Kolmogorov equations in Hilbert spaces 1 · Kolmogorov equations in Hilbert spaces 1 ... The ﬁrst goal would be to construct a general theory of second ... space of test functions

Kolmogorov equations in Hilbert spaces 1

March 18, 2010


We are concerned with a Kolmogorov operator in a separableHilbert space H

N0ϕ(x) :=12

Tr [BB∗D2xϕ(x)] + 〈Ax + b(x),Dxϕ(x)〉

where

A : D(A) ⊂ H → H is the infinitesimal generator of astrongly continuous semigroup etA,B ∈ L(H), B∗ is the adjoint of B,b : D(b)⊂ H → H is nonlinear.


If H is infinite dimensional, N0 can be considered as an ellipticoperator (possibly degenerate) with infinitely many variables.

We want to study the parabolic equation

ut (t , x) = (N0u)(t , x), u(0, x) = u0(x) (1)

and the elliptic equation

λϕ− N0ϕ = f , (2)

where λ is a positive number and u0, f are given in suitablefunctional spaces which we shall discuss later.


The first goal would be to construct a general theory of secondorder elliptic and parabolic equations in Hilbert spaces, that iswith infinitely many variables.

This project was essentially started by

Yu. Daleckij, Dokl. 66

and zz

L. Gross, JFA, 67


For further developments see the monographs,

Yu. Daleckij and S. V. Fomin, Kluwer, 91,

S. Cerrai, Springer 01,

Z. M. Ma and M. Röckner, Springer, 92

DP-Zabczyk, Cambridge 02.


One of the main motivation for studying the Kolmogorovequation (1) comes from the stochastic differential equation.

dX = (AX + b(X ))dt + BdW (t),

X (0) = x ∈ H,(3)

Several equations arising in the application are of this form. Forinstance: reaction-diffusion, Burgers or Navier–Stokes equationand many others.


Let us assume first that problem (3) has a unique solutionX (t , x) (in a sense to be specified). Then there is a naturalcandidate for the solution to (1) given by

u(t , x) = Ptϕ(x), x ∈ H, t ≥ 0,

where Pt is the transition semi-group

Ptϕ(x) = E[ϕ(X (t , x))], ϕ ∈ Bb(H).


There are no chances, however, to check in general, even for avery regular ϕ, that u(t , x) = E[ϕ(X (t , x))] is a solution of (1)because X (t , x) is not differentiable enough.

Also it is not a priori clear for which ϕ the expression of N0ϕ ismeaningful, because A and b are not everywhere defined.

So, one of our main concern will be the choice of a suitablespace of test functions D such that N0ϕ is meaningful for eachϕ ∈ D .


We shall see often that Pt acts on Cb(H), which means that it isFeller.

However, Pt it is not a strongly continuous semi group in Cb(H)in general, because for ϕ ∈ Cb(H) the convergence

limt→0

Ptϕ(x) = ϕ(x)

only holds point-wise.

We shall say that Pt is a π-semigroup.


We can generalize, however, the usual definition of infinitesimalgenerator N of Pt , so that the resolvent of N coincides with theLaplace transform of Pt ,

(λ− N)−1f (x) =

∫ +∞

0e−λtPt f (x)dt . (4)

Since Pt is a contraction semigroup in Cb(H), L is m-dissipative,so that its resolvent set includes the half-line (0,+∞).

Notice that the integral in (4) is defined and convergent onlypointwise, that is for each x ∈ H.


Now the problem arises to see the relationship between theabstract operator N and the Kolmogorov operator N0.

We expect that the space of test function D is chosen in such away then N0ϕ coincides with Nϕ for all ϕ ∈ D .

We shall also wish that the restriction of N to D determines N,in other words that D is a core for N.


We also notice that if the space of test functions D is a corethen we can exploit the explicit properties of the differentialoperator N0.

Let us give an example, assuming that D is an algebra ofregular functions.

