Generalized Mehler Semigroup on White Noise Functionals and White Noise Evolution Equations

In this paper, we study a representation of generalized Mehler semigroup in terms of Fourier–Gauss transforms on white noise functionals and then we have an explicit form of the infinitesimal generator of the generalized Mehler semigroup in terms of the conservation operator and the generalized Gross Laplacian. Then we investigate a characterization of the unitarity of the generalized Mehler semigroup. As an application, we study an evolution equation for white noise distributions with n-th time-derivative of white noise as an additive singular noise.


Introduction
Since the white noise theory initiated by Hida [1] as an infinite dimensional distribution theory, it has been extensively studied by many authors [2][3][4][5][6][7][8] (and references cited therein) with many applications to wide research fields, stochastic analysis, quantum field theory, mathematical physics, mathematical finance and etc. The white noise theory is based on a Gelfand triple: we refer the reader to [20] (see also [21]) which includes major recent achievements and open questions, and in which the generalized Mehler semigroups are briefly discussed.
The objective of this paper is twofold: the first one is to study the generalized Mehler semigroup on the space (E) of the test white noise functionals with its explicit form in terms of the generalized Fourier-Gauss transform. From the representation of the generalized Mehler semigroup, we investigate a characterization of the unitarity of the generalized Mehler semigroup. The second objective is to study a white noise Langevin (type) equation: where Ξ * ∈ L((E) * , (E) * ) is the infinitesimal generator of an equicontinuous semigroup and W (n) t is the n-th time-derivative of Gaussian white noise which is considered as a highly singular noise process. Specially, we are interested in the case of Ξ * which is the infinitesimal generator of the adjoint of the generalized Mehler semigroup (see the Equation (16)). Recently, in [22], the author studied an evolution equation associated with the integer power of the Gross Laplacian ∆ G . We note that the Gross Laplacian ∆ G is a special case of the generator of the generalized Mehler semigroup.
As main results of this paper, we provide a representation of the generalized Mehler semigroup in terms of the generalized Fourier-Gauss transform on the space of the test white noise functionals, and then by applying the properties of the generalized Fourier-Gauss transform, we have an explicit form of the infinitesimal generator of the generalized Mehler semigroup, which is a perturbation of the Ornstein-Uhlenbeck generator. By duality, we study the generalized Fourier-Mehler transform and its infinitesimal generator, which induce the dual semigroup of the generalized Mehler semigroup and its infinitesimal generator, and then as application we investigate the unique weak solution of the Langevin type stochastic evolution equations with very singular noise forcing terms (see Theorem 9). This paper is organized as follows. In Section 2, we recall basic notions for Gaussian space and (Gaussian) white noise functionals. In Section 3, we invite the general theory for white noise operators which is necessary for our main study. In particular, we review the generalized Fourier-Gauss and Fourier-Mehler transforms on white noise functionals. In Section 4, we study the generalized Mehler semigroups on (test) white noise functionals with their representations in terms of generalized Fourier-Gauss transform, and explicit forms of the infinitesimal generators of the generalized Mehler semigroups in terms of the conservation operators and the generalized Gross Laplacian. As the last result of Section 4, we investigate a characterization of the unitarity of the generalized Mehler semigroup. In Section 5, we consider the white noise integrals of white noise operator processes as integrands against with the highly singular noise processes (n-th time-derivatives of white noise). In Section 6, we investigate the unique existence of a weak solution of Langevin (type) white noise evolution equation for white noise distribution whose explicit solution is represented by the adjoint of the generalized Mehler semigroup.

White Noise Functionals
Let H be a separable Hilbert space and let A be a positive, selfadjoint operator in H. Suppose that there exist a complete orthonormal basis {e n } ∞ n=1 for H and an increasing sequence {λ n } ∞ n=1 of positive real numbers with λ 1 > 1 such that (A1) for all n ∈ N, Ae n = λ n e n , (A2) A −1 is of Hilbert-Schmidt type, i.e.
For each p ∈ R, we define a norm | · | p by for any 0 ≤ p ≤ q, where E −p and E * p are topologically isomorphic. Then by taking the projective limit space of E p and the inductive limit space of E −p , we have a Gelfand triple: where from the condition (A2), E becomes a countably Hilbert nuclear space.

