Table Of ContentOrnstein-Uhlenbe
k Equations with
9
0 time-dependent
oe(cid:30)
ients
0
2 and Lévy Noise in (cid:28)nite and in(cid:28)nite dimensions
n
a
Florian Knäble
J
9 Universität Bielefeld
1 E-mail: f.knaeblegooglemail.
om
]
R January 2009
P
.
h
Abstra
t
t
a
We solve a time-dependent linear SPDE with additive Lévy noise in
m
the mild and weak sense. Existen
e of a generalized invariant measure
[ fortheasso
iated transitionsemigroLu2pisestablishedandthegeneratoris
hara
terizedonthe
orresponding -spa
e. Thesquare(cid:28)eldoperatoris
1
al
ulated, allowing to derive a Poin
aré and a Harna
kinequality.
v
7
8
8 1 Introdu
tion
2
. Re
ently, there seems to be a growing interest in the study of semilinear non-
1
0 autonomous sto
hasti
evolution equations in in(cid:28)nite dimensional spa
es, see
9 e.g. [20℄ and the referen
es therein. In the so-
alled semigroup approa
hwhi
h
0 we also pursue in this paper, this is of
ourse
losely
onne
ted to solving the
:
v underlyingnon-autonomousCau
hyproblem,
overedbythetheoryofevolution
i semigroups or evolution families, see e.g. [16℄. So, let us
onsider the following
X
equation:
dX = (A(t)X +f(t))dt+B(t)dZ
r t t t
a X(s) = x (1)
(cid:26)
H A :H H
t
onaHilbertspa
e ,where → arelinearoperatorsandall
oe(cid:30)
ients
are T-periodi
.
Fromamathemati
alpointofview it isnaturalto study (1) withan additional
Lips
hitz non-linearity, be
ause this
ase
an still be
overed if one atta
ks
equation (1) using
ontra
tion methods (e.g. Pi
ard-Lindelöf iteration). Our
main interest, however, in the linear
ase (1), is to exploit a lot of expli
it
formulae (e.g. for the transition semigroup, whi
h are in fa
t time-dependent
versionsofgeneralizedMehlerformulae)togetaspre
iseaspossibleinformation
about the solution and the
orresponding Kolmogorov equations. Con
erning
A
t
the time-dependent (possibly unbounded) operators , for simpli
ity, we only
1
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
onsider the
ase of a
ommon domain. The general
ase is te
hni
ally harder,
butweHexp=e
RtdourreZsultstoholdtrue. Thiswillbethesubje
toffurtherstudy.
t
For and ad-dimensionalBrownianmotion,(1)wasstudiedinten-
sivelyin[7℄. Inspiredbytheirpaper,ourwork
onsistsprimarilyingeneralizing
H Z
t
theirresultstothe
ase in(cid:28)nite-dimensionaland aLévypro
ess. Anumber
of our arguments are adapted from [7℄, although the Lévy setting for
es us to
work more heavily with Fourier transforms and the in(cid:28)nite dimensional setting
requires extra
are. Nevertheless, we su
eed in proving the following results:
existen
eanduniquenessofthesolution,expli
it
al
ulationofits
hara
teristi
fun
tion, proof of the Chapman-Kolmogorov equations, existen
e and unique-
ness of an evolution system of measures, existen
e and form of the generator
(in
luding a result on its spe
trum), pre
ise form of the square (cid:28)eld operator,
proof of a Poin
aré and a Harna
k inequality.
In the autonomous
ase, the semigroups asso
iated to this kind of equation
areknownasthe generalized Mehlersemigroups mentioned aboveand they are
already well understood. Invariant measures are established in [5℄ and [10℄,
generators are examined in [11℄ and the square (cid:28)eld operator is identi(cid:28)ed in
[12℄.
The paper is organized as follows. In Se
tion 2 we shortly review the Lévy-
Ito de
omposition and the Lévy-Khin
hine representation. In Se
tion 3 we
develop the ne
essarytheory of integration to givesense to our solution, whose
existen
e is established in a rather standard way. Then, in Se
tion 4, we
al-
ulate the Fourier transforms of oursolutions and thus determine its transition
semigroupexpli
itly, whi
h is a two-parametersemigroup, sin
e the equationis
non-autonomous. In this
ase the
on
ept of an invariant measure has to be
generalized to allow for a whole
olle
tion of measures - a so-
alled evolution
system of measures- whi
h are invariant in an appropriatesense. We prove the
A
t
existen
e of su
h a system under the
ondition that the generate an expo-
nentially stable semigroup and provided that the Lévy symbol is su(cid:30)
iently
smooth. In
ontrastto the Gaussiansetting, the
onstru
tionoflimit measures
is more deli
ate in our
ase.
