Stochastic evolution equations with multiplicative noise

We study parabolic stochastic partial differential equations (SPDEs), driven by two types of operators: one linear closed operator generating a $C_0-$semigroup and one linear bounded operator with Wick-type multiplication, all of them set in the infinite dimensional space framework of white noise analysis. We prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the stationary case when the time-derivative is equal to zero.


Introduction and definitions
We consider a stochastic Cauchy problem of the form d dt U(t, x, ω) = AU(t, x, ω) + B♦U(t, x, ω) + F (t, x, ω) where t ∈ (0, T ], ω ∈ Ω, and U(t, ·, ω) belongs to some Banach space X. The operator A is densely defined, generating a C 0 −semigroup and B is a linear bounded operator which combined with the Wick product ♦ introduces convolution-type perturbations into the equation. All stochastic processes are considered in the setting of Wiener-Itô chaos expansions. A comprehensive explanation of the action of the operators A and B in this framework will be provided in Section 2. Our investigations in this paper are inspired by [12] where the authors provide a comprehensive analysis of equations of the form d dt u(t, x, ω) = Au(t, x, ω) + δ(Mu(t, x, ω)) = Au(t, x, ω) + Mu(t, x, ω)♦W (x, ω) dx, where δ denotes the Skorokhod integral and W denotes the spatial white noise process. In Proposition 2.8 we prove that for every operator M there exists a corresponding operator B such that B♦u = δ(Mu). On the other hand, the class of operators B is much larger. This holds also for the class of operators A we consider (a comprehensive analysis of all operators is given in Section 2.1). Thus, we extend the results of [12] and [13] to a more general class of stochastic differential equations which are driven by two linear multiplicative operators: A acting with ordinary multiplication, while B♦ is acting with the convolution-type Wick product.
We have studied elliptic SPDEs, in particular the stochastic Dirichlet problem of the form L♦u + f = 0 in our previous papers [11], [18], [19]. As a conclusion to this series of papers we study parabolic SPDEs of the form (3). Such equations also include as a special case equations of the form d dt u = Lu + f and d dt u = L♦u + f , where L is a strictly elliptic second order partial differential operator. These equations describe the heat conduction in random media (inhomogeneous and anisotropic materials), where the properties of the material are modeled by a positively definite stochastic matrix.
Other special cases of (3) include the heat equation with random potential d dt u = ∆u+B♦u, the Schrödinger equation (i ) d dt u = ∆u+B♦u+f , the transport equation d dt u = d 2 dx 2 u + W ♦ d dx u driven by white noise as in [20], the generalized Langevin equation d dt u = Ju+C(Y ′ ), where Y is a Lévy process, J the infinitesimal generator of a C 0 −semigroup and C a bounded operator, which was studied in [1], as well as the equation d dt u = Lu + W ♦u, where L is a strictly elliptic partial differential operator as studied in [3] and [8].
Equations of the form d dt u = Au+BW were also studied in [14] and [15], where A is not necessarily generating a C 0 −semigroup, but an r-integrated or a convolution semigroup.
In order to solve (3) we apply the method of Wiener-Itô chaos expansions, also known as the propagator method. With this method we reduce the SPDE to an infinite triangular system of PDEs, which can be solved by induction. Summing up all coefficients of the expansion and proving convergence in an appropriate weight space, one obtains the solution of the initial SPDE.
We also consider the case of stationary equations AU + B♦U + F = 0. In particular, elliptic SPDEs have been studied in [11], [13], [18] and [19]. With the method of chaos expansions one can also treat hyperbolic SPDEs [9] and SPDEs with singularities [21]. One of its advantages is that it provides explicit solutions in terms of a series expansion, which can be easily implemented also to numerical approximations and computational simulations.

C 0 −semigroups
We recall some well-known facts which will be used in the sequel (see [16]). Let X be a Banach space. If B is a bounded linear operator on X and A is the infinitesimal generator if u is continuous on [0, T ], continuously differentiable on (0, T ], u(t) ∈ D, t ∈ (0, T ] and the equation is satisfied on , and it is called a mild solution. Clearly, a mild solution that is continuously differentiable on (0, T ] is a classical solution. Let The initial value problem has a solution u for every u 0 ∈ D if one of the following conditions is satisfied (see [16]): If the initial value problem has a solution on [0, T ] for some u 0 ∈ D, then v(t) satisfies both (i) and (ii). Note that if f ∈ C 1 ([0, T ], X) then conditions (i) and (ii) are fulfilled. Moreover, if f ∈ C 1 ([0, T ], X) and u 0 ∈ D(A), then for the solution u of (2) we have that u ∈ C 1 ([0, T ], X) and d dt u(0) = Au 0 + f (0).