Then, given ϕ ∈ D we see, by an elementary computation, that

N0(ϕ2) = 2N0ϕϕ+ |B∗Dϕ|2, ∀ ϕ ∈ D . (5)


Assume now that µ is an infinitesimally invariant probabilitymeasure for N0, ∫

HN0ϕdµ = 0, ∀ ϕ ∈ D (6)

Then by (5) we deduce the important Identité du carré duchamps∫

HNϕϕdµ = −1

2

∫H|B∗Dϕ|2dµ, ∀ϕ ∈ D(N).


Another problem which arises now is to study the regularity ofthe strong solution ϕ of (6) following that of f

and also (this is more difficult) to try to characterize the domainof N.

We can follow this program only in particular cases, which weshall discuss later.


We shall also consider another approach, namely to look for arealization of Pt in Lp(H, µ), p ≥ 1, where µ is an invariantmeasure for Pt , that is∫

HPtϕdµ =

∫Hϕdµ, ∀ ϕ ∈ Cb(H).

Then one can uniquely extend Pt to Lp(H, µ) for all p ≥ 1 as astrongly continuous semigroup of contractions, which we stilldenote by Pt , whereas we call Np its infinitesimal generator.


Again the problem arises to find a core of nice functions for Np.As a core we shall often choose the space EA(H) of all linearcombinations of real parts of exponential functions

ϕh(x) = ei〈x ,h〉, h ∈ D(A∗), x ∈ H.

Notice that,

N0ϕh(x) =

(−1

2|B∗h|2 + i〈x ,A∗h〉+ i〈b(x),h〉

)ϕh(x). (7)

So, we must assume at least that∫

H |b(x)|µ(dx) < +∞.


We notice that proving that EA(H) is a core for Np is equivalentto show that the linear operator

N0ϕ(x) =12

Tr [BB∗D2xϕ(x)] + 〈x ,A∗Dxϕ(x)〉

+〈b(x),Dxϕ(x)〉, ϕ ∈ EA(H),

has a unique m–dissipative extension in Lp(H, µ).

We shall prove essential m–dissipativity of N0 for someconcrete example.


The case when problem (3) is ill-posed

There are interesting situations where existence anduniqueness for the stochastic differential equation (3) are notknown.

In this case the transition semigroup Pt is not given in advance.

Then one can try to study directly the Kolmogorov equation

λϕ(x)− 12

Tr [BB∗D2xϕ(x)]− 〈Ax + b(x),Dxϕ(x)〉 = f . (8)


One possibility is to introduce and solve an approximatingequation

λϕn(x)− 12

Tr [BB∗D2xϕn(x)]− 〈Ax + bn(x),Dxϕn(x)〉 = f , (9)

then to find suitable estimates for ϕn and their derivatives,independent of n, and show the convergence of a subsequenceof (ϕn) to a solution, in a suitable sense, of (8).

This method was used in DP–A. Debussche JMPA 03, to proveexistence, but not uniqueness, of a solution of (8) in the case of3D-NS.


A second possibility is to show existence of an infinitesimallyinvariant measure µ of N0 and then to prove the essentialm-dissipativity of N0 in L2(H, µ).

This method was used in DP–Röckner PTRF 02, to study someequations with dissipative and singular coefficients.

If one is able to prove essential m-dissipativity of N0, then onecan try to construct a unique martingale solution of (3).


Finally, an important approach to be mentioned is the one ofDirichlet forms in infinite dimensional spaces started in

S. Albeverio and R. Høegh-Krohn, Z. Wahrsch. verw. Geb, 77

S. Albeverio and M. Röckner, PTRF 91.

See also the monograph

Z. M. Ma and M. Röckner, Springer-Verlag, 92

andW. Stannat, Memoirs AMS, 99.


Plan of the lectures

1) Ornstein–Uhlenbeck operators.

2) Bounded perturbations of O.U.

3) Dissipative perturbations of O.U.

4) Fokker–Planck equations with singular coefficients.

