Stochastic Invariance and Degenerate Elliptic Operators. P. Cannarsa G. Da Prato H. Frankowska

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1 Stochastic Invariance and Degenerate Elliptic Operators P. Cannarsa G. Da Prato H. Frankowska Preprint di Matematica N. 1 Maggio 2008

2 Stochastic Invariance and Degenerate Elliptic Operators P. Cannarsa, G. Da Prato, H. Frankowska May 15, 2008 Abstract This paper is devoted to the invariance of a bounded open domain in R n under a diffusion process with Lipschitz continuous data. We show that such a property holds under the same conditions that insure the invariance of a closed domain. This result is applied to get the wellposedness of a degenerate elliptic equation without imposing explicit boundary conditions, and to study the existence and uniqueness of the invariant measure for the associated transition semigroup. Key words: stochastic differential equation, invariance, degenerate elliptic operator, invariant measure. MSC Subject classifications: 60H10, 47D07, 35K65, 37L40. 1 Introduction The invariance of a closed domain K R n for the stochastic diffusion associated with dx(t) = b(x(t))dt + σ(x(t)) dw (t) t 0 X(0) = x K Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, Roma (Italy), cannarsa@mat.uniroma2.it Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, I Pisa (Italy), daprato@sns.it CNRS, CREA, Ecole Polytechnique, 1 rue Descartes, Paris (France), helene.frankowska@polytechnique.edu 1

3 has been investigated in several papers, in recent years. For instance, in [1], Aubin and the second author introduced the notion of stochastic contingent cone to provide necessary and sufficient conditions for the viability of K as well as its invariance. Another approach to the problem was proposed in [6], using viscosity solutions of a second order Hamilton-Jacobi equation. Later on, Bardi and Goatin [5] used second order jets to study invariance, while the second and third author used the Stratonovich drift to give first order necessary and sufficient conditions for the invariance of a given set for a stochastic control system. Following the above works, in [9, 10, 13] invariance properties were derived using the distance function, the oriented distance function, and the elliptic operator L 0 ϕ(x) = 1 2 Tr [a(x) 2 ϕ(x)] + b(x), ϕ(x), where a(x) = σ(x)σ (x). In particular, from the results of the above papers it can be deduced that, when K is regular, the necessary and sufficient conditions for the invariance of K reduce to (i) L 0 δ K (x) 0 x K, (1.1) (ii) a(x) δ K (x), δ K (x) = 0 where δ K stands for the oriented distance from K. Notice that condition (ii) implies that a(x) is a singular matrix for all x K. In this paper, we will further investigate the above problem showing that conditions (1.1) are necessary and sufficient for the invariance of the interior of K. Moreover, we extend the analysis to compact domains K that can be obtained as finite intersections of smooth domains. It is noteworthy that the above characterization of the invariance of the interior of K strongly depends on the smoothness of data. Even in the purely deterministic case (i.e. σ 0) the result would be false, in general, should the dynamics b be merely continuous, or K be nonsmooth. Using the invariance of the interior of K, we study the transition semigroup P t ϕ(x). = E[ϕ(X(t, x))] showing, first, that its infinitesimal generator on C(K) is given by operator L 0 above. Consequently, for every λ > 0 and every continuous function f K R, we obtain an existence and uniqueness result for the elliptic equation λϕ L 0 ϕ = f in K 2

4 without imposing boundary conditions. Finally, we apply our results to study the existence and uniqueness of invariant measures for P t. Observe that, since K is bounded, P t always admits at least one invariant measure. On the other hand, unlike the semigroups that are associated with operators defined in the whole space R n (see e.g. [18] and the references therein), P t can have several invariant measures, which, moreover, need not be absolutely continuous with respect to Lebesgue s measure. In this paper, taking advantage of the interior invariance result described above, we prove that P t has at most one invariant measure on C(K), in the class of all absolutely continuous measures with respect to Lebesgue s measure. Moreover, strengthening condition (1.1), we are able to prove the existence of such a measure. This paper is organized as follows: section 2 contains notations and all preliminary results; section 3 develops our interior invariance result for smooth domains, and section 4 for piecewise smooth domains. Section 5 provides the characterization of the infinitesimal generator of P t. Finally, section 6 is devoted to the anaysis of the invariant measure for P t, absolutely continuous with respect to Lebesgue s measure: we first prove uniqueness, then existence. We conclude with a few examples. 2 Notation Given a metric space (E, d), B(E ) stands for the Borel σ-algebra in E, and B b (E ) for the space of all bounded Borel functions ϕ : E R. Let n be a positive integer. We denote by:, the Euclidean scalar product in R n ; the Euclidean norm in R n ; j 1 }} e j = ( 0,..., 0, 1, 0,..., 0), where j = 1,..., n, the elements of the canonical base of R n ; x y the tensor product of x, y R n, i.e., (x y)z = y, z x for all z R n ; B(x 0, r) the open ball of radius r > 0, centered at x 0 R n, and we set B r = B(0, r); L (R n, R m ) the space of all linear maps Λ : R n R m, where m is a positive integer (any element of L (R n, R m ) will be identified with the 3