Then (as usual) we turn the problem into an autonomous one by enlarging
the state spa
e, allowing for a one-parameter semigroup. Via the evolution
system of measuresand thanks to the periodi
ity of the
oe(cid:30)
ients weareable
to
onstru
t a unique invariant measure for this semigroup as in [7℄. Thus we
L2
an introdu
e the -spa
e with respe
t to the invariant measure where the
semigroup is then shown to be strongly
ontinuous.
Se
tion5isdedi
atedtoananalysisofthegenerator. Weidentifyanexpli
it
domain of uniqueness for the generator and its a
tion on it, thus identifying it
as an pseudo-di(cid:27)erential operator in in(cid:28)nitely many variables. Subsequently,
in Se
tion 6, we establish the form of its square (cid:28)eld operator whi
h yields a
generalizationtothe
ru
ial(cid:16)integrationbypartsformula(cid:17) in[7℄. Thenweprove
an estimate for the square (cid:28)eld operator, that allows us to obtain a Poin
aré
and a Harna
k inequality for our semigroup.
A
knowledgement I wish to thank Prof. Dr. Rö
kner for introdu
ing me to
this interesting subje
t and for many helpful suggestions.
2
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
2 Lévy Pro
esses
H , :=
In the following let be a separable Hilbert spa
e with s
alar produ
t h· ·i
, := H
H H
h· ·i and norm k·k k·k . An -valued sto
hasti
pro
ess L adapted to a
( )
t t 0
(cid:28)ltration F ≥ is
alled a Lévy pro
ess if it has independent and stationary
P(L = 0) = 1
in
remAents,(His)sto
hasti
ally
ontinu0ou/sA,¯and we have . We say
that ∈B is bounded below if ∈
N(t,A) A t
We denote by the (random) number of "jumps of size " up to time ,
N(t,A):= 0 s t∆L A
s
that is
ard{ ≤ ≤ | ∈ }
A N(t,A) M(A)
If isMbo(uAn)d:e=dEbe[Nlow(1,,tAh)e]n is a Poisson pro
ess, with intensity ,
where .
M H
De(cid:28)nition 2.1 A measure on with :
min(1, x 2)M(dx)<
k k ∞
ZH\{0}
is
alled a Lévy measure.
L H
Theorem 2.2 (Lévy-Ito De
omposition) If is an -valuedLévy pro
ess,
b H R W H W
R R
there is a drift ve
tor ∈ , a -Wiener pro
ess on , su
h that is
N (A) A
t
independent of for any that is bounded below and we have:
L =bt+W (t)+ x(N (dx) tM(dx))+ xN (dx)
t R t t
−
Zkxk<1 Zkxk≥1
N L M
t
where is the Poisson random measure asso
iated to , and the
orre-
sponding Lévy measure.
(cid:4)
Proof See e.g. [1℄ Theorem 4.1
L H
Theorem 2.3 (Lévy-Khin
hine Representation) If isan -valuedLévy
pro
ess with LévEy-[eItihoLtd,ue
i]om=peotsλi(tui)on as in 2.2, then its
hara
teristi
fun
tion
takes the form: and
1
λ(u)=i b,u u,Ru + ei u,x 1 i u,x χ M(dx)
h i− 2h i ZH/{0}h h i− − h i {kxk≤1}i (2)
(cid:4)
Proof See [13℄ Theorem 5.7.3
Sin
e a measure is
hara
terized by its Fourier transform we will say that a
µ [b,R,M]
measure is asso
iated to a triple if its
hara
teristi
exponent has
the form (2).
Remark 2.4 A
tually the Lévy-Khin
hine representation holds not only for
Lévy pro
esses but for any in(cid:28)nitely divisible random variable. (See [18℄ for
3
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
an a
ount of in(cid:28)nite divisibility) Moreover, Lévy pro
esses and in(cid:28)nitely divis-
ible measures
an be brought in a one to one
orresponden
e. In parti
ular the
onverse of 2.3 is true: any fun
tion of the form
1
exp i b,u u,Ru + ei u,x 1 i u,x χ M(dx)
( h i− 2h i ZH/{0}h h i− − h i {kxk≤1}i )
is the
hara
teristi
fun
tion of a measure.
