- Email: [email protected]
Null controllability and strong feller property of markov transition semigroups
\,m,inelrr
-I,dwii
Ihrorl.
Weeihrrd, & 4/Jppl,cor,ons.
voi 25. NOP 9-10. pp w-949, 1995 Copynght 4 1995 Elsevw Science Ltd Prmed III Great Britam All rights reserved 0362-546X/95 $9.50+ ,042
Pergamon 0362-546X(95)00089-5
NULL
CONTROLLABILITY OF MARKOV
AND STRONG FELLER TRANSITION SEMIGROUPS GIUSEPPE
PROPERTY
DA PRATO
Scuola ?rormale Superiore di Piss, Plaza
de1 CavalIeri 7, 56126 Pisa, Italy
Key words and phrases: Markov transition wnigroup. strong Feller semigroups, differential equations, null controllability, Kolmogorov equations. I. IN TROl~U(‘TlON
:‘.Nl)
S1.T l-INC;
OF THE
stochastic
PROBLEM
Let H (norm 1.1, inner product (. , . > be a separable Hilbert space. We are concerned with the differential stochastic equation dX(t)
= (Ax(t)
+ f(r),
dt + Y’Q
t>szo,
dW(r),
(1.1)
X(s) = .\-, under the following
assumptions.
Hypothesis 1. (i) Q is a positive, symmetric bounded operator in H. (ii) f: [0, +03[ + H is square locally Bochner integrable. (iii) A: D(A) c H + H is the infinitesimal generator of a strongly continuous S(f), t 2 0, in H. (iv) W(.) is a cylindrical Wiener process in H of the form WI
= i
Pk(rkk,
r 2 0,
semigroup
(1.2)
k=i
where lek I is a complete orthonormal system on H and IPk(.)] is a sequence of mutually independent real standard Wiener processes on a probability space (C2,5, Ip). (v) The linear operators Q,,, , t > s x E H,
(1.3)
are trace-class. As is well known, see [l, theorem 5.41. problem (1.1) has a unique weak or mild solution given by t L 0, X(f, 5; x) = z-(r, s; x) + w, (f, s), (1.4) where z(f, s; x) = .S(f ~ s)x t 3I U>(f,S) =
,! 5
I S(I -
o-)f(a) da
*5
S(r - 0“j v Q d W(CJ),
tr:sLO.
(1.9
G. DA PRATO
942
Moreover, for any t 1 s 2 0 and any x E H, the random variable X(t, s; x) has a Gaussian law X(z(t, s; x), QI.S) with mean z(t, s; x) and covariance operator Q,, I. Let us recall the definition of transition semigroup P,,,, t 2 s L 0 associated with problem (1.1). For any function p: H + II?belonging to the space B,(H), of all bounded Bore1 mappings H -+ I?, we set p,,,$w
= uq&vt,
= tI ff
x E H,
s; x))),
v(~)‘Wz(r, s; xh Qt,,)(W, VJE 4AH).
(1.6)
equation, it follows that, for all t 2 rr L s
We recall that, by the Chapman-Kolmogorov
P,,,p,,ru? = P,,,% 6 E b(H). It is easy to check that the transition
semigroup P$,,, t 2 s 2 0 is Feller, that is v(.O E C,(H),
&,,V E C,(H)
t > S > 0.
Here C,(H) is the Banach space of all uniformly endowed with the norm
continuous
Ih,d/o = ,“~-$‘“‘I~
C’ E c,(H).
Moreover, &,
I > s 2 0 is said to be strongly Feller if
vp~B,(H),t>s>O.
P,,,v7 E C,(H) When f(t) [l, P. 3151
and bounded mappings H + R,
= 0, t 2 0, the semigroup P, = PO,r, t 2 0 is strongly Feller if and only if, see S(t)(H) c Q:,::(H),
vt > 0.
