Cores for generators of some Markov semigroups

Cores for generators of some Markov semigroups Giuseppe Da Prato, Scuola Normale Superiore di Pisa, Italy and Michael Röckner Faculty of Mathematics, University of Bielefeld, Germany and Department of Statistics, Purdue University, W. Lafayette, 4796, IN, U. S. A. Abstract We consider a stochastic differential equation on R d with Lipschitz coefficients. We find a core for the infinitesimal generator of the corresponding Markov process. Some applications, in particular, to well-posedness of Fokker Planck equations are given. 2 Mathematics Subject Classification AMS: 61, 6J35, 47D7. Key words: stochastic differential equation, Markov process, transition semigroup, core. 1 Introduction We are here concerned with a stochastic differential equation in := R d, dx = b(x)dt + σ(x)dw (t), (1.1) X() = x, Research supported by the DFG through CRC 71 and the I. Newton Institute, Cambridge, UK. 1

where b : and σ : L() are Lipschitz continuous. It is well known that equation (1.1) has a unique solution X(, x). Moreover, the transition semigroup P t ϕ(x) = E[ϕ(X(t, x))], ϕ C b (), (1.2) is Feller. (C b () is the Banach space of all mappings ϕ : R which are uniformly continuous and bounded, endowed with the norm ϕ := sup x ϕ(x).) If b and σ are not bounded P t, t, is not strongly continuous in C b () but only pointwise continuous. We call it a π-semigroup, see later for a precise definition. Though the ille Yosida theory cannot be applied to P t, one can define an infinitesimal generator K following [1] or [5]. Then the problem arises to show the relationship between K and the Kolmogorov operator where K ϕ := 1 2 Tr [a(x)d2 ϕ] + b(x), Dϕ(x), ϕ C 2 b (), (1.3) a(x) = σ(x)σ (x), x. The main result of this paper is that if in addition b and σ are of class C 2 with bounded second derivatives, then K ϕ = K ϕ, ϕ C 2 b () and the space Cb 2 () is a core for K. This result seems to be new when a is not uniformly elliptic, see the monograph [3] and references therein for the case of uniformly elliptic a. 2 Notations and preliminaries Let us precise our assumptions. ypothesis 2.1 (i) b : and there is K 1 > such that b(x) b(y) K 1 x y, x. (2.1) (ii) σ : L() and there is K 2 > such that σ(x) σ(y) S K 2 x y, x, (2.2) where the sub-index S means the ilbert Schmidt norm. 2

We note that by (2.1) and (2.2) it follows that and b(x) K 1 x + b(), x, (2.3) σ(x) S K 2 x + σ() S, x, (2.4) respectively. Sometimes we shall need the following more stringent assumptions. ypothesis 2.2 (i) b and σ are of class C 2 and fulfill ypothesis 2.1. (ii) There is K 3 > such that b (x) + σ (x) S K 3, x. (2.5) The following two propositions are well known. Proposition 2.3 Assume that ypothesis 2.1 is fulfilled. Then for any x and any T > there is a unique solution X(, x) L 2 W (Ω; C([, T ]; )) of equation (1.1). Moreover, for all m N, X(, x) L m W (Ω; C([, T ]; )) and there is C T,m > such that E ( X(t, x) m ) C T,m (1 + x m ), x, t [, T ]. (2.6) Finally, there is C T > such that E X(t, x) X(t, y) C T x y, x, y, t [, T ]. (2.7) By L m W (Ω; C([, T ]; )) we mean the space of all adapted continuous stochastic processes F such that ( E ) sup F (t) m < +. t [,T ] Proposition 2.4 Assume that ypothesis 2.2 is fulfilled and let X(, x) be the solution to (1.1). Then the following statements hold. (i) X(t, x) is continuously differentiable in any direction h and setting η h (t, x) = X x (t, x) h we have dη h = b (X)η h dt + σ (X)(η h, dw (t)), (2.8) η h () = h. Moreover, there exists ω 1 R such that E η h (t, x) 2 e 2ω 1t h 2, h, t >. (2.9) 3