3 Solving the generalized Ornstein-Uhlenbe
k equa-
tion
3.1 Sto
hasti
Integration with respe
t to Lévy martin-
gale measures
In this subse
tion we follow [2℄, where the proofs of all results
an be found.
H
1
Let be R the ring of all Borel subsets of the unit ball of whi
h arebounded
below.
H
De(cid:28)nMit:ioRn 3.1 A LéΩvy mHartingale measure on a Hilbert spa
e is a set fun
-
+ 1
tion ×R × → satisfying:
M(0,A)=0 A
1
• almost surely for all ∈R
M(t, )=0
• ∅ almost surely
M(t,A B) = M(t,A)+M(t,B) t
• almost surely we have: ∪ for all and all
A B
1
disjoint , ∈R
M(t,A) A
t 0 1
• { ≥ } is a square-integrable martingale for ea
h ∈R
A B = M(t,A) M(t,B)
t 0 t 0
• if ∩ ∅ { ≥ } and { ≥ } are orthogonal, that is:
M(t,A),M(t,B) A,B
1
h i is a real-valued martingale for every ∈R
sup E[ M(t,A) 2] A (S ) < n N
n
• { k k | ∈B } ∞ for every ∈
A A (S )
j j n
• for everjy sequen
e dliem
reasinEg[tMo t(hte,Aem)p2ty]<set su
h that ⊂ B
j j
for all we have: →∞ k k ∞
s < t A M(t,A) M(s,A)
1
• for every and every ∈ R we have that − is
s
independent of F
M(t,A) = xN˜ (dx) H
Proposition 3.2 A t is a Lévy martingale measure on
A
1
for every ∈R . R
Similarly as a Wiener pro
ess is
hara
terized by its
ovarian
e operator, we
andes
ribethe
ovarian
estru
tureof aLévymartingalemeasureby afamily
1
of operators parametrized by our ring R .
4
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
Proposition 3.3
E[ M(t,A),v 2]=t v,T v
A
|h i| h i
t 0, v H A T
1 A
for all ≥ ∈ ∈R , where the operators are given by
T v := T vν(dx) T v := x,v x
A A x and x h i .
R
We will establish only a limited theory of integration, as for our purposes it
will be su(cid:30)
ient to integrate deterministi
operator valued fun
tions. We do
not even need them to depend on the jump size. The pro
edure is the same
as for Brownian motion, so let us introdu
e the spa
e of our integrands, the
approximatingsimple fun
tions, and state howthe integralis de(cid:28)ned forthem.
M([s,t],A):=M(t,A) M(s,A)
For
onvenien
e, we set − .
H
′
De(cid:28)nition 3.4 Let be another real separable Hilbert spa
e.
2 := 2(T ,T ) R : [T ,T ] (H,U) R
+ +
Let H H − be the spa
e of all − → L su
h that
is strongly measurable and we have:
1
T+ 2
R := tr(R(t)T R (t))ν(dx)ds <
2 x ∗
k kH ZT− Zkxk<1 ! ∞
R 2
Let S be the spa
e of all ∈H su
h that
n
R= R χ χ
i (ti,ti+1] A
i=0
X
T = t < t < ... < t = T n N R
0 1 n+1 + i
where − for some ∈ , where ea
h ∈
(H,H ) A
′
L and where ∈R
R , t [T ,T ]
+
For ea
h ∈S ∈ − de(cid:28)ne the sto
hasti
integral as follows:
n
I (R):= R M([t t,t t],A)
t i i i+1
∧ ∧
i=0
X
2
Proposition 3.5 The spa
e H with inner produ
t
T+
R,U := tr(R(t)T U (t))ν(dx)ds
x ∗
h i
ZT− Zkxk<1
is a Hilbert spa
e.
2
Proposition 3.6 The spa
e S is dense in H .
R :E[I (R)]=0
t
Proposition 3.7 We have for any ∈S and
t
E[ I (R) 2]= (R(s)T R (s))ν(dx)ds = χ R(t) 2
k t k ZT−ZAtr x ∗ k [T−,t] kH2
t I : L2(Ω, ,P;H)
t
So for (cid:28)xed, S → F is an isometry.