(1.7)
The strong Feller property is very useful in studying Kolmogorov equations, see [l], and uniqueness and ergodicity of invariant measures, see [2]. We remark that condition (1.7) is equivalent to null controllability in any time T > 0 of the deterministic system tro z’(t) = AZ(f) + \~?&(t),
(1.8)
Z(0) = x E H,
see Section 2. This means that for any T > 0 and for any x E H there exists a control (not unique in general), u E L’(0, T; H) such that
Z(T) = Z(T, x; u) = S(T)x + L,u = 0,
(1.9)
where ‘7
L, =
I .o
S( T - s)&&(s)
ds.
(1.10)
Section 2, is devoted to recalling some results about null controllability, involving in particular the controllability operator QI,, . We give also a sufficient condition in order that Q,,,(H) C D(A). In Section 3 we prove that the null controllability condition (1.7) implies strong Feller property for the semigroup Ps,f, I 2 s 2 0. Finally, in Section 4 we study the asymptotic behaviour of l’,,, when s + -CCJ.
Markov 2. NULL
transition
943
semigroups
CONTROLLABILITY
We will need the following well-known operators, see e.g. [l, proposition B. 11.
result on the comparison
of images of linear
PROPOSITION 2.1. Let I/ and H be Hilbert spaces and let A: CT-+ H and G: H 4 H be linear bounded operator. Then the following statements are equivalent (i) G(H) C NW, (ii) there exists C > 0 such that
lG*xI,z, 5 CIA*xl,,
VXEH,
(2.1)
where G* and A* are the adjoint operators of G and A, respectively. Let G E S(H). We denote by X;, the kernel of G and by rr6 the orthogonal Moreover, we consider the linear operator G,:K; Obviously,
--) H,
projector on K&.
x + Gx.
the linear operator GKA: Kh --t H, is one-to-one. We will set G-lx
= (GK;)-‘x,
VXEH.
G-’ is called the pseudo-inverse of G. For any x E H and T > 0 let us consider the energy functional JT(u) =
The following
” lu(s)l* ds,
II 0
u E L*(O, T; H).
(2.2)
result is well known, see e.g. [I].
2.2. Assume that hypothesis 1 holds. Then system (1.8) is null controllable In this case the following statements hold (i) for any T > 0 we have
THEOREM
in any time
t > 0 if and only if (1.7) is fulfilled.
L.(L ‘(0, T; H)) = Q:(:VO, (ii) setting u* = -Q&L;x
(2.3)
we have J,(u*)
= min]J,(u)
: u E L2(0, T; H), Z(T, x; u) = 01.
(2.4)
(iii) we have ll~*/li~~O,~~H~= /Q;1’2W)~12. In the following
we shall denote by T(T) the linear bounded operator H -+ H, defined as r(T)
where Q,y’
(2.5)
= Qi,:/2S(T),
is the pseudo-inverse of Qd(+.
T> 0,
(2.6)
G.
944
COROLLARY 2.3. Assume that hypothesis
DA
PRATO
1 holds and that Q = 1. Then we have (2.7)
(2.8) Proof. Let x E H, T > 0, and set u(t) = - + S(t)x. Then we have u(T) = 0, so that
and the conclusion
follows from (2.5).
n
We conclude this section by giving some properties of the images of Qxg and Qo,r. To formulate the results we have to recall some properties on Cauchy problem
f E 10,Tl,
Y’(f) = A Y(2) + f(t), Y(0) = 0.
PROPOSITION 2.4. Let A: D(A) c H + H be the infinitesimal
(2.9)
generator of an analytic semi-
group. Then the following statements hold: (i) if f E L’(O, T; H) then problem (2.9) has a unique solution Y E W’32(0, T: H) fl L’(O, T; D(A)), and YE C([O, T]; DA(1/2, 2)); (ii) if f E L’(O, T; DA (l/2, 2)) then the solution
Y of (2.9) fulfills
Y E W’-2(0, T; DA(1/2, 2)) fl L’(0, T; DA(3/2, 2)), and YE C([O, T]; 0,&l/2, (iii) iffe
L’(O, T; Da(l/2,
2));
2)) and in addition there exists w E R such that I/S(t)l! i e”‘,
t 2 0,
then we have
Y E C(K), Tl; D(A)). We recall that D,s (l/2,2) that
is the real interpolation
space (D(A), H)1,2,2 introduced
D,z,(3/2,2) = (x E D(A) : Ax E DA(1/2, 2)).