Generalized stochastic processes
Denote by (Ω, F , P ) the Gaussian white noise probability space (S ′ (R), B, µ), where S ′ (R) denotes the space of tempered distributions, B the Borel sigma-algebra generated by the weak topology on S ′ (R) and µ the Gaussian white noise measure corresponding to the characteristic function given by the Bochner-Minlos theorem. We recall the notions related to L 2 (Ω, µ) (see [7]) where Ω = S ′ (R) and µ is Gaussian white noise measure. Define the set of multi-indices I to be (N N 0 ) c , i.e. the set of sequences of non-negative integers which have only finitely many nonzero components. Especially, we denote by 0 = (0, 0, 0, . . .) the multi-index with all entries equal to zero. The length of a multi-index is |α| = ∞ i=1 α i for α = (α 1 , α 2 , . . .) ∈ I, and it is always finite. Similarly, α! = ∞ i=1 α i !, and all other operations are also carried out componentwise. We will use the convention that α − β is defined if α n − β n ≥ 0 for all n ∈ N, i.e., if α − β ≥ 0, and leave α − β undefined if α n < β n for some n ∈ N.
Then, every F ∈ L 2 (Ω, µ) can be represented via the so called chaos expansion Denote by ε k = (0, 0, . . . , 1, 0, 0, . . .), k ∈ N the multi-index with the entry 1 at the kth place. Denote by H 1 the subspace of L 2 (Ω, µ), spanned by the polynomials H ε k (·), k ∈ N. The subspace H 1 contains Gaussian stochastic processes, e.g. Brownian motion is given by the chaos expansion Denote by H m the mth order chaos space, i.e. the closure of the linear subspace spanned by the orthogonal polynomials H α (·) with |α| = m, m ∈ N 0 . Then the Wiener-Itô chaos expansion states that L 2 (Ω, µ) = ∞ m=0 H m , where H 0 is the set of constants in L 2 (Ω, µ). It is well-known that the time-derivative of Brownian motion (white noise process) does not exist in the classical sense. However, changing the topology on L 2 (Ω, µ) to a weaker one, T. Hida [6] defined spaces of generalized random variables containing the white noise as a weak derivative of the Brownian motion. We refer to [6], [7] for white noise analysis (as an infinite dimensional analogue of the Schwartz theory of deterministic generalized functions).
Let (2N) α = ∞ n=1 (2n) αn , α = (α 1 , α 2 , . . . , α n , . . .) ∈ I. We will often use the fact that the series α∈I (2N) −pα converges for p > 1. Define the Banach spaces Their topological dual spaces are given by The Kondratiev space of generalized random variables is (S) −1 = p∈N 0 (S) −1,−p endowed with the inductive topology. It is the strong dual of (S) 1 = p∈N 0 (S) 1,p , called the Kondratiev space of test random variables which is endowed with the projective topology. Thus, forms a Gelfand triplet. The time-derivative of the Brownian motion exists in the generalized sense and belongs to the Kondratiev space (S) −1,−p for p ≥ 5 12 . We refer to it as to white noise and its formal expansion is given by . We extended in [17] the definition of stochastic processes also to processes of the chaos expansion form U(t, ω) = α∈I u α (t)H α (ω), where the coefficients u α are elements of some Banach space X. We say that U is an X-valued generalized stochastic process, i.e.
and the nth Wick power is defined by F ♦n = F ♦(n−1) ♦F , In [18] we proved that differentiation of a stochastic process can be carried out componentwise in the chaos expansion, i.e. due to the fact that (S) −1 is a nuclear space it holds that This means that a stochastic process U(t, ω) is k times continuously differentiable if and only if all of its coefficients u α (t), α ∈ I are in C k [0, T ].
The same holds for Banach space valued stochastic processes i.e. elements of C k ([0, T ], X)⊗ (S) −1 , where X is an arbitrary Banach space. By the nuclearity of (S) −1 , these processes can be regarded as elements of the tensor product space