5 unique n m matrix that represents σ with respect to the canonical bases of R n and R m ); Λ the operator norm of Λ L (R n, R m ), i.e., Λ = max x =1 Λx ; Tr [Λ] the trace of Λ L (R n, R n ), i.e., Tr [Λ] = j Λe j, e j ; µ n the Lebesgue measure on B(R n ); 1l S the characteristic function of a set S; ϕ, 2 ϕ and ϕ the gradient vector, the Hessian matrix, and the Laplacian of the function ϕ, respectively. Given a positive integer m and Lipschitz continuous maps b : R n R n and σ : R n L (R n ; R m ), consider the stochastic differential equation dx(t) = b(x(t))dt + σ(x(t)) dw (t) t 0 (2.1) X(0) = x where W (t) is a standard m-dimensional Brownian motion on a complete filtered probability space (Ω, F, F t } t 0, P). It is well-known that, for any x R n, problem (2.1) has a unique solution that we shall denote by X(, x). Moreover, X(, x) is P a.s. continuous. Let S R n be a nonempty set. We denote by d S the Euclidean distance function from S, that is, d S (x) = inf y S x y x Rn. It is well-known that d S is a Lipschitz function of constant 1. If S is closed, then the above infimum is a minimum, which is attained on a set that will be called the projection of x R n onto S, labeled proj S (x), that is, proj S (x) = y S x y = d S (x)} x R n. We say that S is invariant for X(, ) iff x S = X(t, x) S P a.s. t 0. (2.2) For every x S, the hitting time of S is the random variable defined by τ S (x) = inf t 0 : X(t, x) S }. 4

6 Let K be a closed subset of R n with nonempty interior K and boundary K. A well-known function in metric analysis is the so-called oriented distance from K, that is, the function d K (x) if x K δ K (x) = d K (x) if x K c. In what follows we will use the following sets, defined for any ε > 0: N ɛ = x R n : δ K (x) < ɛ} K ε = K N ε K ε = K N ε In this paper, we will use the following function spaces: C b (A): all bounded continuous functions on the open set A; C 2,1 (A): all twice differentiable functions on A, with bounded Lipschitz second order derivatives; C 2,1 loc (A): all twice differentiable functions on A, with locally Lipschitz second order derivatives; C(K): all continuous functions ϕ : K R; C 1 (K): all continuously differentiable functions in a neighborhood of K; H 2 (A): the Sobolev space of all Borel functions ϕ : A R that are square integrable on A, together with their second order derivatives in the sense of distributions. H 2 loc (A): all Borel functions ϕ : A R that belong to H2 (A ) for every open set A such that A A. We say that K is a closed domain of class C 2,1 if it is a closed connected subset of R n such that, for some r > 0 and all x K, there is a function φ x : B(x, r) R of class C 2,1 (B(x, r)) satisfying the following: (i) K B(x, r) = y B(x, r) φ x (y) = 0} (ii) the Lipschitz norm of 2 φ x is bounded, uniformly in x K. 5

7 It is well-known that K closed domain of class C 2,1 ε 0 > 0 : δ K C 2,1 (N ε0 ) (2.3) see, e.g., [15]. A useful consequence of the above property is that (a)! x K : δ K (x) = x x x K ε0 (b) δ K (x) = δ K ( x) = ν K ( x) (2.4) where ν K ( x) stands for the outward unit normal to K at x. Any bounded closed domain K of class C 2,1 will be called a compact domain of class C 2,1. It is easy to see that, if K is such a domain, then there is a sequence Q i } of compact domains of class C 2,1 such that Q i Q i+1 and Q i = K. (2.5) Indeed, owing to (2.3), it suffices to take, for all i large enough, Q i = x R n δ K (x) 1 }. i More generally, the existence of a sequence Q i } of compact domains of class C 2,1 satisfying (2.5) is also guaranteed when K is a compact set such that m K = K j, j=1 where K j are closed domains of class C 2,1. Finally, we observe that, if K is a compact set and Q i } is an increasing sequence of compact domains of class C 2,1 satisfying (2.5), then, for all x K, τ Qi (x) τ K (x) P a.s. as i. (2.6) Indeed, since X(, x) is continuous, τ Qi (x) is an increasing sequence of random variables bounded above by τ K (x). So, τ Qi (x) τ(x) for some random variable τ(x) τ K (x). Moreover, if P(τ(x) < τ K (x)) > 0, then for some t 0 > 0. Consequently, So, i=1 P(τ(x) < t 0 < τ K (x)) > 0 X(t 0, x) / i Q i P a.s. on τ(x) < t 0 < τ K (x)}. X(t 0, x) K P a.s. on τ(x) < t 0 < τ K (x)}, in contrast with the definition of τ K (x). 6

8 3 Invariance of the interior of K In this section, we will study the invariance properties of a compact domain of class C 2,1, K, with respect to the flow X(, ) associated with the stochastic differential equation (2.1) with Lipschitz continuous coefficients b and σ. Necessary and sufficient conditions for the invariance of the compact set K were established in [9] in terms of the differential operator D(L 0 ) = ϕ C(K) ϕ H 2 loc ( K), L 0 ϕ C(K) } L 0 ϕ(x) := 1 2 Tr [a(x) 2 ϕ(x)] + b(x), ϕ(x) x K, (3.1) where a(x) is the symmetric matrix associated with the diffusion coefficient σ, i.e., a(x) = σ(x)σ (x) 0 x K. From [13] it follows that, when K is convex, K is invariant with respect to the X(, ) if and only if the following conditions are satisfied: (i) L 0 δ K (x) 0 x K (3.2) (ii) a(x) δ K (x), δ K (x) = 0 Notice that, on account of (2.4), the above conditions imply that the elliptic operator L 0 is necessarily degenerate on K, in the normal direction to K. Our next result implies that, under condition (3.2), the open set K is invariant too. Such a property heavily relies on the Lipschitz continuity of b and σ, as well as on the smoothness of K: the conclusion of Theorem 3.1 is not true, in general, if b and σ are just continuous, see [16]. Theorem 3.1 Assume (3.2). Then P(τ K (x) < ) = 0 x K. (3.3) Proof: Having fixed ε 0 as in (2.3), consider the function V (x) := log δ K (x) x K ε0. By a straightforward computation we have that L 0 V (x) = 1 δ K (x) L 0δ K (x) + 1 δ 2 K (x) σ (x) δ K (x) 2 x K ε0. (3.4) 7