in [3] and
Markov
transition
semigroups
945
Proof. For the proof of (i) and (ii) see [4, 5, theorem 23, (iv)]. Let us prove (iii). We first remark that, since A - w is m-accretive, then A has bounded imaginary powers, see [6]. Consequently, by theorem 1.15.3 in [7], it follows that the complex interpolation space [X, D(A2)]2 coincides with D(A). We recall, however, that in a Hilbert space complex and real interpolation spaces coincide, see [7, remark 3, p, 1431, so that [X, D(A2)], = (X, D(fQ)), We can prove now the following
n
result.
PROPOSITION 2.5. Assume besides hypothesis
Q::;(H) If in addition,
= D(A).
1 that the semigroup S(a) is analytic. Then (2.10)
c 4z, (l/2,2).
there exists o E R such that IiS
5 e”‘,
Qo,rU?~(l/L2)) c
tI
Q,(l/Z
0, and (2.11)
3,
then we have
Qo,rW)c
(2.12)
D(A).
Proof. The first statement, proved in [l, remark B.81, follows immediately from proposition 2.4(ii). To prove the second one it is enough to replace H with the Hilbert space DA (l/2,2) and to apply proposition 2.4(iii). 4 3. REC;L1_.4RLTY
PROPERTIES
OF THE~TRANSITION
SEMIGROUP
THEOREM 3.1. Assume that, besides the hypothesis
1, inclusions (1.7) are fulfilled. Let, moreover, P,., , t > s 2 0 be the family of linear operators defined by (1.6). Then, for any p E C,(H) we have P,,,rp E C;(H),? and the derivative DP,,,y, is given by (Dp,.,dx),
h)
=
\
, ff
ddf,
S; X)
+ Y)
<
r(r -- Q/Z,
Q.;:"Y > WO, Q,,,)W,
(3.1)
for all h E H. Moreover,
~DP,,,(o(x)I 5 IlUf - s)IIih&~
t > 0, x E H, (P E C’,(H).
Proof. We proceed as in [ 11. First step. z(t, s; x) E Q,‘:;(H). Let t > s > 0, x-H, then
x E H, t > s > 0.
Z(t, s; x) = s(t - s)x +
t CL(H) is the set of all mappings II - P continuously
S(t - a)f(a) da. .I F
differentiable
and bounded with their first derivative.
(3.2)
G.DAPRATO
946
First of all, by (1.6) it follows that s(r - s)x E Q,:<‘(N) = Q~~f-,(~. Moreover, by theorem 2.2(i), sf
S(r - a)f(o)do
=
a,--’ S(t ~ s - p)f(p + s) dp E Z+,(L2(0, t - s; H)) II (1
= Q[f-s(H)
= Qt.‘:(H).
Secondstep. Conclusion. Let r > s > 0, x E H, then by the Cameron-Martin formula we have
dfWz(t, s; 4, Qy,,) (Y) = d(r, s; x, Y), d'W0, Qs,,)
(3.3)
Since for any h E H we have dd(t’;;-y’ ‘), h = d(t, s; x, y)[(r(t
- s)h, Q,:‘*y - (Q,;‘“z(t, s; x), r(t - s)h))],
it follows that
(DP,,,dx), h) = \ cp(z(t,s; xl + Y) d(tt s; x, Y) .H [(r(f
-
=
0,
Q,:‘2~
-
(Q;:“‘z(r,
xl,
Ur
-
Q,,,)(W
s)h))lWO,
. N c@A~,s; 4 + y)[(Ut - sP, Q,:‘% - (Q;:“z(t, s; xl, Ut - s)h))l I
Wz(r, s; XL Q,.,)(dy) = , H lo(z(t,s; x) + Y) < Ut - s)h, Q,:“% > W, Q,,,)(dy). I Finally, (3.2) follows easily by using HGlder’s inequality.