Example 1.
As a prototype of the Gelfand triple given as in (1), we consider the Hilbert space H = L 2 (R, dt) of square integrable complex-valued functions of R with respect to the Lebesgue measure and the harmonic oscillator A = − d 2 dt 2 + t 2 + 1 (see [7,8]). Then the projective limit space E coincides with the Schwartz space S(R) of rapidly decreasing C ∞ -functions and E * coincides with the space S (R) of tempered distributions and then we have which will be used for the concrete construction of Brownian motion and its higher-order time derivatives.
By the Bochner-Minlos theorem, we see that there exists a probability measure µ, called the standard Gaussian measure, on E * R such that where E * R is the real nuclear space such that E = E * R + iE * R and ·, · is the canonical complex-bilinear form on E * × E. Note that the inner product on the Hilbert space H is given by ·, · . The probability space (E * R , µ) is called a standard Gaussian space and we put (L 2 ) := L 2 (E * R , µ) the space of square integrable complex-valued (Gaussian) random variables.
By the celebrated Wiener-Itô-Segal isomorphism, (L 2 ) is unitarily isomorphic to the (Boson) Fock space Γ(H) defined by where H ⊗n is the n-fold symmetric tensor product of H and | · | 0 is the Hilbertian norm on H ⊗n again. Note that the Wiener-Itô-Segal (unitary) isomorphism between (L 2 ) and Γ(H) is determined by the following correspondence: where φ ξ is called an exponential vector (or coherent vector) associated with ξ ∈ E. The second quantization Γ(A) of A is defined in Γ(H) by see (5). From the (Boson) Fock space Γ(H) and the positive, selfadjoint operator Γ(A) in Γ(H), by using the arguments used to construct the Gelfand triple given as in (1), we construct a chain of (Boson) Fock spaces: for any p ≥ 0, and by taking the projective and inductive limit spaces, we have the Gelfand triple: and then (E) becomes again a countably Hilbert nuclear space (see [4,7,8]). Then from the Gelfand triple (4), by using the Wiener-Itô-Segal unitary isomorphism, we have the Gelfand triple of Gaussian white noise functionals: which is referred as to the Hida-Kubo-Takenaka space [6]. For each p ∈ R, the Hilbertian norm on Γ(E p ) is denoted by · p and given by The canonical complex-bilinear form on (E) * × (E) is denoted by ·, · and then for each Φ = (F n ) ∈ (E) * and φ = ( f n ) ∈ (E), we have Note that {φ ξ : ξ ∈ E} spans a dense subspace of (E). Therefore, every Φ ∈ (E) * is uniquely determined by the function SΦ : E → C defined as which is called the S-transform of Φ. In fact, for each Φ ∈ (E) * and F := SΦ, we can easily see that The converse is also true as given in the next theorem, which is called the analytic characterization theorem for S-transform.

Theorem 1 ([23]).
A complex-valued function F on E is the S-transform of an element in (E) * if and only if F satisfies the conditions (S1) and (S2).

Remark 1.
Theorem 1 is originally from [23] and the proof of Theorem 1 in [23] had an essential gap and then the gap has been corrected later (see [24]). For a corrected proof of Theorem 1, we refer to [7,8] (see also [24]).

White Noise Operators
For locally convex spaces X and Y, the space of all continuous linear operators from X into Y is denoted by L(X, Y). A continuous linear operator Ξ ∈ L((E), (E) * ) is called a white noise operator (or a generalized operator).
As an operator version of the S-transform, the symbol Ξ : Then since the exponential vectors span a dense subspace of (E), every white noise operator Ξ ∈ L((E), (E) * ) is uniquely determined by the operator symbol Ξ. In fact, for each Ξ ∈ L((E), (E) * ) and Θ = Ξ, we can easily see that Furthermore, if Ξ ∈ L((E), (E)), then Θ = Ξ satisfies the following condition: (Θ2 ) for any p ≥ 0 and > 0, there exist constants K ≥ 0 and q ≥ 0 such that The converse is also true as given in the next theorem, which is called the analytic characterization theorem for operator symbol. Throughout this paper, for a white noise operator Ξ ∈ L((E), (E) * ), the adjoint operator of Ξ with respect to the canonical complex-bilinear form ·, · is denoted by Ξ * . Then for each Ξ ∈ L((E), (E) * ), we have Ξ * ∈ L((E), (E) * ) and for any φ, ψ ∈ (E), Then we can easily check that Θ 1 satisfies the conditions (Θ1) and (Θ2 ) and then by Theorem 2, there exists a unique white noise operator, denoted by ∆ G (S) and called the generalized Gross Laplacian (see [27]), in L((E), (E)) such that ∆ G (S) = Θ 1 . In fact, the generalized Gross Laplacian ∆ G (S) is uniquely determined by the action on exponential vectors: In particular, for S = I (the identity operator), ∆ G (I) is called the Gross Laplacian and denoted by ∆ G . For the adjoint operator of Then we can easily check that Θ 2 satisfies the conditions (Θ1) and (Θ2) and then by Theorem 2, there exists a unique white noise operator, denoted by Λ(S) and called the conservation operator (see [8,27] In fact, if S ∈ L(E, E) is an equicontinuous generator (see Section 4 or [28]), then for the equicontinuous semigroup {e tS } t∈R generated by S, the conservation operator Λ(S) is uniquely determined by the action on exponential vectors: see [29]. Then we have Λ(S) * = Λ(S * ).