9 For every x K ε0 let x denote the point of K satisfying (2.4). Then, owing to (3.2) (ii), σ (x) δ K (x) = (σ (x) σ (x)) δ K (x) + σ (x) δ K (x) = (σ (x) σ (x)) δ K (x) σ (x) σ (x) c 1 x x = c 1 δ K (x) where c 1 is a Lipschitz constant for σ. Consequently, 1 δ 2 K (x) σ (x) δ K (x) 2 c 2 1 x K ε0. (3.5) Also, observe that, since δ K C 2,1 (N ε0 ), the function L 0 δ K (x) = 1 2 Tr [σ(x)σ (x) 2 δ K (x)] + b(x), δ K (x) is Lipschitz continuous in K ε0. Therefore, (3.2) (i) yields, for all x K ε0, 1 δ K (x) L 0δ K (x) = 1 ( L0 δ K (x) L 0 δ K (x) ) 1 δ K (x) L 0δ K (x) δ K (x) 1 L0 δ K (x) L 0 δ K (x) δ K (x) c 2 δ K (x) x x = c 2 (3.6) where c 2 is a Lipschitz constant for L 0 δ K. So, combining (3.4), (3.5) and (3.6), we conclude that, for some positive constant c 0, L 0 V (x) c 0 x K ε0. (3.7) Next, by a standard localization argument, it is easy to construct a nonnegative function U C 2,1 loc ( K) such that U(x) = V (x) x K ε0 /2 L 0 U(x) M x K (3.8) for some positive constant M. Let us set U(x) = lim U(y) = x K. K y x 8

10 Now, let Q i } be a sequence of compact domains of class C 2,1 satisfying (2.5) and consider their stopping times τ Qi. By Itô s formula we have, for all x Q i and t 0, U(X(t τ Qi (x), x)) = U(x) + + t τqi (x) 0 t τqi (x) 0 (L 0 U)(X(s, x))ds Hence, taking expectation and recalling (3.8), we have E [ U(X(t τ Qi (x), x)) ] = U(x) + E U(X(s, x)), σ(x(s, x))dw (s). t τqi (x) 0 L 0 U(X(s, x))ds U(x) + Mt. Thus, owing to (2.6) and Fatou s lemma, the above inequality yields E [ U(X(t τ K (x), x)) ] U(x) + Mt t 0, x K. (3.9) Since the function in the right-hand above is finite on K, we deduce that P(τ K (x) t) = P ( U(X(t τ K (x), x)) = ) = 0 t 0, x K. To complete the proof, it suffices to take a sequence t k and observe that 0 = P(τ K (x) t k ) P(τ K (x) < ) x K. Under a slightly stronger assumption, one can improve the estimates of the above proof, to obtain the following result that will be applied in section 6.2. Proposition 3.2 Assume (i) lim L 0 δ K (x) supδk (x) 0 δ K (x) log δ K (x) < 0 (ii) a(x) δ K (x), δ K (x) = 0 x K Then, there is a nonnegative function U C 2,1 loc ( K) and such that U(x) = log δ K (x) x K ε0 /2 L 0 U(x) M αu(x) x K (3.10) (3.11) for some constants α > 0 and M 0. 9

11 Remark 3.3 In particular, assumption (3.10) (i) holds under the following stronger version of assumption (3.2) (i): L 0 δ K (x) > 0 x K. Observe that the above condition was used in [10] to prove the invariance of K for a diffusion process with a continuous drift. In the proof of Proposition 3.2 given below, we only underline the main differences with the proof of Theorem 3.1. Proof: the reasoning goes in the same way as above up to (3.5). Then, in view of (3.10) (i), there exist positive numbers α and ρ such that 1 δ K (x) L 0δ K (x) α log δ K (x) = αv (x) x K ε0. (3.12) So, combining (3.4), (3.5) and (3.12), we conclude that L 0 V (x) c 2 1 αv (x) x K ε0. Finally, by the usual localization argument, one can easily construct a nonnegative function U C 2,1 loc ( K) satisfying (3.11). 4 Piecewise smooth domains In this section we will generalize and improve the above results assuming that K is a compact set such that K = m K j (4.1) j=1 where K j are closed domains of class C 2,1 with the following property: for some ε 1 > 0 and all j 1,..., m} proj K j(x) K x K K j ε 1, (4.2) where we recall that K j ε 1 = x K j : δ K j(x) < ε 1 }. For every x K we denote by J(x) the set of all active indeces at x, that is, j J(x) x K j. 10

12 K 2 K proj x K 2 (x) K 1 proj K 1 (x) Figure 1: Assumption (4.2). Using, for simplicity, the abbreviated notation δ j for the oriented distance δ K j, let us also assume that 0 / co δ j (x) j J(x) } x K. (4.3) Then, Clarke s tangent cone to K at every x K has nonempty interior. For this reason, K coincides with the closure of K. Moreover, according to [4, chapter 4], Clarke s normal cone to K at any point x K is given by N K (x) = j J(x) R + δ j (x). (4.4) Example 4.1 A typical example of a piecewise smooth domain satisfying conditions (4.1), (4.2), and (4.3) is the cube Q 1 = x R n } max x j 1. 1 j n Notice that Q 1 = x R n max j x j = 1}, and J(x) = j 1,..., n} x j = 1 } x Q 1 δ j (x) = x j x j e j x Q 1, j J(x). (4.5) We now give an extension of Theorem 3.1 to piecewise smooth domains. Theorem 4.2 Assume (4.1), (4.2), and (4.3). statements are equivalent: Then the following three (a) K is invariant; (b) for all x K and j J(x) (i) L 0 δ j (x) 0 (ii) a(x) δ j (x), δ j (x) = 0 ; 11