(ii) X(t, x) is twice continuously differentiable in any couple of directions h, k and setting ζ h,k (t, x) = X x (t, x)(h, k) we have dζ h,k = b (X)ζ h,k dt + b (X)(η h, η k )dt +σ (X)(ζ h,k, dw (t)) + σ (X)(η h, η k, dw (t)), (2.1) ζ h,k () =. Moreover, there exists ω 2 R and C 2 > such that E ζ h,k (t, x) 2 C 2 e 2ω 2t h 2 k 2, h, k, t >. (2.11) 2.1 Transition semigroup Let us introduce some notations. For any k N by Cb k () we denote the space of all mappings ϕ : R which are uniformly continuous and bounded together with their derivatives of order lesser than k. Cb k (), endowed with the norm ϕ k := ϕ + k D j ϕ, is a Banach space. Moreover, for any m N by C b,m () we denote the space of all mappings ϕ : R such that the mapping j=1 R, x ϕ(x) 1 + x m belongs to C b (). C b,m (), endowed with the norm ϕ(x) ϕ b,m := sup x 1 + x, m is a Banach space. Taking into account (2.6) we can define the transition semigroup P t ϕ(x) = E[ϕ(X(t, x))], ϕ C b,m (). (2.12) We know that P t ϕ C b () for all t and all ϕ C b (). It follows from (2.6) that also P t ϕ C b,m () for all ϕ C b,m (). Furthermore, P t, t, is a semigroup. 4

Proposition 2.5 Assume that ypothesis 2.2 is fulfilled. Then Cb 1 () and Cb 2() are stable for P t, t. Moreover there exist positive constants K 1 and K 2 such that P t ϕ j K j ϕ j, ϕ C j b (), j = 1, 2, t, (2.13) where j denotes the norm in C j b (). Proof. The assertions follow from Propositions 2.4 and the identities and DP t ϕ(x), h = E[ Dϕ(X(t, x)), η h (t, x) ], t, h, x, D 2 P t ϕ(x)h, k = E[ Dϕ(X(t, x)), ζ h,k (t, x) ] 2.2 Itô s formula +E[D 2 ϕ(x(t, x))(η h (t, x), η k (t, x))], t, h, k, x. Let us consider the Kolmogorov operator (1.3). By ypothesis 2.1 it follows that there exists M > such that for all ϕ Cb 2 () we have K ϕ(x) M(1 + x 2 ), x, (2.14) so that K ϕ C b,2 (). Proposition 2.6 (Itô s formula) Assume that ypothesis 2.1 is fulfilled. Then for all ϕ Cb 2 () we have E[ϕ(X(t, x))] = ϕ(x) + Proof. Write X(t, x) = x + t By (2.3) and (2.6) we see that t E[K ϕ(x(s, x))]ds. (2.15) b(x(s, x))ds + t σ(x(s, x))dw (s). E b(x(t, x)) C T,1 K 1 (1 + x ) + b(). so that b(x(, x)) C W ([, T ]; L 1 (Ω, R d )). 5

Moreover, by (2.4) and (2.6) so that, E σ(x(t, x)) 2 S 2K 2 (1 + x ) 2 + 2 σ() 2 S, σ(x(, x)) C W ([, T ]; L 2 (Ω, R d )). Then we may apply Itô s formula and the conclusion follows. Note that if ϕ C 2 b () we have K ϕ C b,2 () by (2.14). Therefore, d dt P tϕ = P t K ϕ, ϕ C 2 b (). (2.16) Proposition 2.7 Assume that ypothesis 2.2 is fulfilled. Then d dt P tϕ = K P t ϕ, ϕ C 2 b (). (2.17) Proof. By the semigroup property, Proposition 2.5 and (2.16) we have for all ϕ C 2 b () d dt P tϕ = d dɛ P t+ɛϕ ɛ= = d dɛ P ɛ(p t ϕ) ɛ= = K P t ϕ 3 The infinitesimal generator of P t Let us introduce some notations. Given a sequence (ϕ n ) C b () and ϕ C b (), we say that (ϕ n ) is π-convergent to ϕ and write ϕ π n ϕ, if the following conditions are fulfilled. (i) For each x we have (ii) sup ϕ n < +. n N lim ϕ n(x) = ϕ(x). n Proposition 3.1 Assume that ypothesis 2.1 is fulfilled and let (ϕ n ) C b (), ϕ C b () such that ϕ n π ϕ. Then for all t we have P t ϕ n π P t ϕ. 6