Example 4.
Let K ∈ L(E, E * ) and S ∈ L(E, E) be given.
(1) Consider a function Θ 3 : Then we can check that Θ 3 satisfies the conditions (Θ1) and (Θ2 ). Therefore, by Theorem 2, there exists a unique white noise operator, denoted by G K,S and called the generalized Fourier-Gauss transform (see [7,27]), in L((E), (E)) such that G K,S = Θ 3 . In fact, the generalized Fourier-Gauss transform G K,S is uniquely determined by the action on exponential vectors: and so we have (2) The adjoint operator of G K,S with respect to ·, · , denoted by F K,S , i.e., F K,S = G * K,S and called the generalized Fourier-Mehler transform (see [7,27]), belongs to L((E) * , (E) * ). Then we have

Generalized Mehler Semigroup
Let {S * t } t≥0 and {T * t } t≥0 be families of continuous linear operators on E * R . For each ξ ∈ E, we define In fact, from (2), we obtain that Therefore, since µ is a Gaussian measure, we obtain that and so we have as an element of L((E), (E)).
On the other hand, from Examples 3 and 4, for any ξ ∈ E, we obtain that Therefore, from (9), we obtain that Hence we have the following characterization of semigroup property. holds.

Remark 2.
A general result for a characterization of the semigroup property of {P t } t≥0 can be found in Proposition 2.2 of [12]. In particular, a characterization of the semigroup property of {P t } t≥0 for a general Gaussian case on Hilbert space can be found in Proposition 4.1 of [12]. Furthermore, a definition of a generalized Mehler semigroup can be found in Definition 2.4 of [12].
In (6), we used the fact that for any ξ ∈ E, On the other hand, by applying the Bochner-Minlos theorem, there exists a unique Gaussian measure µ t with the covariance operator T * t T t such that Since the Gaussian measure µ on E * R is symmetric, from (6) we have and so from the continuities of P t and the integral transform given as in the right hand side of (11), we see that  (10).

Remark 3.
Let S * ∈ L(E * , E * ) be given. Then S * is said to be real if S * E * R ⊂ E * R . Consider the generalized Mehler semigroup {P t } t≥0 given by for ξ ∈ E. If we consider P t as the left hand side of (12), then the exponential vector φ ξ is defined on E * R and so we have to consider the families {S * t } t≥0 ⊂ L(E * , E * ) and {T * t } t≥0 ⊂ L(E * , E * ) which are real. However, if we consider P t as the right hand side of (12), then we do not need such restriction. Throughout this paper, we consider the generalized Mehler semigroup {P t } t≥0 defined as the right hand side of (12). (8), if S t and T t satisfy that T * t T t + S * t S t − I = 0, i.e., T t = (I − S * t S t ) 1/2 , then we have P t = Γ(S t ) and so in this case, {P t } t≥0 becomes the Ornstein-Uhlenbeck semigroup and then we have the Mehler's formula of the Ornstein-Uhlenbeck semigroup (or second quantization) in the infinite dimensional case:

Remark 4. In
which can be found in Theorem 6.1.1 of [2] (see also Theorem 4.5 of [16] where V = d dt T * t T t t=0 .
Proof. Suppose that {P t } t≥0 is a generalized Mehler semigroup. For notational convenience, put Then from (10), by taking s = t = 0, we have 2V(0) = V(0), and so V(0) = 0. Furthermore, we obtain that from which we have (14). The proof of the converse is straightforward.

Remark 5.
A similar explicit form of T * t T t for the generalized Mehler semigroup {P t } t≥0 for a general Gaussian case on a Hilbert space can be found in Proposition 4.3 of [12].