m
Remark 3.2. By using well-known properties of Gaussian measures, we see that the formula (1.6) is meaningful for all Bore1 functions p with polynomial grow. Then formula (3.1) can be written as DP,,,cp = P>,,U-*U - 4Qs::'%o(x) - U-*(f - s)W - SW’,,,@‘(X), for all q E C,(H). COROLLARY
3.3. Under the hypotheses of theorem 3.1, P,,l, t 2 s 2 0 is strongly Feller.
(3.5)
941
Proof. Let p E B,(H) and let [qn\ be a sequence in C,(H) such that (i) there exists M > 0 such that n E h,
lIdlO 5 M,
(3.6)
(ii) lim p,(x) = v(x), Vx E H. n-x Let t > 0, by applying formula (3.2) to V, we find
Letting n tend to infinity
we find that P,,,q is Lipshitz continuous. -1. -2SYfvlPTOTIC~
w
BEH/\VIOLR
We assume, besides hypothesis 1 that we have the following
hypothesis.
Hypothesis 2. (i) There exists M, (L, > 0 such that t20
/~s(t)ll I Me?‘,
(4.1)
(ii) f: R + H is bounded.
Remark 4.1. Under hypotheses 1 and 2 there exists the limit
and
x E H.
Q,x = I* S(s)QS*(s)x ds,
.0
Moreover,
Qm is trace class. We have in fact
Qm = f
.W)Q,,, ,S*(k),
k=l
and so Tr Qm % M2 Tr Q,,, , f
eek4’.
k=l
We now consider equation (1.1) for t 2 s > - cx). For this we have to define the Wiener process in R by setting iftr0 Wf) w(t) = if t 5 0, i U’,(-l) where WI(t), t 2 0 is another cylindrical Wiener process on (n, 5, P) independent on IV(t), t 2 0. Moreover, we denote by f the unique bounded solution in R of the equation 4, T’(t) = AZ(t) + f(f),
tE I?.
As is well known, .f is given by the formula f(t)
= (’
S(t - &f(a)
da.
(4.2)
G.DAPRATO
948 LEMMA
4.2. For all t > 0, x E H we have lim X(1, s; .u) = fct) + e(t), y---m
(4.3)
where S(t - a)&~d@(a)
Ir(f) =
do.
(4.4)
Proof. Fix x E H, t E R and set >f
.f Y(s) = X(f, s, x) -
S(t ~ a)f(a) .I --m
da -
L.,
S(t - a)adw(a).
Then Y(s) = S(f - s)x -
S(f - o)f(a)do
-
S(t - a)ad@‘(o).
It follows that
Consequently,
and the conclusion
follows.
n
We remark that q(t), t E R is a stationary variable X(0, Qm).
process, moreover, q(f) is a Gaussian random
PROPOSITION4.3. Assume that hypotheses 1 and 2 hold. Let q E C,(H), and lets < t < B. Then we have
Proof. By lemma 4.2, it follows that
= lim P,,.cp(x) = ~H WWt(fW, y---m I
Qm)(dy).
n
Markov
transition
semigroups
949
REFERENCES 1. DA PRATO G. & ZABCZYK J., Stochastic equations in infinite dimensions, in Encyclopedia of Mathematics and its Appliculions. Cambridge University Press, Cambridge (I 992). 2. PESZAT S. & ZABCZYK J., Strong property and irreducibility for diffusions on Hilbert spaces, Ann. Prob. (to appear). 3. LIONS J.-L. & PEETRE J., Sur une classe d’espaces d’interpolation, Pub/. Math. 1’I.H.E.S. 19, 5-68 (1964). 4 DA PRATO G. & GRISVARD P., Sommes d’operateurs lineaires et equations differentielles optrationnelles, J. Math. pures appl. 54, 305-387 (1975). 5. DI BLASIO G., Linear parabolic evolution equations in f.p-spaces, Anna/i Mat. puru appl. CXXXVIII, 55-104 (1983. 6. PRUSS J. & SOHR H., On operators with bounded imaginary powers in Banach spaces, Marh. Z. 203, 429-452 (1990). 7. TRIEBEL H., Interpolation Theorv, Funcrion Spaces, llifferentral Operators. North-Holland, Amsterdam (1986).