I 2
t
So we
an isometri
ally extend the operator from S to its
losure H .
5
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
3.2 Sto
hasti
Convolution
We want to give meaning to the integral
t
X := U(t,r)B(r)dL(r)
U,B
Zs
L H
whi
h we will
all a sto
hasti
onvolution. Here is a -valued Lévy pro
ess
U(t,r) (H),B(r) (H) s r t
and we have ∈ L ∈ L ∀ ≤ ≤ . In anti
ipation of the
assumptions in se
tion 4 we will pose the following
onditions:
sup B(r) <
• r∈Rk kL(H) ∞
M >0,ω >0 U(t,r) <Me ω(t r)
(H) − −
• there is su
h that : k kL
r B(r) r U(t,r) t
• 7→ is measurable and 7→ is measurable for any (cid:28)xed
U B
Proposition 3.8 If and are as above, the sto
hasti
onvolution exists in
the following sense:
t
U(t,r)B(r)dL(r)
Zs
t t
= U(t,r)B(r)b dr+ U(t,r)B(r)x N (dx)
r
Zs Zs Zkxk≥1 (3)
t t
+ U(t,r)B(r)dW (r)+ U(t,r)B(r)x N˜ (dx)
Q r
Zs Zs Zkxk<1
U(t,s)=
Proof The proof is analogous to the one of Theorem 6 in [2℄ where
S(t s) S
− for a strongly
ontinuous semigroup : The (cid:28)rst term in (3) is well
de(cid:28)ned as a simple Bo
hner integral, and the se
ond as a (cid:28)nite random sum.
For the other two terms, it is straightforward to
he
k under our assumptions,
(cid:4)
that the integrands are su
h that the respe
tive isometries apply.
3.3 Existen
e of the Mild Solution
In the following we will have to deal with a non-autonomous abstra
t Cau
hy
problem- non-autonomousmeanswearenot in the frameworkofstrongly
on-
tinuous semigroups anymore. This implies in parti
ular, that we have no easy
hara
terizationofwell-posednessinthesenseoftheHille-Yosidatheoremavail-
able. There are di(cid:27)erent, yet te
hni
al, approa
hes (see [15℄ and the referen
es
therein for a re
ent overview), but sin
e this subje
t is not in the primary in-
terest of our work, we assume that the problem is well posed. This is
losely
related to the notion of evolution semigroups. Our de(cid:28)nition is taken from [4℄
We
onsider the following non-autonomous generalisation of the Langevin
equation:
dX = (A(t)X +f(t))dt+B(t)dL
t t t
X(s) = x (4)
(cid:26)
6
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
B : R (H)
wfh:eRre H → L is Lst(rto)ngly
oHntinuous and bounded in operator noAr(mt),
→ is
ontinuous, is an -valued Lévy-pro
ess and where the
H D(A)
aAre:lRinearDo(pAe)ratorHs on with
ommon domain and
× → is su
h that we
an solve the asso
iated non-autonomous
abstra
t Cau
hy problem
dX = (A(t)X +f(t))dt
t t
X(s) = x (5)
(cid:26)
a
ording to the following de(cid:28)nitions:
H
De(cid:28)nition 3.9 An exponentially bounded evolution family on is a two pa-
U(t,s) H
t s
rameterfamily { } ≥ of boundedlinearoperators on su
h thatwe have:
U(s,s)=Id U(t,s)U(s,r)=U(t,r) r s t
(i) and whenever ≤ ≤
x H (t,s) U(t,s)x s t
(ii) for ea
h ∈ , 7→ is
ontinuous on ≤
M >0 ω >0 U(t,s) Me ω(t s) , s t
− −
(iii) there is and su
h that : k k≤ ≤
Assumption 3.10 There is a unique solution to (5) given by an exponentially
U(t,s)
bounded evolution family so that the solution takes the form:
t
X =U(t,s)x+ U(t,r)f(r)dr
t
Zs
Moreover, we assume that :
d
U(t,s)x=A(t)U(t,s)x
dt
A
t
Remark 3.11 Note that in the (cid:28)nite dimensional
ase, where ea
h is au-
tomati
ally bounded we get the existen
e of an evolution family that solves (5),
t A
t
under the reasonable assumption that 7→ is
ontinuous and bounded in the
operator norm, by solving the following matrix-valued ODE:
∂ U(t,s)=A(t)U(t,s)
∂t
U(s,s)=Id
(cid:26)
(t,M) A(t)M
Existen
eanduniquenessareassuredsin
e 7→ isglobally Lips
hitz
M
in . This result even holds in in(cid:28)nite dimensions, see [6℄.