13 (c) K is invariant. Proof: Let ε 1 > 0 be such that, for every j 1,..., m}, there exist functions g j C 2,1 (R n ) satisfying 0 g j 1 on K j 0 < g j on K j \ Kε j 1 (4.6) g j δ j on Kε j 1 1. Assume (a). Then, according to [11], for every x K and all p N K (x), we have σ (x)p = 0. Consequently, owing to (4.4), property (b)(ii) holds true. To obtain (i), fix x K and let j J(x). Then, g j (X(t, x)) 0 P a.s. for all t 0, and g j (x) = 0. Therefore, d dt E[ g j (X(, x)) ] t=0 0. Consequently, applying Itô s formula (see, e.g., [8, p. 61]), d dt E[ g j (X(, x)) ] t=0 = E[ L 0 g j (x) ] 0 Since g j δ j on a neighborhood of K j, we deduce (i). 2. The proof of the fact that (b) (c), is quite similar to the proof of Theorem 3.1. Let us consider the function Then, L 0 V (x) = V (x). = m j=1 m log g j (x) x K. j=1 1 g j (x) L 0g j (x) + We claim that, for all j = 1,..., m, m j=1 1 g 2 j (x) σ (x) g j (x) 2. (4.7) 1 gj 2 (x) g j (x) 2 1 (x) σ g j (x) L 0g j (x) c x K for some constant c 0. Indeed, the above estimate holds true when δ j (x) ε 1 since g j is strictly positive on x K δ j (x) ε 1 }. Moreover, by assumption (4.2) and the same argument as in the proof 12

14 of Theorem 3.1, one can show that it also holds true on K K j ε 1, because g j = δ j on such a set. Therefore, L 0 V (x) M x K for some constant M 0. Observing that lim V (y) = x K, K y x we obtain (c) exactly as in the proof of Theorem Finally, suppose K is invariant and fix x K. Recalling that K coincides with the closure of K, let x k } be a sequence in K such that x k x. Then, by our invariance assumption, X(t, x k ) K, P a.s. for all t 0. Since X(t, x k ) X(t, x), P a.s. for all t 0, we conclude that X(t, x) K, P a.s., for all t 0. Since x is an arbitrary point in K, K is invariant. Proposition 4.3 Assume (4.1), (4.2), and (4.3), and suppose x K, j J( x) (i) lim sup K x x L 0 δ j (x) δ j (x) log δ j (x) < 0 (ii) a( x) δ j ( x), δ j ( x) = 0 Then, there is a nonnegative function U C 2,1 loc ( K) such that U(x) = m j=1 log δ j(x) x K ε1 /2 L 0 U(x) M αu(x) x K (4.8) for some constants α > 0 and M 0. We skip the proof which goes as for Proposition Transition semigroup In this section we will assume the following without further notice: K is a compact set satisfying (4.1), (4.2) and (4.3); Q i } is a sequence of compact domains of class C 2,1 satisfying (2.5); condition (b) of Theorem 4.2 holds true. 13

15 Then, we know that K and K are invariant for the stochastic flow X. So, as recalled above, the elliptic operator L 0 defined in (3.1) is degenerate on K in the sense specified by condition (b). Later on, we will further assume that L 0 is uniformly elliptic on all compact subsets of K, that is, det a(x) > 0 x K. (5.1) The main objective of our analysis is the study of the transition semigroup P t associated with the stochastic flow X(, ), that is, the semigroup on B b (K) defined by P t ϕ(x) := E[ϕ(X(t, x))] ϕ B b (K), x K, t 0. (5.2) As is easily seen, P t is a Markov semigroup, that is, (i) ϕ ψ = P t ϕ P t ψ (ii) P t 1l K = 1l K We begin with some preliminary properties of P t. Proposition 5.1 P t is a Feller semigroup on B b (K), and its restriction to C(K) is strongly continuous. Proof: The Feller property of P t is easy to check. Indeed, ϕ C(K) = P t ϕ C(K) t 0 owing to the continuity of the map x X(t, x). Notice that we will use the same symbol P t to denote the restriction of P t to C(K). In order to prove that P t is a strongly continuous semigroup on C(K), observe that, since C 1 (K) is dense in C(K) and P t 1 (1), it is enough to show that lim P t ϕ = ϕ uniformly in K (5.3) t 0 for every ϕ C 1 (K). Now, for any such function ϕ we have that P t ϕ(x) ϕ(x) ϕ C 1 (K) E[ X(t, x) x 2 ] } 1/2 x K, t 0. (5.4) Moreover, by Hölder s inequality, X(t, x) x 2 t 2 b(x(s, x))ds 2t 0 t b(x(s, x)) 2 ds + 2 t 0 t 0 σ(x(s, x))dw (s) σ(x(s, x))dw (s) (1) Here, P t denotes the norm P t regarded as a bounded linear operator on C(K)

