Stroock, S. R. S. Varadhan; Multidimensional diffusion processes, reprint of the 1997 edition, Classics in Mathematics, Springer, Berlin 2006. [20] J. A. van Casteren; Feller semigroups and evolution equations, Series on Concrete and Ap- plicable Mathematics, 12, World Scientific Publishing Co., Pte. Ltd., Hackensack, 2011. [21] N. Th. Varopoulos, L. Saloff-Coste, T. Coulhon; Analysis and geometry on groups, Vol. 100 of Cambridge Tracts in Math., Cambridge Univ. Press, 1992. EJDE-2022/04 POISSON MEASURES ON HILBERT SPACES 15 Richard C. Penney Department of Mathematics, Purdue University, 150 N. University St, West Lafayette, IN 47907, USA Email address: rcp@math.purdue.edu Roman Urban Institute of Mathematics, Wroclaw University, Plac Grunwaldzki 2/4, 50-384 Wroclaw, Poland Email address: roman.urban@math.uni.wroc.pl 1. Introduction 2. Gaussian measures on 2 3. Vertical component 4. Horizontal component 4.1. Ad-invariance 5. A skew-product formula 6. Construction of the Poisson measure References