De(cid:28)nition 3.12 Given assumption 3.10 we
all the pro
ess:
t t
X(t,s,x)=U(t,s)x+ U(t,r)f(r)dr+ U(t,r)B(r)dL
r
Zs Zs
a mild solution for (4).
7
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
3.4 Existen
e of the Weak Solution
Wehave
alledtheaboveexpressionamildsolution,thoughthereisnoobvious
relation to the equation yet. Now, we will show that our
andidate solution
a
tually solves our equation in the weak sense. The following de(cid:28)nition makes
this pre
ise, but (cid:28)rst we need to strengthen our assumption
on
erning the
A(t)
ommon domain of the a little:
A (t)
∗
Assumption 3.13 Werequirethattheadjointoperators alsohavea
om-
t D(A )
∗
mon domain independent of whi
h we will denote by . Furthermore, we
D(A ) H
∗
assume that is dense in and that we have:
d
U (t,s)y =U (t,s)A y
dt ∗ ∗ ∗t
y D(A )
∗
for every ∈ .
H X
t
De(cid:28)nition 3.14 An -valued pro
ess is
alled a weak solution for (4) if
y D(A )
∗
for every ∈ we have:
t t t
X ,y = x,y + X ,A y dr+ f(r),y dr+ B (r)ydL
h t i h i h r ∗r i h i ∗ r (6)
Zs Zs Zs
(B (r)y)(h) := B (r)y,h B (r)y (H,R)
∗ ∗ ∗
Here h i so that ∈ L and the integral is
well de(cid:28)ned, sin
e 1
B y 2 (t s)sup B(r) 2 y 2 T2e 2ν(dx)<
k ∗ kH2 ≤ − rk k k k k kxk<1k x kk ∞.
P R
X
t
Theorem 3.15 The mild solution from de(cid:28)nition 3.12 is also a weak solu-
tion for (4). Moreover, it is the only weak solution.
Proof Analogous to [2℄ Theorem 7. After some relatively straightforward
al-
ulations the problem is redu
ed to proving the following equality:
t t r t
U(t,r)B(r)dL ,y = U(r,u)B(u)dL ,A y dr+ B (u)ydL
r u ∗r ∗ u
(cid:28)Zs (cid:29) Zs (cid:28)Zs (cid:29) Zs
(7)
and we will do so with the help of two lemmas.
(M, ,µ)
Proposition 3.16 [sto
hasti
Fubini℄ Let be M a measure spa
e with
µ G2(M) (H,H ) R
′
(cid:28)nite. By denote the spa
e of all L - valued mappings on
[s,t] M (r,m) R(r,m)y y H
R 2× s:=u
htthat R(r,m7→)T21 2ν(dx)iµs(dmmea)sdurr<able for ea
h ∈ and
k kG2(M) s Mk x k ∞ Then we have:
R Rt t
R(u,m)dL µ(dm)= R(u,m)µ(dm) dL
u u
ZM(cid:18)Zs (cid:19) Zs (cid:18)ZM (cid:19)
Proof see [2℄ Theorem 5
8
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
R 2 y H
Lemma 3.17 Let be ∈H and ∈ . Then we have:
t t
R(r)dL ,y = R (r)ydL
r ∗ r
(cid:28)Zs (cid:29) Zs
(cid:4)
Proof see [2℄ Theorem 4
Now we are able to (cid:28)nish our proof of 3.15:
t r t r
U(r,u)B(u)dLu,A∗ry dr 3=.17 B∗(u)U∗(r,u)A∗ry dLu dr
Zs (cid:28)Zs (cid:29) Zs (cid:18)Zs (cid:19)
t t t t d
3=.16 B∗(u)U∗(r,u)A∗ry dr dLu = B∗(u) drU∗(r,u)y dr dLu
Zs (cid:18)Zu (cid:19) Zs (cid:18) Zu (cid:19)
t t
= B∗(u)[U∗(t,u) Id]y dLu 3=.17 [U(t,u) Id]B(u)dLu,y
− −
Zs (cid:28)Zs (cid:29)
t t
= U(t,u)B(u)dL ,y B(u)dL ,y
u u
−
(cid:28)Zs (cid:29) (cid:28)Zs (cid:29)
(cid:4)
and that is pre
isely what we had to show.