16 So, taking expectation yields E( X(t, x) x 2 ) 2t 2 b 2 + 2t σ 2, where we have set b = max x K b(x) and σ = max x K σ(x). Thus, (5.3) follows recalling (5.4). Remark 5.2 As a corollary of Theorem 4.2, we have that the transition semigroup P t defined in (5.2) satisfies P t ϕ(x) = E[ϕ(X(t, x))1l t τk (x)] t 0, x K (5.5) for every ϕ B b (K). semigroups Now, for all i N consider the so-called stopped P i t ϕ(x) = E[ϕ(X(t, x))1l t τqi (x)] (t 0, x Q i ) (5.6) associated with stopping times τ Qi (x). Then, by (5.5) and (2.6), we conclude that P i t approximate P t on K in the sense that, for every ϕ B b (K), lim P t i ϕ(x) = P t ϕ(x) t 0, x K. (5.7) i Remark 5.3 Under hypothesis (5.1) we have that L 0 is uniformly elliptic on Q i for all i N. So, by classical results (see, e.g., [17]), for any ϕ C(K) the Dirichlet problem t u(t, x) = L 0 u(t, x) t 0, x Q i u(t, x) = 0 t > 0, x Q i (5.8) u(0, x) = ϕ(x) x Q i. has a unique solution u i C([0, ); L p (Q i )) for every p 1, which satisfies t u i (t, ), h k u i (t, ) L p (Q i ) t > 0, h, k = 1,..., n (5.9) Moreover, u i is given by the formula u i (t, x) = P i t ϕ(x) (t 0, x Q i ), (5.10) where Pt i is the semigroup defined in (5.6), see, e.g., [7, section 6.2.2]. Observe, however, that u i can be also represented by the formula u i (t, x) = G i (t, x, y)ϕ(y)dy (t 0, x Q i ), (5.11) Q i 15

17 where G i (t, x, y) is the Green function of the parabolic operator in (5.8). It is well-known that G i (t, x, y) is strictly positive for all t 0 and x, y Q i (see, e.g., [17]). By the maximum principle we conclude that G i (t, x, y) G(t, x, y) t 0, x, y K, x y Therefore, G(t, x, y) > 0 (t 0, x, y K, x y). (5.12) Also, on account of (5.10), (5.11) and (5.7), for all ϕ C(K) we have P t ϕ(x) = G(t, x, y)ϕ(y)dy t 0, x K. (5.13) K Let L be the infinitesimal generator of the strongly continuous semigroup P t on C(K). We will show next that L coincides with operator L 0. Theorem 5.4 Assume (5.1) and let λ > 0. Then, for every f C(K) there exists a unique solution ϕ f D(L 0 ) of the equation Moreover, ϕ f D(L) and Lϕ f = L 0 ϕ f. Proof. We will split the proof into three steps. λϕ L 0 ϕ = f in K. (5.14) 1. Existence and regularity. Observe that, since one can argue with the positive and negative part of f separately, it suffices to prove the existence of a solution to (5.14) for f 0. Having fixed f, define ϕ f (x) = Then, as is well-known, ϕ f D(L) and Now, for i N large enough, let 0 e λt P t f(x)dt x K. (5.15) Lϕ f = λϕ f f in K. (5.16) ϕ f i (x) := e λt Pt i f(x)dt (x Q i ), 0 where P i t are the stopped semigroups defined in (5.6). Owing to (5.7), ϕ f i (x) ϕf (x) (i ) x K. (5.17) 16

18 Moreover, on account of (5.6), ϕ f i (x) = e λt E[f(X(t, x)1l t τqi (x)]dt x Q i. 0 Since our diffusion process is nondegenerate in Q i and Q i is a compact domain of class C 2,1, it is well-known that ϕ f i satisfies λϕ f i L 0ϕ f i = f in Q i ϕ f i = 0 on Q (5.18) i. Thus, by classical elliptic theory we conclude that, for all i N, ϕ f i belongs to H 2 (Q i ). Also, for any open subset A of K such that Ā K, ϕ f i H 2 (A) C A (5.19) for a suitable constant C A, independent of i (see, e.g., [18, Appendix A]). So, from (5.19) and (5.17) we deduce that ϕ f H 2 loc ( K), and (5.18) yields L 0 ϕ f = λϕ f f in K. (5.20) Since the right-hand side above is continuous in K, (5.20) holds on the closed domain K and L 0 ϕ f C(K). Therefore, ϕ f D(L 0 ) and, in view of (5.16), L 0 ϕ f = Lϕ f. 2. An auxilary problem. Let ϕ 1 D(L 0 ) be the solution of (5.14) for f 1, that we constructed in the previous step. Since P t 1 = 1, by (5.15) we conclude that ϕ 1 (x) = 1 λ x K. Moreover, owing to (5.17), ϕ 1 i (x) 1 λ (i ) x K, (5.21) where ϕ 1 i is the solution of (5.18) for f Uniqueness. We will show that, if u D(L 0 ) λu L 0 u = 0 in K, 17

19 then u 0. Let v(x) := 1 λ u(x) λ(1 + u C(K) ) x K. Then v D(L 0 ) v(x) > 0 x K λv L 0 v = 1 in K. Therefore, comparing v and the solution ϕ 1 i of (5.18) for f 1 on Q i, we obtain v(x) ϕ 1 i (x) x Q i for all i N large enough. Hence, in view of (5.21), v(x) 1 λ x K, which in turn implies that u(x) 0 for all x K. argument applied to u we conclude that u 0. By the same 6 Invariant measure for P t In this section, we will study the existence and uniqueness of the invariant measure µ for the transition semigroup P t defined by (5.2), in the class of all absolutely continuous measures with respect to Lebesgue s measure µ n. We will make, without further notice, the following assumptions: K is a compact set satisfying (4.1), (4.2) and (4.3); Q i } is a sequence of compact domains of class C 2,1 satisfying (2.5); condition (b) of Theorem 4.2 holds true; the interior ellipticity condition (5.1) is satisfied. Let us recall that a probability measure µ on (K, B(K)) is said to be invariant for P t if, for any t 0, P t ϕ(x)µ(dx) = ϕ(x)µ(dx) ϕ C(K). (6.1) K K 18