Example 5.
Let b ∈ C and S t = e bt for t ≥ 0. Then from (14), we have where V ∈ L(E, E * ) is a given operator, and so we have Therefore, we have where G C,D be a generalized Fourier-Gauss transform (see [27,30], and also Example 4).
Let X be a barrelled locally convex Hausdorff space whose topology is generated by a family of seminorms {| · | p } p∈N X . An operator S ∈ L(X, X) is called an equicontinuous generator if for any r > 0, the family {(rS) n /n!} ∞ n=0 is equicontinuous, i.e., for any p ∈ N X , there exist C ≥ 0 and q ∈ N X such that | ((rS) n /n!) x| p ≤ C|x| q for all x ∈ X (see [8,28]). For such equicontinuous generator S, we can prove that the series and furthermore, if the map t → T * t T t ∈ L(E, E * ) is differentiable at 0, then by Lemma 1, we have and hence we have the following theorem for the explicit representation of the generalized Mehler semigroup.
Theorem 3. Let {S * t } t≥0 ⊂ L(E * , E * ) and {T * t } t≥0 ⊂ L(E * , E * ) be families of continuous linear operators. Suppose that {S t } t≥0 ⊂ L(E, E) is an equicontinuous semigroup with the infinitesimal generator S ∈ L(E, E) and the map t → T * t T t ∈ L(E, E * ) is differentiable at 0. Then the generalized Mehler semigroup {P t } t≥0 given as in (8) has the following representation in terms of the generalized Fourier-Gauss transform:

Theorem 4.
Under the assumptions given as in Theorem 3, the infinitesimal generator of the generalized Mehler semigroup {P t } t≥0 is given by Λ(S) + 1 2 ∆ G (V + S * + S).
Proof. We now give a sketch of the proof. A detailed proof of this theorem is a simple modification of the proof of Theorem 5.3.11 of [8] (also, see the proof of Theorem 4.3 of [30]). Consider the symbol of P t and then for any ξ, η ∈ E, we obtain that from which we have the desired assertion.
For a Gelfand triple X ⊂ K ⊂ X * , where X = X R + iX R with the real vector space X R , and for L ∈ L(X , X * ), the complex conjugation L of L is defined by and then for the Hermitian adjoint L † of L, we have L † = (L * ) = L * .
Theorem 5. Let {S t } t≥0 ⊂ L(E, E) be an equicontinuous semigroup with the infinitesimal generator S ∈ L(E, E) and {T * t } t≥0 ⊂ L(E * , E * ) be a family of continuous linear operators such that the map t → T * t T t ∈ L(E, E * ) is differentiable at 0. Suppose that for each t ≥ 0, S t can be extended to H such that {S t } t≥0 ⊂ L(H, H) is a strongly continuous semigroup. Let {P t } t≥0 be the generalized Mehler semigroup defined as in (7). Then the followings are equivalent: (i) For each t ≥ 0, P t can be extended to Γ(H) as a unitary operator, (ii) S t is unitary and K t := T * t T t + S * t S t − I = 0 for all t ≥ 0, (iii) S † = −S and V + S + S * = 0, where V = d dt T * t T t t=0 .
Proof. (i) ⇔ (ii) We first observe that for any ξ, η ∈ E, Since the exponential vectors span a dense subspace of Γ(H) and ξ, η are chosen arbitrarily, P † t P t = I holds if and only if K t = 0 and S † t S t = I. Therefore, we have P t = Γ(S t ) and Hence S t is unitary and K t = 0. Conversely, K t = 0 implies P t = Γ(S t ) and so S t is unitary implies that P t is unitary.
(ii) ⇔ (iii) Note that, by Stone's theorem (Theorem 10.8 in [31]), S t is unitary if and only if iS is selfadjoint if and only if S † = −S. Moreover, we observe that Hence (ii) and (iii) are equivalent.

Remark 6.
The unitarity of the generalized Fourier-Gauss transform has been studied in [32,33] as transform from the space of Gaussian functionals onto another space of Gaussian functionals with different covariance operator.