Stochastic operators
Definition 2.1. Let X be a Banach space and O : X ⊗ (S) −1 → X ⊗ (S) −1 an operator acting on the space of stochastic processes. We will call O to be a coordinatewise operator if there exists a family of operators o α : Clearly, not all operators are coordinatewise, for example O(F ) = F ♦2 can not be written in this form. For example, the operator of differentiation [18] and the Fourier transform [21] are simple coordinatewise operators. The Ornstein-Uhlenbeck operator is a coordinatewise operator but it is not a simple coordinatewise operator.
Note that even if all o α , α ∈ I, are bounded linear operators, the coordinatewise operator O itself does not need to be bounded. If o α , α ∈ I, are uniformly bounded by some C > 0, then O is also a bounded operator. This follows from This condition is sufficient, but not necessary, and can be loosened by the embedding Note that the condition o α L(X) ≤ R(2N) rα for some r, R > 0 is actually equivalent to stating that there exists r > 0 such that Throughout the paper we will consider the equation where both operators A and B are assumed to be coordinatewise operators, i.e. composed out of a family of operators {A α } α∈I , {B α } α∈I , respectively. The operators A α , α ∈ I, are assumed to be infinitesimal generators of C 0 −semigroups with a common domain D dense in X and the action of A is given by The operators B α , α ∈ I, are assumed to be bounded and linear on X, and the action of the operator B♦ : In the next two lemmas we provide two sufficient conditions that ensure the operator B♦ to be well-defined. Both conditions are actually equivalent to the fact that B α , α ∈ I, are polynomially bounded, but they provide finer estimates on the stochastic order (Kondratiev weight) of the domain and codomain of B♦.

Special cases and relationship to other works
Some of the most important operators of stochastic calculus are the operators of the Malliavin calculus. We recall their definitions in the generalized S ′ (R) setting [10].
• The Malliavin derivative, D, as a stochastic gradient in the direction of white noise, is a linear and continuous mapping D : In terms of quantum theory it corresponds to the annihilation operator reducing the order of the chaos space ( D : H m → H m−1 ).
• The Skorokhod integral, δ, as an extension of the Itô integral to non-anticipating processes, is a linear and continuous mapping δ : It is the adjoint operator of the Malliavin derivative and in terms of quantum theory it corresponds to the creation operator increasing the order of the chaos space (δ : • The Ornstein-Uhlenbeck operator, R, as the composition of the previous ones δ • D, is the stochastic analogue of the Laplacian. It is a linear and continuous mapping In terms of quantum theory it corresponds to the number operator. It is a selfadjoint operator R : H m → H m with eigenvectors equal to the basis elements H α , α ∈ I, i.e. R(H α ) = |α|H α , α ∈ I. Thus, Gaussian processes with zero expectation are the only fixed points for the Ornstein-Uhlenbeck operator.
Clearly, the Ornstein-Uhlenbeck operator is a coordinatewise operator, while the Malliavin derivative and the Skorokhod integral are not coordinatewise operators. The Ornstein-Uhlenbeck operator is the infinitesimal generator of the semigroup It is also closely connected to the Ornstein-Uhlenbeck process. The Ornstein-Uhlenbeck process is the solution of the SDE [2]. The solution of the generalized heat equation d dt u + R(u) = 0, u(0) = u 0 , is given by u = T t (u 0 ), i.e. u(t, x) = (T t u 0 )(x) and (T t ϕ)(x) = E(ϕ(u(t, x)) for any ϕ ∈ C b (R) and u is the Ornstein-Uhlenbeck process. Now we turn to our equation where A and B are coordinatewise operators as described in Section 2, composed out of a family of operators {A α } α∈I , {B α } α∈I , respectively, where A α are infinitesimal generators on X and B α are bounded linear operators on X, both families being polynomially bounded, and their actions given by Some important special cases include the following: I) Special cases for A:

1)
A is a simple coordinatewise operator, i.e. A α = A, α ∈ I, where A is the infinitesimal generator of a C 0 −semigroup on X. Such operators are, for example the Laplacian ∆ on X = W 2,2 (R n ) or any strictly elliptic linear partial differential operator of even order P (x, D) = |ι|≤2m a ι (x)D ι . For example, second order elliptic operators can be written in divergence form L = ∇·(Q∇·+b)+c∇·, where Q is a positively definite function matrix.
2) A α = A + R α , α ∈ I, where A is as in 1), while R α , α ∈ I, are bounded linear operators on X so that R is a coordinatewise operator Especially, if we take A = 0 and R α to be multiplication operators R α (x) = r α ·x, x ∈ X, then the resulting operator R is a self-adjoint operator with eigenvalues r α corresponding to the eigenvectors H α and thus represents a natural generalization of the Ornstein-Uhlenbeck operator. For r α = |α|, α ∈ I, we retrieve the Ornstein-Uhlenbeck operator R.
Finally, we note that every bounded linear coordinatewise operator R can be written in the form Ru = δ(Mu) where M is a generalization of the Malliavin derivative. This will be done in Proposition 2.6.

II) Special cases for B:
1) B is an operator acting as a multiplication operator with a deterministic function, i.e. B α = b for α = (0, 0, 0, 0, . . .) and B α = 0 for all other α ∈ I. Its action is thus For example, we may take X = L 2 (R n ) and b = b(x), x ∈ R n , for an essentially bounded function b.
2) B is multiplication with spatial white noise on X = L 2 (R n ). Clearly, Multiplication with spatial white noise is important for applications since it describes stationary perturbations.
3) B is of the form B ε k = B k , k ∈ N, and B α = 0 for α = ε k , where B k : X → X, k ∈ N, are bounded linear operators. Note that in this case there is a one-to-one correspondence between operators of the form B♦ and operators of the form δ(Mu) where M is a simple coordinatewise operator. This will be done in Proposition 2.8.

4)
B is a simple coordinatewise operator, i.e. B α = B, α ∈ I, where B is a bounded linear operator on X. Alternatively, we may also regard operators as B : X → X ′ in order to make them bounded; such operators are for example the divergence ∇· as a mapping from X = W 1,2 (R n ) to X ′ = W −1,2 (R n ).

There exists an operator
holds.
2. Especially, if R is a selfadjoint operator, then M is a generalization of the Malliavin derivative.
Proof. a) In [10] we proved that the Skorokhod integral is invertible, i.e. there exists a unique solution to equations of the form δ(v) = f . Considering the equation δ(Mu) = α∈I R α u α H α and applying the result from [10], we obtain Mu in the form By defining we obtain the assertion. b) Let R be a self-adjoint operator with eigenvalues r α and eigenvectors H α , α ∈ I, i.e., an operator of the form Ru = α∈I r α u α H α . Assume that r α = k∈N r k,α for some r k,α ∈ R, k ∈ N, α ∈ I, is an arbitrary decomposition of the value r α .
Remark 2.7. The converse is not true. Even if each M k , k ∈ N, is a simple coordinatewise operator (and so is M), R := δ • M does not need to be a coordinatewise operator. This would require that the system R α (u α ) = k∈N m k (u α−ε k ), α ∈ I, is solvable for R α (·) given the functions m k (·), k ∈ N, which is not true in general.
for some simple coordinatewise operators M k : Then, there exists a coordinatewise operator B such that B α = 0 for α = ε k , k ∈ N, and δ(Mu) = B♦u holds.
Conversely, for any coordinatewise operator B such that B α = 0 for α = ε k , k ∈ N, there exists an operator M of the form Mu = ∞ k=1 M k u ⊗ ξ k for some simple coordinatewise operators M k , k ∈ N, such that δ(Mu) = B♦u holds.
Proof. Let M be an operator as stated above and since M k are simple coordinatewise operators, we can write them as for some operators m k : X → X, k ∈ N. Thus, On the other hand, if B is such that B α = 0 for α = ε k , k ∈ N, and we denote by B k := B ε k , k ∈ N, the operators acting on X, then From (8) and (9) it follows that δ(Mu) = B♦u if and only if m k = B k for all k ∈ N. Thus, there is a one-to-one correspondence between the operators B♦ and δ • M.
Remark 2.9. In [12] and [13] Rozovskii and Lototsky considered the equation d dt = Au + δ(Mu) + f , where M is of the form (7). They implicitly assumed that all their operators A and M k , k ∈ N, belong to our class of simple coordinatewise operators. This corresponds to our special cases I-1) and II-3).
Some special cases of stochastic differential equations covered by (4) include the following: • The heat equation with random potential In particular, if the random potential is modeled by stationary perturbations, we may take spatial white noise as a model and obtain d dt u = ∆u + W ♦u. This corresponds to the special choice of operators I-1) and II-2).
• The heat equation in random (inhomogeneous and anisotropic) media, where the physical properties of the medium are modeled by a stochastic matrix Q. This corresponds to the case I-1) with A = 0 and II-5) leading to an equation of the form • Taking A = 0 and B k := B ε k = ξ k ∇·, k ∈ N, (see special cases II-2) and II-4)) we obtain the transport equation driven by white noise • The Langevin equation λ > 0, corresponding to the case I-1) with A = −λ, f = W and B = 0. Its solution is the Ornstein-Uhlenbeck process describing the spatial position of a Brownian particle in a fluid with viscosity λ.