4 Semigroup and Invariant Measure
T >0
Assumption 4.1 From now on, we assume that there exists su
h that
A,f B T
the
oe(cid:30)
ients and in (4) are -periodi
.
Re
all that the weak solution for (4) takes the following form:
t t
X(t,s,x)=U(t,s)x+ U(t,r)f(r)dr+ U(t,r)B(r)dL
r
Zs Zs
As opposed to the Gaussian
ase we are no longer able to give an easy
X(t,s,x)
representationofthelawof ,butwe
an
al
ulateitsFouriertransform:
Lemma 4.2 (
hara
teristi
fun
tion)
E[exp(i h,X(t,s,x) )]=
h i
t t
exp i h,U(t,s)x+ U(t,r)f(r)dr exp λ(B (r)U (t,r)h)dr
∗ ∗
(cid:26) (cid:28) Zs (cid:29)(cid:27) (cid:26)Zs (cid:27)
λ L
where is the Lévy symbol of .
Proof : Straightforward,byusingtheisometriestoapproximatethesto
hasti
integral by a sum, and then using independen
e of in
rements and the Lévy-
(cid:4)
Khin
hin formula. Details are in
luded in the appendix.
The following lemma is a straightforward generalization of the standard
monotone
lass theorem.
9
F. Knäble Non-Autonomous Ornstein-Uhlenbe
k Equations
Lemma 4.3 (
omplex monotone
lasses) Let H be a
omplex ve
tor spa
e
of
omplex-valued bounded fun
tions, that
ontains the
onstants and is
losed
under
omponentwise monotone
onvergen
e. Let M⊂H be
losed under mul-
σ( )
tipli
ation and
omplex
onjugation. Then, all bounded M -measurable fun
-
tions belong to H.
The last and the next result in
ombination will be parti
ularly useful:
:= ei h,x ,h H
h i
Lemma 4.4 The fun
tions M { ∈ } form a
omplex multipli
a-
σ H
tive systems that generates the Borel -algebra of .
Proof It isobviousthat M is
losedunder multipli
ation and
omplex
onju-
gation.
σ( ) = (H)
To show that indeed M B we make use of the following lemma: (see
[19℄ page 108)
X
Lemma 4.5 A
ountable family of real-valued fun
tions on a Polish spa
e
X X
separating the points of already generates the Borel-sigma-algebra of .
Our
ountable family will be {fn,k(x) := sin(hn1ek,xi)}k,n∈N ⊂ M where {ek}
H
is an orthonormal basis of .
S1ine
e,xth)e sine fun
tion is inje
tHive in a neighfborhoodof zero, and the fun
tions
hn k i separatethepointsof ,sodothe n,k. Asrealandimaginarypartsof
fun
tionsinM,itis
lear,thatthesigma-algebrageneratedbythemisin
luded
σ( ) (cid:4)
in M .
Now we will show that our solution indu
es a two-parameter semigroup,
de(cid:28)ned as follows:
f :H C
De(cid:28)nition 4.6 Whenever → is measurable and bounded, de(cid:28)ne
P(s,t)f(x):=E[f(X(t,s,x))]
P(s,t)
will be
alled the two-parameter semigroup (asso
iated to the solution
X
).
f P(s,t)
Lemma 4.7 For as above, we have the following (cid:29)ow property, i.e.
satis(cid:28)es the Chapman-Kolmogorov equations:
P(r,s)P(s,t)f(x)=P(r,t)f(x)
f (x) = ei h,x
h h i
Proof We will show the equality for the fun
tions and extend
it with the help of 4.3. First note, that by 4.2 we have
t t
P(s,t)fh(x)=exp i h,U(t,s)x+ U(t,r)f(r)dr + λ(B∗(r)U∗(t,r)h)dr
(cid:26) (cid:28) Zs (cid:29) Zs (cid:27)
so that:
P(r,s)P(s,t)f (x)=E[P(s,t)f (X(s,r,x)]
h h
=E[exp i U (t,s)h,X(s,r,x) ]
∗
{ h i}
t t
exp i h, U(t,r)f(r)dr + λ(B (r)U (t,r)h)dr
∗ ∗
×
(cid:26) (cid:28) Zs (cid:29) Zs (cid:27)
10