20 6.1 Uniqueness We will show the following uniqueness result. Theorem 6.1 Semigroup P t possesses at most one invariant measure in the class of all measures that are absolutely continuous with respect to µ n. For the proof of the above theorem we will need several intermediate steps. To begin, let us introduce the following metric ρ K in K : ρ K (x, y) = 1 δ K (x) 1 δ K (y) + x y, x, y K. (6.2) It is easy to see that ( K, ρ K ) is a complete metric space. Remark 6.2 It is worth noting that a set Q K is compact in ( K, ρ K ) if and only Q is compact in R n with the Euclidean metric. Indeed, suppose that Q is compact in R n. Then, Q Q i for some positive integer i. Thus, for all x Q, δ K (x) > 0. Consider any sequence x k Q and x Q such that x k x 0. Then, ρ K (x, x k ) 0. Consequently, Q is also compact in ( K, ρ K ). Conversely, assume that Q is compact in ( K, ρ K ) and let x, x k Q be such that ρ K (x, x k ) 0. Then, x k x 0. So, Q is compact in the Euclidean metric. Taken ( K, ρ K ), consider the semigroup P t ϕ(x) := E[ϕ(X(t, x))] ϕ B b ( K), x K, t 0. (6.3) Recall that a probability measure µ on ( K, B( K)) is invariant for P t iff P t ϕ(x)µ(dx) = ϕ(x)µ(dx) ϕ C b ( K), t 0. (6.4) K K Our next result is intended to compare the notion of invariant measure for P t with the one for P t. Lemma 6.3 (a) If µ << µ n is an invariant measure for P t, then its restriction to ( K, B( K)) is an invariant measure for P t. (b) If µ << µ n is an invariant measure for P t, then it can be uniquely extended to an invariant measure for P t. Proof. First of all, we observe that, in view of definitions (5.2) and (6.3), P t ϕ(x) = P t ϕ(x) t 0, x K, (6.5) where ϕ denotes any function in C(K) as well as its restriction to K. 19

21 (a) Let µ << µ n be an invariant measure for P t, and let ϕ C b ( K). We shall localize φ in a neighborhood of each domain Q i : take the positive sequence ε i = min x Q i δ K (x) i 1 and define ϕ i (x). = ϕ(x) [ 1 1 ε i δ Qi (x) ] if x K + 0 if x K where [s] + = maxs, 0}. Then, ϕ i C(K). Moreover, Therefore, lim i lim i ϕ i (x) = ϕ(x) and ϕ i (x) ϕ(x) x K. (6.6) K ϕ i (x)µ(dx) = K ϕ(x)µ(dx) = K ϕ(x)µ(dx), (6.7) where the last equality follows since µ n ( K) = 0 and µ << µ n. Also, owing to (6.6) and definition (6.3), we obtain P t ϕ i (x) sup ϕ and lim Pt ϕ i (x) = P t ϕ(x) x K. i K So, recalling (6.5), by the Dominated Convergence theorem we obtain P t ϕ i (x)µ(dx) = P t ϕ i (x)µ(dx) i P t ϕ(x)µ(dx). (6.8) K K On account of (6.7) and (6.8) µ is invariant for P t. (b) Let µ be an invariant measure for P t, absolutely continuous with respect to Lebesgue s measure, and let ϕ C(K). Then ϕ restricted to K is a bounded continuous function. Thus, again by (6.5), K ϕ(x)µ(dx) = K So, µ is invariant for P t. ϕ(x)µ(dx) = P t ϕ(x)µ(dx) K = P t ϕ(x)µ(dx) = Our next result establishes important properties of P t. K K K P t ϕ(x)µ(dx). 20

22 Lemma 6.4 The transition semigroup P t is irreducible and strongly Feller. Proof: Let us first prove that P t is irreducible, that is, for every open subset A of ( K, ρ K ), P t 1l A (x) > 0 t > 0, x K. Let x 0 A and let B(x 0, r) be contained in A together with its closure. Then, B(x 0, r) Q i for some integer i. So, recalling (5.11), by the maximum principle we obtain P t 1l B(x0,r)(x) Pt i 1l B(x0,r)(x) = G i (t, x, y)dy > 0. Q i So, P t is irreducible. Let us now show that P t is strongly Feller, that is, P t ϕ C b ( K) t > 0, ϕ B b ( K). For any ϕ B b ( K), all t > 0 and all positive integers i, we know that Pt i ϕ Qi C(Q i ) since the stopped semigroup Pt i is strongly Feller by wellknown regularity properties of solutions to parabolic equations. On the other hand, for any compact set Q in ( K, ρ K ) or, equivalently (according to Remark 6.2), in K, we have that [ P t ϕ(x) Pt i ϕ(x) E ϕ(x(t, x)) ( ) ] 1 1l t τqi (x) sup ϕ P(τ Qi (x) < t) x Q. K Now, in view of Theorem 4.2, property (2.6) ensures that, for any t (0, ), P(τ Qi (x) < t) 0 (i ) x K. So, Dini s Theorem implies that the above convergence is uniform on Q, which yields, in turn, the continuity of P t ϕ on K. Proof of Theorem 6.1: Let µ and µ be two invariant measures for P t, both absolutely continuous with respect to Lebesgue s measure. Then, in view of Lemma 6.3 (a), their restrictions to ( K, B( K)) still labeled µ and µ are invariant for P t. Therefore, Khas minskii s regularity result (see, e.g., [14, Proposition 4.1.1]) and Doob s uniqueness theorem (see, e.g., [14, Theorem 4.2.1]) ensure that µ and µ coincide on ( K, B( K)). So, they coincide on K as well, since µ, µ << µ n. We conclude this section with two useful properties of P t. 21