In [1] the authors considered the generalized Langevin equation leading to generalized
Ornstein-Uhlenbeck operators driven by Lévy processes where Y is a Lévy process, J the infinitesimal generator of a C 0 −semigroup and C a bounded operator. All processes are Hilbert space valued. This corresponds to our case with X being this Hilbert space, A = J, B = 0 and f = C(Y ′ ).
• The equation where L is a strictly elliptic partial differential operator as studied in [3] and [8]. This corresponds to the special case I-1) and II-2).

Using (18) we obtain
The first term on the right-hand side, for all t ∈ [0, T 0 ], having in mind (12) and (17), satisfies α∈In,m Similarly, for all t ∈ [0, T 0 ], using (14) and (17), the third term satisfies Note that in (23) we took the supremum over the whole interval [0, T ]. For the second term, using (11), (17), (21) and the generalized Minkowski inequality, we obtain Finally, for all n, m ∈ N, we obtain Let (m n ) n∈N be an arbitrary sequence of positive integers tending to infinity. Then, since it is a series of positive numbers and thus does not depend on the order of summation. Now we show that In order to acomplish that, we differentiate (18) with respect to t, and obtain In the sequel we estimate partial sums of α∈I sup t∈[0, According to (12) and (13), we obtain α∈I (A α + B 0 )u 0 α H α (ω) ∈ X ⊗ (S) −1,−p . So the first term on the right-hand side can be evaluated by α∈In,m The third term, for all t ∈ [0, T 0 ], satisfies The fourth term, using (11), (12), (17) and the generalized Minkowski inequality, can be estimated by α∈In,m For the fifth term, using (14) and (17), we have α∈In,m Finally, for the second term, using (11), (17), (21) and the generalized Minkowski inequality, Finally, for all n, m ∈ N, we obtain Again, taking (m n ) n∈N to be an arbitrary sequence of positive integers tending to infinity, we have Therefore, we obtain Next, we consider in (33) supremums over the interval [T 0 , 2T 0 ]. On [T 0 , 2T 0 ] one can rewrite the initial value problem (15) in the following equivalent form: The semigroup corresponding to the generator A α + B 0 in (34) is again the semigroup (S t ) α , t ≥ 0. Using (15) and (33), we have that U(t, ω) ∈ Dom(A), for all t ∈ [0, T 0 ], and When approximating partial sums of α∈I sup t∈[0,T 0 ] v α (t) 2 X (2N) −pα , comparing to the previous calculations for u α (t), only the constant Q 1 will be different, and here, we denote it by Q 2 , so we again obtain Similarly, for the derivative d dt V (t, ω) we obtain where, comparing to the estimates of d dt U(t, ω), only the constants Q ′ 1 and H ′ 1 have changed and we denoted them here by Q ′ 2 and H ′ 2 . For arbitrary T > 0, one can cover the interval [0, T ] by intervals of the form [kT 0 , (k + 1)T 0 ], k ∈ N 0 , in finitely many steps (say in l steps). So we have Therefore, U(t, ω) ∈ C 1 ([0, T ], X) ⊗ (S) −1,−p and thus, U is a solution of (3) in the sense of Definition 3.1. The solution U is unique due to the uniqueness of the coordinatewise (classical) solutions u α in (18) and due to uniqueness in the Wiener-Itô chaos expansion.
where A is a simple coordinatewise operator A α = A, α ∈ I, generating a C 0 −semigroup, B α = 0 only for α = ε k , k ∈ N, are such that k∈N B ε k (2k) − p 2 < ∞, and U 0 and F are deterministic functions, i.e. u 0 α = 0 and f α = 0 for all α ∈ I \ {0}. The solution of this system, according to Theorem 3.2, is the same form as it was obtained in [12].
We provide two generalisations of Theorem 3.2: one possibility is to allow the operators B α to depend on the time variable t (except for B 0 which must be free of t). This embraces for example SPDEs driven by space-time noises which have zero expectation (and are thus free of t). The other possibility is to allow B 0 to be unbounded but satisfying certain properties so that A α + B 0 are infinitesimal generators of C 0 −semigroups. For example, if A α = ∂ 2 ∂x 2 and B 0 = ∂ ∂x , then although B 0 is unbounded, A α + B 0 is the generator of a contraction semigroup. Following [4] we will enlist some sufficient conditions which ensure that A α + B 0 is the generators of a C 0 −semigroup.  The solution is (37) The proof can be performed in the same manner as in Theorem 3.2, now taking T 0 ∈ (0, T ] to be small enough so that C(T 0 ) < 1 6K 2 , since now we have six summands in (38) instead of the previous five in (26).
Remark 3.6. In Theorem 3.2 one can consider the operator B 0 to be unbounded, densely defined on D (the same domain which is common for all A α ) so that either of the following holds: (i) A α , α ∈ I, are generating contraction semigroups (i.e. M = 1, w = 0), and B 0 is dissipative, , for all α ∈ I, (ii) B 0 is closable, dissipative and A α −compact (i.e. B : (D, · Aα ) → X is compact where · Aα denotes the graph norm), for all α ∈ I, (iii) A α are generating analytic semigroups (i.e. w < 0), α ∈ I, and B 0 is closable and A α −compact .
Then, A α + B 0 is the infinitesimal generator of a C 0 −semigroup (denote it (S t ) α ) for all α ∈ I. If the semigroups (T t ) α corresponding to A α are uniformly bounded in α, then so will be (S t ) α . Retaining all other assumptions of Theorem 3.2, now we follow the same proof pattern with the semigroup (S t ) α , (S t ) α ≤M ew t , for someM ≥ 1,w ∈ R, independent of α.
Finally we note that in case (i) and (ii) A α + B 0 will be generating contraction semigroups, while in case (iii) they will be generating analytic semigroups.