-I,dwii
Ihrorl.
Weeihrrd, & 4/Jppl,cor,ons.
voi 25. NOP 9-10. pp w-949, 1995 Copynght 4 1995 Elsevw Science Ltd Prmed III Great Britam All rights reserved 0362-546X/95 $9.50+ ,042
Pergamon 0362-546X(95)00089-5
NULL
CONTROLLABILITY OF MARKOV
AND STRONG FELLER TRANSITION SEMIGROUPS GIUSEPPE
PROPERTY
DA PRATO
Scuola ?rormale Superiore di Piss, Plaza
de1 CavalIeri 7, 56126 Pisa, Italy
Key words and phrases: Markov transition wnigroup. strong Feller semigroups, differential equations, null controllability, Kolmogorov equations. I. IN TROl~U(‘TlON
:‘.Nl)
S1.T l-INC;
OF THE
stochastic
PROBLEM
Let H (norm 1.1, inner product (. , . > be a separable Hilbert space. We are concerned with the differential stochastic equation dX(t)
= (Ax(t)
+ f(r),
dt + Y’Q
t>szo,
dW(r),
(1.1)
X(s) = .\-, under the following
assumptions.
Hypothesis 1. (i) Q is a positive, symmetric bounded operator in H. (ii) f: [0, +03[ + H is square locally Bochner integrable. (iii) A: D(A) c H + H is the infinitesimal generator of a strongly continuous S(f), t 2 0, in H. (iv) W(.) is a cylindrical Wiener process in H of the form WI
= i
Pk(rkk,
r 2 0,
semigroup
(1.2)
k=i
where lek I is a complete orthonormal system on H and IPk(.)] is a sequence of mutually independent real standard Wiener processes on a probability space (C2,5, Ip). (v) The linear operators Q,,, , t > s x E H,
(1.3)
are trace-class. As is well known, see [l, theorem 5.41. problem (1.1) has a unique weak or mild solution given by t L 0, X(f, 5; x) = z-(r, s; x) + w, (f, s), (1.4) where z(f, s; x) = .S(f ~ s)x t 3I U>(f,S) =
,! 5
I S(I -
o-)f(a) da
*5
S(r - 0“j v Q d W(CJ),
tr:sLO.
(1.9
G. DA PRATO
942
Moreover, for any t 1 s 2 0 and any x E H, the random variable X(t, s; x) has a Gaussian law X(z(t, s; x), QI.S) with mean z(t, s; x) and covariance operator Q,, I. Let us recall the definition of transition semigroup P,,,, t 2 s L 0 associated with problem (1.1). For any function p: H + II?belonging to the space B,(H), of all bounded Bore1 mappings H -+ I?, we set p,,,$w
= uq&vt,
= tI ff
x E H,
s; x))),
v(~)‘Wz(r, s; xh Qt,,)(W, VJE 4AH).
(1.6)
equation, it follows that, for all t 2 rr L s
We recall that, by the Chapman-Kolmogorov
P,,,p,,ru? = P,,,% 6 E b(H). It is easy to check that the transition
semigroup P$,,, t 2 s 2 0 is Feller, that is v(.O E C,(H),
&,,V E C,(H)
t > S > 0.
Here C,(H) is the Banach space of all uniformly endowed with the norm
continuous
Ih,d/o = ,“~-$‘“‘I~
C’ E c,(H).
Moreover, &,
I > s 2 0 is said to be strongly Feller if
vp~B,(H),t>s>O.
P,,,v7 E C,(H) When f(t) [l, P. 3151
and bounded mappings H + R,
= 0, t 2 0, the semigroup P, = PO,r, t 2 0 is strongly Feller if and only if, see S(t)(H) c Q:,::(H),
vt > 0.