23 Proposition 6.5 Let µ be an invariant measure for P t. Then (a) µ << µ n ; (b) for any ϕ C b ( K) lim P t ϕ(x) = t K ϕ(y)µ(dy) x K. Proof: Let µ be an invariant measure for P t, and let B B( K) be such that µ n (B) = 0. Since µ is a regular measure, from (6.4) we deduce, by a standard approximation argument, that P t 1l B (x)µ(dx) = 1l B (x)µ(dx) = µ(b). Then, owing to (5.13), µ(b) = K K ( B K ) G(t, x, y)dx µ(dx), where G is Green s function. Since B G(t, x, y)dx = 0, (a) follows. Finally, property (b) is an immediate consequence of Doob s theorem (see, e.g., [14, Theorem 4.2.1]). 6.2 A sufficient condition for existence In this section we will give sufficient conditions for the existence of an invariant measure µ for P t, absolutely continuous with respect to µ n. Let us recall that a family µ t } t 0 of probability measures on a complete metric space E is said to be tight if, for any ε > 0, there exists a compact subset Q ε of E such that µ t (Q ε ) 1 ε for every t 0. Now, denote by π t (x, ) the law of X(t, x), that is, the measure π t (x, A) = P(X(t, x) A) A B(K). (6.9) Lemma 6.6 Let x 0 K be such that [ m E log δ j (X(t, x 0 )) ] C t 0 (6.10) j=1 for some C 0. Then π t (x 0, dy)} t 0 is tight. 22

24 Proof: For any integer i, let Q c i = K \Q i and consider the positive sequence Since δ K (x) < ε i on Q c i, we have π t (x 0, Q c i) = Q c i π t (x 0, dy) = ε i = min x Q i δ K (x). 1 log εi m log δ j (X(t, x 0 )) π t (x 0, dy) K j=1 1 [ m log εi E log δ j (X(t, x 0 )) ] C log εi. Since ε i 0 as i, the above inequality implies that, given ɛ > 0, j=1 π t (x 0, Q i ) = 1 π t (x 0, Q c i) > 1 ɛ t 0, for all i large enough. So, π t (x 0, dy)} t 0 is tight. Our next result completes the analysis of the existence and uniqueness of the invariant measure for P t. Theorem 6.7 Assume x K, j J( x) (i) lim sup K x x L 0 δ j (x) δ j (x) log δ j (x) < 0 (ii) a( x) δ j ( x), δ j ( x) = 0 Then P t possesses a unique invariant measure µ << µ n. Proof: Since uniqueness is granted by Theorem 6.1, let us concentrate on existence. Suppose we can find an invariant measure for the semigroup P t that we introduced in (6.3). Then, µ would be absolutely continuous with respect to µ n in view of Proposition 6.5 (a). Thus, on account of Lemma 6.3 (b), µ would also be extendable to an invariant measure for P t, which would obviously remain absolutely continuous with respect to µ n. So, to complete the proof it is enough to construct an invariant measure for P t. Now, the Krylov-Bogoliubov theorem (see, e.g. [8, Theorem 7.1]) ensures that P t possesses an invariant measure if, for some x 0 K, the family of probability measures π t (x 0, dy)} t 0 is tight. So, thanks to Lemma 6.6, it suffices to obtain (6.10). Let α > 0 and U be given by Proposition 4.3. Fix x 0 K, apply Itô s formula to U(X(t, x 0 )), and take expectation to obtain E [ U(X(t, x 0 )) ] = U(x 0 ) + E t 0 23 (L 0 U)(X(s, x 0 ))ds t 0.

25 Then, taking into account (3.11), d dt E[ U(X(t, x 0 )) ] = E [ (L 0 U)(X(t, x 0 )) ] M αe [ U(X(t, x 0 )) ]. This yields E [ U(X(t, x 0 )) ] e αt U(x 0 ) + M t 0. α Since U coincides with m log δj (X(t, x 0 )) near K, (6.10) follows. j=1 6.3 Examples We conclude with three examples describing possible applications of our invariance result. Example 6.8 Let us consider the stochastic differential equation (2.1) in the closed unit ball K = B 1 R 2, where b : B 1 R 2 is a Lipschitz vector field and σ is defined as follows. Let ν(x) = (x 1, x 2 ) x and let θ C 1 ([0, 1]) be such that, ξ(x) = (x 2, x 1 ) x 0 θ 1, θ 1 on B 1/3 x B 1 \ B 2/3, 0 on B 1 \ B 2/3 Define, for every x B 1, σ(x) = θ( x )I + ( 1 θ(x) )[ ( 1 x 2 ) ] ν(x) ν(x) + λξ(x) ξ(x) where I is the identity matrix and λ R. Then, it is easy to check that L 0 δ K (x) = λ2 2 b(x), x x K. Therefore, by Theorem 3.1 (see also Theorem 4.2) we have that B 1 is invariant for X if and only if max b(x), x λ2 x =1 2. Moreover, owing to Theorem 6.1, semigroup P t has at most one invariant measure µ << µ 2. Furthermore, such a measure does exist if max b(x), x < λ2 x =