Stationary equations
In this section we consider stationary equations of the form where A : X ⊗ (S) −1 → X ⊗ (S) −1 and B♦ : X ⊗ (S) −1 → X ⊗ (S) −1 are coordinatewise operators as in (5) and (6). We assume that {A α } α∈I and {B α } α∈I are bounded operators and that A α are of the form where B 0 and A α , α ∈ I are compact operators and C α are self adjoint operators for all α ∈ I. Denote by r α the eigenvalue corresponding to the orthogonal family of eigenvectors H α , i.e. C α (H α ) = r α H α , α ∈ I. Using classical results of elliptic PDEs and the Fredholm alternative (see [5]) we prove existence and uniqueness of the solution to (39).  (6), where B 0 : X → X is a compact operator. Assume there exists K > 0 such that: and 2. B is of the form (6), where B β : X → X, β ∈ I \ {0}, are bounded operators and there exists p > 0 such that 3. For every α ∈ I Ker A α + (1 + r α )Id + B 0 = {0}.
By (43) it follows that for each γ ∈ I the homogeneous equation has only trivial solution u γ = 0. Since the operator A γ + (1 + r γ )Id + B 0 is compact, the classical Fredholm alternative implies that for each γ ∈ I there exists a unique u γ that solves (44) and it is of the form u γ = (Id − ((r γ + 1) Id + A γ + B 0 )) −1 f γ + β>0 B β (u γ−β ) , γ ∈ I, so that 1. If A α = 0 for all α ∈ I and B α , α ∈ I are second order strictly elliptic partial differential operators in divergent form with essentially bounded coefficients, then equation (39) reduces to the elliptic equation B♦U = F, which was solved in [18] and [19].
2. Let A α = 0 for all α ∈ I and let B α , α ∈ I, be second order strictly elliptic partial differential operators in divergent form (45). Let C = c P (R), for some c ∈ R, where R is the Ornstein-Uhlenbeck operator, P a polynomial of degree m with real coefficients and P (R) the differential operator P (R) = p m R m + p m−1 R m−1 + ... + p 1 R + p 0 Id. Then, the corresponding eigenvalues are r α = cP (|α|), α ∈ I. Hence, equation (39)