(1.7)
The strong Feller property is very useful in studying Kolmogorov equations, see [l], and uniqueness and ergodicity of invariant measures, see [2]. We remark that condition (1.7) is equivalent to null controllability in any time T > 0 of the deterministic system tro z’(t) = AZ(f) + \~?&(t),
(1.8)
Z(0) = x E H,
see Section 2. This means that for any T > 0 and for any x E H there exists a control (not unique in general), u E L’(0, T; H) such that
Z(T) = Z(T, x; u) = S(T)x + L,u = 0,
(1.9)
where ‘7
L, =
I .o
S( T - s)&&(s)
ds.
(1.10)
Section 2, is devoted to recalling some results about null controllability, involving in particular the controllability operator QI,, . We give also a sufficient condition in order that Q,,,(H) C D(A). In Section 3 we prove that the null controllability condition (1.7) implies strong Feller property for the semigroup Ps,f, I 2 s 2 0. Finally, in Section 4 we study the asymptotic behaviour of l’,,, when s + -CCJ.
Markov 2. NULL
transition
943
semigroups
CONTROLLABILITY
We will need the following well-known operators, see e.g. [l, proposition B. 11.
result on the comparison
of images of linear
PROPOSITION 2.1. Let I/ and H be Hilbert spaces and let A: CT-+ H and G: H 4 H be linear bounded operator. Then the following statements are equivalent (i) G(H) C NW, (ii) there exists C > 0 such that
lG*xI,z, 5 CIA*xl,,
VXEH,
(2.1)
where G* and A* are the adjoint operators of G and A, respectively. Let G E S(H). We denote by X;, the kernel of G and by rr6 the orthogonal Moreover, we consider the linear operator G,:K; Obviously,
--) H,
projector on K&.
x + Gx.
the linear operator GKA: Kh --t H, is one-to-one. We will set G-lx
= (GK;)-‘x,
VXEH.
G-’ is called the pseudo-inverse of G. For any x E H and T > 0 let us consider the energy functional JT(u) =
The following
” lu(s)l* ds,
II 0
u E L*(O, T; H).
(2.2)
result is well known, see e.g. [I].
2.2. Assume that hypothesis 1 holds. Then system (1.8) is null controllable In this case the following statements hold (i) for any T > 0 we have
THEOREM
in any time
t > 0 if and only if (1.7) is fulfilled.
L.(L ‘(0, T; H)) = Q:(:VO, (ii) setting u* = -Q&L;x
(2.3)
we have J,(u*)
= min]J,(u)
: u E L2(0, T; H), Z(T, x; u) = 01.
(2.4)
(iii) we have ll~*/li~~O,~~H~= /Q;1’2W)~12. In the following
we shall denote by T(T) the linear bounded operator H -+ H, defined as r(T)
where Q,y’
(2.5)
= Qi,:/2S(T),
is the pseudo-inverse of Qd(+.
T> 0,
(2.6)
G.
944
COROLLARY 2.3. Assume that hypothesis
DA
PRATO
1 holds and that Q = 1. Then we have (2.7)
(2.8) Proof. Let x E H, T > 0, and set u(t) = - + S(t)x. Then we have u(T) = 0, so that
and the conclusion
follows from (2.5).
n
We conclude this section by giving some properties of the images of Qxg and Qo,r. To formulate the results we have to recall some properties on Cauchy problem
f E 10,Tl,
Y’(f) = A Y(2) + f(t), Y(0) = 0.
PROPOSITION 2.4. Let A: D(A) c H + H be the infinitesimal
(2.9)
generator of an analytic semi-
group. Then the following statements hold: (i) if f E L’(O, T; H) then problem (2.9) has a unique solution Y E W’32(0, T: H) fl L’(O, T; D(A)), and YE C([O, T]; DA(1/2, 2)); (ii) if f E L’(O, T; DA (l/2, 2)) then the solution
Y of (2.9) fulfills
Y E W’-2(0, T; DA(1/2, 2)) fl L’(0, T; DA(3/2, 2)), and YE C([O, T]; 0,&l/2, (iii) iffe
L’(O, T; Da(l/2,
2));
2)) and in addition there exists w E R such that I/S(t)l! i e”‘,
t 2 0,
then we have
Y E C(K), Tl; D(A)). We recall that D,s (l/2,2) that
is the real interpolation
space (D(A), H)1,2,2 introduced
D,z,(3/2,2) = (x E D(A) : Ax E DA(1/2, 2)).