White Noise Integrals
Let T > 0 be fixed and let Φ : [0, T] → (E) * be a function. If the map [0, T] t → S (Φ(t)) (ξ) is measurable for all ξ ∈ E and there exist nonnegative constants K, c ≥ 0 and p ≥ 0 such that T 0 |S (Φ(t)) (ξ)|dt ≤ Ke c|ξ| 2 p for all ξ ∈ E, then by applying Theorem 13.4 of [7], we see that Φ is Pettis integrable and for any ξ ∈ E, we have dt exists and belongs to (E) * . If there exist nonnegative constants K, c ≥ 0 and p ≥ 0 such that for all ξ ∈ E, then by the discussion above we see that U(t) (Φ (t)) is Pettis integrable and In such a case, we write From now on, we consider the case H := L 2 (R, dt) and E := S(R) the Schwartz space of rapidly decreasing C ∞ functions on R. Then we have E * = S (R) the space of tempered distributions (see Example 1).
For each t ≥ 0, we define B t ∈ (L 2 ) by where we used the approximation procedure to define B t , i.e, since E is dense in H and 1 [0,t] ∈ H, there exists a sequence {ξ n } ∞ n=1 ⊂ E such that lim n→∞ ξ n = 1 [0,t] in H and which implies that ·, 1 [0,t] := lim n→∞ ·, ξ n exists in L 2 (E * R , µ). Then we can easily see that from which we see that {B t } is a Brownian motion and it is called a realization of a Brownian motion.

Theorem 6 ([34]
). For each n = 0, 1, 2, · · · , the map t → δ (n) t ∈ E −p is continuous for By applying Theorem 6, we see that the map t → B t ∈ (E) * is a C ∞ -map and we have is an equicontinuous semigroup with the infinitesimal generator S ∈ L(E, E) and the map t → T * t T t ∈ L(E, E * ) is differentiable at 0. Then for the generalized Mehler semigroup {P t } t≥0 given as in (8) and any n ∈ N, is Pettis integrable over [0, t] for all t > 0.
Proof. From Theorem 3, the explicit form of the generalized Mehler semigroup {P t } t≥0 is given by For notational convenience, we put Q := V + S * + S ∈ L(E, E * ).
Since {S t } t≥0 is an equicontinuous semigroup with the infinitesimal generator S ∈ L(E, E), we can write S t = e tS and then we have Therefore, for any ξ ∈ E, we obtain that S P * t−s W On the other hand, since {S t } t≥0 is an equicontinuous semigroup, by applying Lemma 2.1 of [28], we see that for any p ≥ 0, there exist a constant C ≥ 0 and q ≥ p such that where the constants C and p are depending on t > 0. Hence by applying Theorem 6 and the continuity of the differential operator on E = S(R), we see that there exist nonnegative constants K, c ≥ 0 and p ≥ 0 such that for all ξ ∈ E, then by the discussion above we see that P * t−s W (n) s is Pettis integrable.

Stochastic Evolution Equations
In this section, motivated by the results obtain in Section 4 (see Theorem 4), we study the following stochastic evolution equation (for white noise distribution): where Ψ ∈ (E) * is given and W (n) t is the n-th distributional derivative of the white noise process {W t } t≥0 (see (15)). As an abstract extension of (16), we study the stochastic evolution equation: where Ξ * ∈ L((E) * , (E) * ) is a given operator.
In the white noise theory, the integral equation given as in (18) can be represented by a white noise integral equation as following: In fact, we have Hence as a general case, we consider the following white noise integral equation: where f is a function from [0, T] × (E) * into (E) * (see (13.74) in [7]). Then as a special case of Theorem 13.43 of [7], i.e., by taking β = 0 in Theorem 13.43 of [7], we have the following theorem.
for all ξ ∈ E and 0 ≤ s ≤ t with 0 ≤ t ≤ T. Then the stochastic evolution equation of (17) has a unique weak solution Φ t ∈ (E) * given by Moreover, Φ t is given by Proof. Uniqueness. If U(t) and V(t) are weak solutions of (18), then Y(t) = U(t) − V(t) is a solution of the equation: from which, since Ξ is an equicontinuous generator, we have Y(t) = 0 and so U(t) = V(t) for all 0 ≤ t ≤ T. Existence. From the condition given as in (24), for each t > 0, we can see that the white noise integral:   . We now prove that Φ t given as in (25) is a weak solution of the stochastic evolution equation of (17). For any ϕ ∈ (E), we have which proves that Φ t is a weak solution of (25). The representation of Φ t given as in (26) can be obtained by applying the integration by parts formula from (25). a very singular noise forcing term of which the unique weak solution is given as in Theorem 9 which is one of main results of this paper. We should emphasis that our approach provides a useful tool to study the generalized Mehler semigroup and associated Langevin type stochastic evolution equations with singular noise forcing terms, and can be applied to study more general Langevin type equations associated with the perturbations of Ornstein-Uhlenbeck generators.