We say that P t is a π-semigroup. Proof of Proposition 3.1. For any x we have lim P tϕ n (x) = lim E[ϕ n (X(t, x))] = P t ϕ(x), n n thank s to the dominated convergence theorem. Moreover P t ϕ n ϕ n sup ϕ n <. n N We follow here [5]. Definition 3.2 We say that ϕ belongs to the domain of the infinitesimal generator K of P t in C b () if (i) For each x there exists the limit (ii) and K ϕ C b (). 1 sup ɛ (,1] ɛ P ɛϕ ϕ < +. 1 lim ɛ ɛ (P ɛϕ(x) ϕ(x)) =: K ϕ(x) K is called the infinitesimal generator of P t. In the following we set ɛ := 1 ɛ (P ɛ 1). Proposition 3.3 Let ϕ D(K ) and let t. Then P t ϕ K and we have K P t ϕ(x) = P t K ϕ(x), x. (3.1) Moreover, P t ϕ(x) is differentiable in t and d dt P tϕ(x) = K P t ϕ(x) = P t K ϕ(x), x. (3.2) Proof. Let ϕ K. Then we have ɛ P t ϕ(x) = P t ɛ ϕ(x). 7

Since ɛ ϕ π K ϕ, by Proposition 3.1 it follows that ɛ P t ϕ π P t K ϕ. So, P t ϕ D(K ) and K P t ϕ = P t K ϕ. On the other hand, D + P t ϕ(x) = lim ɛ P t ɛ ϕ(x) = P t K ϕ(x) and (3.1) follows because K ϕ is continuous. We shall denote by ρ(k ) the resolvent set of K and by R(λ, K ) its resolvent. The following result is proved in [5] under slightly different assumptions. We sketch here the proof for the reader s convenience. Proposition 3.4 ρ(k ) (, + ). f C b () we have Moreover, for any λ > and any Proof. Set R(λ, K )f(x) = e λt P t f(x)dt, x. (3.3) ɛ := 1 ɛ (P ɛ 1). Let f C b () and for any λ >, x set F (λ)f(x) = + e λt P t f(x)dt. Then it is not difficult to see that F (λ)f C b (). Let us show that λ ρ(k ). Since ɛ F (λ)f(x) = 1 [ ] ɛ eλɛ e λt P t f(x)dt e λt P t f(x)dt = 1 ɛ (eλɛ 1) ɛ we easily deduce that F (λ)f D(K ) and Therefore ɛ e λt P t f(x)dt 1 ɛ ɛ eλɛ e λt P t f(x)dt, lim ɛ F (λ)f(x) = λf (λ)f(x) f(x). (3.4) ɛ K F (λ)f = λf (λ) f. (3.5) 8

It remains to show that F (λ)(λ K )ϕ = ϕ, ϕ D(K ). (3.6) This will complete the proof that λ ρ(k ). To prove (3.6) choose ϕ D(K ), then, taking into account Proposition 3.3, we have e λt P t K ϕ(x)dt = = ϕ(x) + λf (λ)ϕ(x), x. which implies (3.6). Remark 3.5 By (3.3) it follows that e λt d dt P tϕ(x)dt R(λ, K )f 1 λ f, f C b (). Therefore K is m dissipative in C b (). We are going to investigate the relationship between K and K. Proposition 3.6 Assume that ypothesis 2.1 is fulfilled. If ϕ D(K ) C 2 b () then K ϕ = K ϕ. Proof. By Proposition 2.6 we have As t we find Now we show that 1 t (P tϕ(x) ϕ(x)) = t E[K ϕ(x(s, x))]ds. K ϕ(x) = K ϕ(x), x. D := {ϕ C 2 b () : K ϕ C b ()}, is a core for K. Let us first show that D D(K ). Proposition 3.7 Assume that ypothesis 2.1 is fulfilled and let ϕ D. Then ϕ D(K ) and we have K ϕ = K ϕ. 9

Proof. Let ϕ D. By Itô s formula (2.15) we have Therefore, 1 t (P tϕ(x) ϕ(x)) = t E[K ϕ(x(s, x))]ds. 1 lim t t (P tϕ(x) ϕ(x)) = K ϕ(x), x, and 1 t (P tϕ ϕ) K ϕ, which implies that ϕ D(K ) and K ϕ = K ϕ. Theorem 3.8 Assume that ypothesis 2.2 is fulfilled. Then D is a core for K, that is, for any ϕ D(K ) there exists a sequence (ϕ n ) D such that ϕ n ϕ, K ϕ n K ϕ, in C b (). Proof. Let ϕ D(K ). Fix λ > and set f := λϕ K ϕ. Since Cb 2 () is dense in C b () there exists a sequence (f n ) Cb 2() such that f n f in C b (). Set ϕ n := (λ K ) 1 f n, n N. By Proposition 3.4 it is clear that ϕ n D(K ) and lim ϕ n = ϕ, n lim K ϕ n = K ϕ, n in C b (). It remains to show that ϕ n D for any n N. In fact, taking into account Proposition 2.5, we see that ϕ n D(K ) Cb 2 (), so that by Proposition 3.6 we have K ϕ n = K ϕ n C b (), and hence ϕ n D for any n N. So, D is a core as claimed. Remark 3.9 The above result can be generalized. In fact ypothesis 2.2 and also the global Lipschitz assumption in ypothesis 2.1 are too strong. E. g. assumptions as ypotheses 1.1 and 1.2 in [2] are sufficient. Details will be the subject of future work. 4 Applications We start with an important identity K (ϕ 2 ) = 2ϕK ϕ + σ Dϕ 2, ϕ C 2 b (), (4.1) whose proof is straightforward. 1