in [3] and
Markov
transition
semigroups
945
Proof. For the proof of (i) and (ii) see [4, 5, theorem 23, (iv)]. Let us prove (iii). We first remark that, since A - w is m-accretive, then A has bounded imaginary powers, see [6]. Consequently, by theorem 1.15.3 in [7], it follows that the complex interpolation space [X, D(A2)]2 coincides with D(A). We recall, however, that in a Hilbert space complex and real interpolation spaces coincide, see [7, remark 3, p, 1431, so that [X, D(A2)], = (X, D(fQ)), We can prove now the following
n
result.
PROPOSITION 2.5. Assume besides hypothesis
Q::;(H) If in addition,
= D(A).
1 that the semigroup S(a) is analytic. Then (2.10)
c 4z, (l/2,2).
there exists o E R such that IiS
5 e”‘,
Qo,rU?~(l/L2)) c
tI
Q,(l/Z
0, and (2.11)
3,
then we have
Qo,rW)c
(2.12)
D(A).
Proof. The first statement, proved in [l, remark B.81, follows immediately from proposition 2.4(ii). To prove the second one it is enough to replace H with the Hilbert space DA (l/2,2) and to apply proposition 2.4(iii). 4 3. REC;L1_.4RLTY
PROPERTIES
OF THE~TRANSITION
SEMIGROUP
THEOREM 3.1. Assume that, besides the hypothesis
1, inclusions (1.7) are fulfilled. Let, moreover, P,., , t > s 2 0 be the family of linear operators defined by (1.6). Then, for any p E C,(H) we have P,,,rp E C;(H),? and the derivative DP,,,y, is given by (Dp,.,dx),
h)
=
\
, ff
ddf,
S; X)
+ Y)
<
r(r -- Q/Z,
Q.;:"Y > WO, Q,,,)W,
(3.1)
for all h E H. Moreover,
~DP,,,(o(x)I 5 IlUf - s)IIih&~
t > 0, x E H, (P E C’,(H).
Proof. We proceed as in [ 11. First step. z(t, s; x) E Q,‘:;(H). Let t > s > 0, x-H, then
x E H, t > s > 0.
Z(t, s; x) = s(t - s)x +
t CL(H) is the set of all mappings II - P continuously
S(t - a)f(a) da. .I F
differentiable
and bounded with their first derivative.
(3.2)
G.DAPRATO
946
First of all, by (1.6) it follows that s(r - s)x E Q,:<‘(N) = Q~~f-,(~. Moreover, by theorem 2.2(i), sf
S(r - a)f(o)do
=
a,--’ S(t ~ s - p)f(p + s) dp E Z+,(L2(0, t - s; H)) II (1
= Q[f-s(H)
= Qt.‘:(H).
Secondstep. Conclusion. Let r > s > 0, x E H, then by the Cameron-Martin formula we have
dfWz(t, s; 4, Qy,,) (Y) = d(r, s; x, Y), d'W0, Qs,,)
(3.3)
Since for any h E H we have dd(t’;;-y’ ‘), h = d(t, s; x, y)[(r(t
- s)h, Q,:‘*y - (Q,;‘“z(t, s; x), r(t - s)h))],
it follows that
(DP,,,dx), h) = \ cp(z(t,s; xl + Y) d(tt s; x, Y) .H [(r(f
-
=
0,
Q,:‘2~
-
(Q;:“‘z(r,
xl,
Ur
-
Q,,,)(W
s)h))lWO,
. N c@A~,s; 4 + y)[(Ut - sP, Q,:‘% - (Q;:“z(t, s; xl, Ut - s)h))l I
Wz(r, s; XL Q,.,)(dy) = , H lo(z(t,s; x) + Y) < Ut - s)h, Q,:“% > W, Q,,,)(dy). I Finally, (3.2) follows easily by using HGlder’s inequality.