26 Example 6.9 In the closed cube Q 1 R n (see Example 4.1) let us consider the stochastic differential equation (2.1), where b(x) = (b 1 (x),..., b n (x)) is a Lipschitz vector field and σ(x) = n (1 x 2 j)e j e j x Q 1. j=1 Then, conditions (4.1), (4.2), and (4.3) hold true, and L 0 ϕ = 1 2 n (1 x 2 j) j 2 ϕ + b(x), ϕ. j=1 Therefore, recalling (4.5), we conclude that Q 1 is invariant if and only if b j (x) x j x j 0 x Q 1, j J(x). Under the above assumption we have that semigroup P t has at most one invariant measure µ << µ 2, whose existence is guaranteed if b j (x) x j x j < 0 x Q 1, j J(x). Example 6.10 Let us consider the stochastic differential equation in the closed unit ball K = B 1 R n, dx(t) = b(x(t))dt + (1 X(t) 2 ) dw (t), t 0 (6.11) X(0) = x, The corresponding Kol- where b : B 1 R n is a Lipschitz vector field. mogorov operator is L 0 ϕ = 1 2 (1 x 2 ) 2 ϕ + b(x), ϕ. Applying Theorems 4.2 and 6.1, one checks easily that B 1 is invariant for X if and only if max b(x), x 0, x =1 and that P t has at most one invariant measure µ << µ 2 under the above assumption. Moreover, by Theorem 6.7, such a measure does exist if lim inf x 1 b(x), x (1 x ) log(1 x ) > 0. (6.12) 25

27 Now, let us compute the density ρ of µ with respect to Lebesgue s measure, in the case when b(x) = βx (where β is a given real number). Note that, for such a vector field, (6.12) β < 0. Differentiating both sides of equation (6.1) with respect to t, the problem reduces to finding an integrable function ρ such that div [ (1 x 2 ) 2 ρ(x) 2(1 x 2 )ρ(x)x 2βρ(x)x ] = 0 x B 1. Therefore, it suffices to solve the equation (1 x 2 ) 2 ρ(x) 2(1 x 2 )ρ(x)x 2βρ(x)x = 0 x B 1, that is easily seen to possess the solution ρ(x) = 1 1 x 2 e β 1 x 2 x B 1. (6.13) The above integrable function being integrable since β < 0, (6.13) gives the required density. References [1] AUBIN J.-P. and DA PRATO G., Stochastic viability and invariance, Annali Scuola Norm. Sup. Pisa (IV), 17, , [2] AUBIN J.-P. and DA PRATO G, Stochastic Nagumo s viability theorem, Stochastic Anal. Appl., 13, 1 11, [3] AUBIN J.-P. and DA PRATO G., The viability theorem for stochastic differential inclusion, Stochastic Anal. Appl., 16, 1 15, [4] AUBIN J.-P. and FRANKOWSKA H., Set-Valued Analysis, Birkhäuser, Boston, [5] BARDI M. and GOATIN P., Invariant sets for controlled degenerate diffusions: a viscosity solution approach, in Stochastic analysis, control, optimization and application (W.M.M. Mc Eneaney, G.G. Yin, Q. Zhang Eds.), Birkhäuser, , [6] BUCKDAHN R., PENG S., QUINCAMPOIX M. and RAINER C., Existence of stochastic control under state constraints, C. R. Acad. Sci. Paris, Série I, 327, 17 22,

28 [7] CESSENAT M., DAUTRAY R., LEDANOIS G., LIONS P.-L., PAR- DOUX E., and SENTIS R., Méthodes probabilistes pour les équations de la physique, Eyrolles, Paris, [8] DA PRATO G., An introduction to infinite-dimensional analysis, Springer-Verlag, Berlin, [9] DA PRATO G. and FRANKOWSKA H., Stochastic viability for compact sets in terms of the distance function, Dynam. Systems Appl., 10, , [10] DA PRATO G. and FRANKOWSKA H., Existence, uniqueness and regularity of the invariant measure for a class of elliptic degenerate operators, J. Diff. and Integral Eqs. Volume 17, , [11] DA PRATO G. and FRANKOWSKA H., Invariance of stochastic control systems with deterministic arguments, J. Differential Equations, 200, 18 52, [12] DA PRATO G. and FRANKOWSKA H., Invariant measure for a class of parabolic degenerate equations, NoDEA, Volume 12, Number 4, , [13] DA PRATO G. and FRANKOWSKA H., Stochastic viability of convex sets, J. Math. Anal. Appl. 333, no. 1, , [14] DA PRATO G. and ZABCZYK J., Ergodicity for infinite dimensional systems, London Mathematical Society Lecture Notes, n.229, Cambridge University, [15] DELFOUR M.C. and ZOLESIO J.-P., Shape analysis via oriented distance function, J. Funct. Anal., Volume 123, , [16] FELLER W., On second order differential operators, Ann. of Math. (2) 61, , [17] LADYZHENSKAJA O.A, SOLONNIKOV V. A. and URAL CEVA N.N., Linear and quasilinear equations of parabolic type, Transl. Math. Monographs, Amer. Math. Soc., [18] LORENZI L. and BERTOLDI M., Analytical methods for Markov semigroups, Chapman & Hill/CRC,

29 Stampato in proprio Maggio 2008 Scuola Normale Superiore, Piazza dei Cavalieri, 7 - PISA