Proposition 4.1 Assume, besides ypothesis 2.2, that σ is bounded. Then for any ϕ D we have ϕ 2 D and K (ϕ 2 ) = 2ϕK ϕ + σ Dϕ 2. (4.2) Proof. Let ϕ D. Then ϕ 2 C 2 b () and by (4.1) it follows that K (ϕ 2 ) C b (). Corollary 4.2 Let ϕ D. Given λ > set f = λϕ K ϕ. Then we have ϕ 2 = (2λ K ) 1 (2ϕf σ Dϕ 2 ). (4.3) 4.1 Invariant measures In this subsection we assume that ypothesis 2.2 holds. Let µ P(). We say that µ is invariant if P t ϕdµ = ϕdµ, ϕ C b (). (4.4) Proposition 4.3 µ is invariant for P t if and only if K ϕdµ =, ϕ D. (4.5) Proof. Since the only if part is obvious, let us show the if part. Assume that (4.5) is fulfilled. Then K ϕdµ =, ϕ D(K ). Then if ϕ D(K ) we have by Proposition 3.3, that P s ϕ D(K ), for all s and P t ϕ(x) ϕ(x) = t K P s ϕ(x)ds. Integrating with respect to µ, yields (P t ϕ ϕ)dµ =, ϕ D(K ). Therefore, µ is invariant. 11

Proposition 4.4 Assume that µ is an invariant measure for P t such that x 2 µ(dx) < +. (4.6) Then we have ϕk ϕdµ = 1 2 σ Dϕ 2 dµ, ϕ C 2 b (). (4.7) Proof. Step 1 We have K ϕdµ =, ϕ Cb 2 (). (4.8) The proof follows by (2.14), (2.17), (4.6) and Lebesgues dominated convergence theorem Step 2 Conclusion. By (4.1) we have = K (ϕ 2 )dµ = 2 so, the conclusion follows. ϕk ϕdµ + 4.2 Application to Fokker Planck σ Dϕ 2 dµ, ϕ C 2 b (), In this subsection we assume that ypothesis 2.2 holds. Let B() be the Borel σ-algebra of. We recall that a measure µ(dt, dx) = µ t (dx)dt on B([, T ]) B(), where µ t are probability measures on B(), measurable in t [, T ], is called a solution to the Fokker Planck equation for (K, D) if t ϕ dµ t = ϕ dµ + K ϕ dµ s ds, a.e. t, ϕ D. (4.9) Theorem 4.5 For every probability measure ζ on B() there exists a unique measure µ(dt, dx) = µ t (dx)dt (as above) solving (4.9) with µ = ζ. Proof. Existence is trivial. Just define the measure µ t by ϕ(x) µ t (dx) := E[ϕ(X(t, x))]ζ(dx), a.e. t, ϕ : R +, B()-measurable. Then (4.9) follows by Proposition 2.6 (i.e., by Itô s formula). To show uniqueness we note that Theorem 3.8 implies that (4.9) with µ = ζ holds for all ϕ K. ence uniqueness follows from [4, Theorem 2.12]. 12

References [1] S. Cerrai, A ille-yosida theorem for weakly continuous semigroups, Semigroup Forum, 49, 349-367, 1994. [2] S. Cerrai, Second order PDE s in finite and infinite dimensions. A probabilistic approach, Lecture Notes in Mathematics, 1762, Springer-Verlag, 21. [3] L. Lorenzi and M. Bertoldi, Analytical methods for Markov semigroups, Chapman & ill/crc, 26. [4] L. Manca, Kolmogorov operators in spaces of continuous functions and equations for measures. Theses of Scuola Normale Superiore di Pisa (New Series)], 1. Edizioni della Normale, Pisa, 28. [5] E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions, Studia Math., 136, 271-295, 1999. 13