m
Remark 3.2. By using well-known properties of Gaussian measures, we see that the formula (1.6) is meaningful for all Bore1 functions p with polynomial grow. Then formula (3.1) can be written as DP,,,cp = P>,,U-*U - 4Qs::'%o(x) - U-*(f - s)W - SW’,,,@‘(X), for all q E C,(H). COROLLARY
3.3. Under the hypotheses of theorem 3.1, P,,l, t 2 s 2 0 is strongly Feller.
(3.5)
941
Proof. Let p E B,(H) and let [qn\ be a sequence in C,(H) such that (i) there exists M > 0 such that n E h,
lIdlO 5 M,
(3.6)
(ii) lim p,(x) = v(x), Vx E H. n-x Let t > 0, by applying formula (3.2) to V, we find
Letting n tend to infinity
we find that P,,,q is Lipshitz continuous. -1. -2SYfvlPTOTIC~
w
BEH/\VIOLR
We assume, besides hypothesis 1 that we have the following
hypothesis.
Hypothesis 2. (i) There exists M, (L, > 0 such that t20
/~s(t)ll I Me?‘,
(4.1)
(ii) f: R + H is bounded.
Remark 4.1. Under hypotheses 1 and 2 there exists the limit
and
x E H.
Q,x = I* S(s)QS*(s)x ds,
.0
Moreover,
Qm is trace class. We have in fact
Qm = f
.W)Q,,, ,S*(k),
k=l
and so Tr Qm % M2 Tr Q,,, , f
eek4’.
k=l
We now consider equation (1.1) for t 2 s > - cx). For this we have to define the Wiener process in R by setting iftr0 Wf) w(t) = if t 5 0, i U’,(-l) where WI(t), t 2 0 is another cylindrical Wiener process on (n, 5, P) independent on IV(t), t 2 0. Moreover, we denote by f the unique bounded solution in R of the equation 4, T’(t) = AZ(t) + f(f),
tE I?.
As is well known, .f is given by the formula f(t)
= (’
S(t - &f(a)
da.
(4.2)
G.DAPRATO
948 LEMMA
4.2. For all t > 0, x E H we have lim X(1, s; .u) = fct) + e(t), y---m
(4.3)
where S(t - a)&~d@(a)
Ir(f) =
do.
(4.4)
Proof. Fix x E H, t E R and set >f
.f Y(s) = X(f, s, x) -
S(t ~ a)f(a) .I --m
da -
L.,
S(t - a)adw(a).
Then Y(s) = S(f - s)x -
S(f - o)f(a)do
-
S(t - a)ad@‘(o).
It follows that
Consequently,
and the conclusion
follows.
n
We remark that q(t), t E R is a stationary variable X(0, Qm).
process, moreover, q(f) is a Gaussian random
PROPOSITION4.3. Assume that hypotheses 1 and 2 hold. Let q E C,(H), and lets < t < B. Then we have
Proof. By lemma 4.2, it follows that
= lim P,,.cp(x) = ~H WWt(fW, y---m I
Qm)(dy).
n
Markov
transition
semigroups
949
REFERENCES 1. DA PRATO G. & ZABCZYK J., Stochastic equations in infinite dimensions, in Encyclopedia of Mathematics and its Appliculions. Cambridge University Press, Cambridge (I 992). 2. PESZAT S. & ZABCZYK J., Strong property and irreducibility for diffusions on Hilbert spaces, Ann. Prob. (to appear). 3. LIONS J.-L. & PEETRE J., Sur une classe d’espaces d’interpolation, Pub/. Math. 1’I.H.E.S. 19, 5-68 (1964). 4 DA PRATO G. & GRISVARD P., Sommes d’operateurs lineaires et equations differentielles optrationnelles, J. Math. pures appl. 54, 305-387 (1975). 5. DI BLASIO G., Linear parabolic evolution equations in f.p-spaces, Anna/i Mat. puru appl. CXXXVIII, 55-104 (1983. 6. PRUSS J. & SOHR H., On operators with bounded imaginary powers in Banach spaces, Marh. Z. 203, 429-452 (1990). 7. TRIEBEL H., Interpolation Theorv, Funcrion Spaces, llifferentral Operators. North-Holland, Amsterdam (1986).