MAXIMAL DISSIPATIVITY OF A CLASS OF ELLIPTIC DEGENERATE OPERATORS IN WEIGHTED L 2 SPACES

We consider a degenerate elliptic Kolmogorov–type operator arising from second order stochastic differential equations in Rn perturbed by noise. We study a realization of such an operator in L2 spaces with respect to an explicit invariant measure, and we prove that it is m-dissipative.

1. Introduction.We are concerned with a Kolmogorov operator in R 2n = R n x × R n y , Kϕ(x, y) = 1 2 ∆ x ϕ(x, y) − M y + x + D y U (y), D x ϕ(x, y) + x, D y ϕ(x, y) , (1) where M is a symmetric positive definite matrix, and U ∈ C 1 (R n , R) is a nonnegative function satisfying suitable assumptions.We stress that U and its derivatives may be unbounded, and even grow exponentially as |y| → +∞.
The operator K arises in the study of the second order stochastic initial value problem in R n , (2) See [6] for a discussion and several developements.Setting Y (t) = X(t), problem (2) is equivalent to the system 752 G. DA PRATO AND A. LUNARDI and K is precisely the Kolmogorov operator in R 2n associated to (3).
Here we study a realization of K in L 2 (R 2n , µ).The main result of this paper is that ) is closable, and its closure is m-dissipative.C 2 b (R 2n ) denotes the space of all bounded and twice continuously differentiable functions from R 2n to R with bounded first and second order derivatives.
Maximal dissipativity of Kolmogorov operators was studied in the last few years by several authors, but most results concern nondegenerate elliptic operators.See for instance the papers [9,10,11,3], the books [5,2] and the references therein.
Let us explain our method.Our assumptions imply that ), and integrating by parts we obtain Consequently it is closable, and we denote by K its closure.To prove that K is m-dissipative, we have to show that for each λ > 0 the range of λI − K is dense in L 2 (R 2n , µ), i.e. we have to solve the resolvent equation for every f in a dense set in L 2 (R 2n , µ).This is not obvious, because K is a degenerate elliptic operator with unbounded coefficients.If the coefficients of K were smooth enough and had bounded derivatives, for smooth f (say ) would be easily obtained by the classical stochastic characteristics method.Therefore, we assume that U has good bounded approximations U α ≤ U , α > 0, and for every α > 0 we solve where Of course, (6) may be rewritten as and our aim is to prove that The simplest way to reach this goal is to assume that |D y U α − D y U | goes to 0 in L 4 (R 2n , µ), and to prove that the derivatives with a constant C independent of α.Here R 2n e −( M y ,y +|x | 2 ) e −2U α (y ) dx dy := c α e −( M y,y +|x| 2 ) e −2Uα(y) .
(10) Note that, even for n = 1, estimate (8) cannot follow from L 2 estimates on the second order x-derivatives of ϕ α through Sobolev embedding, because our measures µ α are of Gaussian type, and no Sobolev embeddings are available.
The core of the paper is in fact the proof of estimate (8) for f ∈ C 3 b (R 2n ), which is dense in L 2 (R 2n , µ).To this aim we need further assumptions on the approximations U α , that turn out to be assumptions on U .Such assumptions are far from being optimal, however they are satisfied if U is any polynomial such that lim |y|→∞ U (y) = +∞, or any positive smooth function such that U and its derivatives up to the third order have polynomial (resp.exponential) growth as |y| → +∞.See next section.
We note that proving m-dissipativity in L 2 (R 2n , µ) instead of in L 1 (R 2n , µ), which is the setting considered in [4], is an important starting point for the study of further properties, such as asymptotic behavior as t → +∞ of the semigroup generated by K.This study will be the object of a future paper.
2. The approximating operators.We denote by C k b (R 2n ) the space of all bounded and k times continuously differentiable functions from R 2n to R with bounded derivatives up to the order k.Moreover we shall use the following notation: for functions ϕ that have the derivatives appearing in the formulas.
Throughout the paper we shall assume that the following conditions are satisfied.
We notice that Hypothesis 2.1 is fulfilled if U is a C 4 nonnegative function having polynomial or exponential growth together with his derivatives up to the order 3, in the sense that there are m 0 , m 1 , c 0 , c 1 > 0, such that for large |y| and for every i, j, k = 1, . . ., n we have We fix here α > 0 and consider the approximating equation (6), where λ > 0 and Proof.To solve equation ( 6) we shall use the classical stochastic characteristics method.It is based on the solution of the following system of stochastic differential where x, y ∈ R n and W (t) is a standard Brownian motion in R n .By Hypothesis 2.1-(ii), D y U α is Lipschitz continuous, so that problem (13) has a unique global solution (X α (•, x, y), Y α (•, x, y)).Moreover, since D y U α has bounded derivatives up to the order 3, in view of a classical result on the dependence of the solution of (13) upon initial data (see e.g.[7, Theorem 1, page 61], [1, Proposition 1.3.3]), it follows that (X α (t, x, y), Y α (t, x, y)) is thrice continuously differentiable with respect to (x, y), with bounded derivatives up to the third order.
By the Itô formula it follows that the parabolic problem x, y))] as a solution, and consequently equation ( 6) has a solution ϕ α ∈ C 3 b (R n ), given by The classical maximum principle may be easily adapted to elliptic operators with Lipschitz continuous coefficients; for a detailed proof see e.g.[4].It implies estimate (12) as well as uniqueness of the solution.
3. Integral estimates for the solutions of the approximating problems.
Here we fix α > 0 and we derive several estimates on the solutions of ( 6), which will be used in the next section to prove that The starting point is that Hypothesis 2.

Proposition 3.2. There exists a constant
Proof.Denote by m ij = M e j , e i = M e i , e j the entries of the matrix M , so that M y, D x ϕ α = n i,j=1 y i m ij D xj ϕ α .Differentiating (6) with respect to y i yields Multiplying both sides by D yi ϕ α , integrating with respect to µ α over R 2n , taking into account (15) and summing up yields

n, let us estimate the integral
Integrating by parts with respect to y i , we obtain m ij y j ρ α dx dy.
Taking into account (12) and using the Hölder inequality, we find . By Hypothesis 2.1 and Proposition 3.1 we get for any ε > 0. Coming back to (20) we get |I ik |, and the statement follows using estimate (21 and the statement is obvious.

Corollary 3.3. There exists a constant
Proof.Differentiating (6) with respect to x i yields Multiplying both sides by D y i ϕ α , integrating with respect to µ α over R 2n , taking into account (15) and summing up yields The conclusion follows using (19).
Proposition 3.4.There exists a constant Proof.For i = 1, . . ., n let us estimate Integrating by parts with respect to Now, using (12) and the Hölder inequality, we obtain and . Consequently, setting and using (22) we obtain where Now using the Hölder inequality we obtain, for all i = j, i, j = 1, ..., n, and the statement follows.
4. m-dissipativity.We prove here the main result of the paper.
Theorem 4.1.Assume that Hypothesis (2.1) is fulfilled.Then the closure K of the operator K : ) and let ϕ α be the solution of (6).Then In fact, taking into account Proposition 3.4 and using the Hölder inequality, we get Now the claim follows from Hypothesis 2.1-(iii), recalling that 1/c α is bounded by a constant independent of α.Since C 3 b (R 2n ) is dense in L 2 (R 2n , µ), (27) implies that the range of λI − K is dense in L 2 (R 2n , µ) and the statement of the theorem follows from the Lumer-Phillips Theorem.
In its turn, this implies that, denoting by T (t) the semigroup generated by K, i.e. µ is an invariant measure for T (t).
Open problems.Several natural questions about K, T (t) and µ arise now.A first one is about the regularity properties of the functions in D(K).Do their second order derivatives D xixj ϕ exist and belong to L 2 (R 2n , µ)?A related question is about the smoothing properties of T (t).In the case U ≡ 0 we have the nice representation formula (T (t)ϕ)(z) = 1 (2π) n/2 (det Q t ) 1/2 R 2n e − Q −1 t ξ,ξ /2 ϕ(e tB z − ξ)dξ, t > 0, z ∈ R 2n , (29) where Q t is the matrix Using (29) it is not hard to see that T (t) maps L 2 (R 2n , µ) into C ∞ (R 2n ) for t > 0. But a similar formula is not available for general U .A third interesting question is whether the kernel of K consist of constant functions.As well known, this is equivalent to ergodicity of µ with respect to T (t).Formula (28) implies immediately that every ϕ in the kernel of K depends only on the variables y.If ϕ were weakly differentiable, we would obtain that x, D y ϕ(y) = 0 for almost all x, y ∈ R n , so that ϕ ≡ constant.But regularity of ϕ is not obvious, and we cannot conclude.
Last, but not least: under which conditions the domain of K is compactly embedded in L 2 (R 2n , µ)?For nondegenerate Kolmogorov-type elliptic operators, under reasonable assumptions the domain is continuously embedded in H 1 (R 2n , µ), the space of the weakly differentiable functions with derivatives in L 2 (R 2n , µ), which is compactly embedded in L 2 (R 2n , µ) because logarithmic Sobolev inequalities hold.See e.g.[3].But in our case the embedding D(K) ⊂ H 1 (R 2n , µ) is out of reach.
We hope to be able to answer (a part of) these questions in a future paper.