5) Porous media equations.


1 Ornstein–Uhlenbech operators

Here we consider the Kolmogorov operator in H of the form

L0ϕ(x) :=12

Tr [BB∗D2xϕ(x)] + 〈x ,A∗Dxϕ(x)〉

where

A : D(A) ⊂ H → H is the infinitesimal generator of astrongly continuous semigroup etA,B ∈ L(H), B∗ is the adjoint of B,The linear operator

Qt =

∫ t

0esABB∗esA∗ds, t ≥ 0, (10)

is of trace class for any t > 0.


Condition (10) is necessary. In fact, setting u(t , x) = v(t ,etAx)we transform the problem

ut (t , x) = (L0u)(t , x), u(0, x) = u0(x), (11)

in

vt (t , x) =12

Tr [etABB∗etA∗D2x v(t , x)].

This can be easily solved

v(t , x) =

∫H

u0(y)NQt (dy)

where Qt is given by (10).


The stochastic differential equation corresponding to L0 is thefollowing

dX = AXdt + BdW (t),

X (0) = x ∈ H,(12)

which can be solved by the variation of constants formula

X (t , x) = etAx +

∫ t

0e(t−s)AdW (s) =: etAx + WA(t).

WA(t) is called the stochastic convolution.


The stochastic convolution

We are given

(ek )k∈N complete othonormal system in H.

(Ω,F , (Ft )t≥0,P) filtered probability space.

(Wk )k∈N sequence of real independent standard Brownianmotions in (Ω,F , (Ft )t≥0,P).


The we define

WA(t) : =

∫ t

0e(t−s)ABdW (s)

=∞∑

k=1

∫ t

0e(t−s)ABekdWk (s), t ≥ 0.

(13)


Since

E|WA(t)|2 =∞∑

k=1

∫ t

0|e(t−s)ABek |2ds

=

∫ t

0Tr [esABB∗esA∗ ]ds

the series in (13) is convergent.


The Ornstein–Uhlenbeck process

For each x ∈ H we consider the process

X (t , x) := etAx + WA(t), t ≥ 0.

X (t , x) is a Gaussian random variable NetAx ,Qt,

where

Qt =

∫ t

0esABB∗esA∗ds, t ≥ 0,


The corresponding transition semigroup Rt is

Rtϕ(x) : = E[ϕ(X (t , x))]

=

∫Hϕ(y)NetAx ,Qt

(dy)

=

∫Hϕ(etAx + y)NQt (dy), ϕ ∈ Bb,n(H).

(14)


Notation

For any n ∈ N, Bb,n(H) is the Banach space of all Borelfunctions ϕ : H → R such that

‖ϕ‖0,n := supx∈H

|ϕ(x)|1 + |x |n

< +∞.

Cb,n(H) is the closed subspace of Bb,n(H) of all functions ϕsuch that |ϕ(·)|

1+|·|n is uniformly continuous.

It is easy to see that the semigroup Rt acts both on Bb,n(H)and in Cb,n(H).

If n = 0 we write Cb,n(H) = Cb(H) and ‖ϕ‖0,n = ‖ϕ‖0.


Infinitesimal generator of the semigroup Rt

We first consider Rt acting in Cb(H). In this case we have

‖Rtϕ‖0 ≤ ‖ϕ‖0.

If A 6= 0, Rt is not a strongly continuous semigroup.

In fact if ϕh(x) = ei〈x ,h〉 the limit

limt→0

Rtϕh(x) = limt→0

e−12 〈Qt h,h〉ei〈etAx ,h〉 = ϕh(x), x ∈ H,

is not uniform in x for any h 6= 0 (unless A = 0).


What we can say is that for any ϕ ∈ Cb(H) one has

limt→0

Rtϕ(x) = ϕ(x), ∀ x ∈ H

and

supt≥0‖Rtϕ‖0 ≤ ‖ϕ‖0.

We say that Rt is a π-semigroup.


Notation, π-convergence

Let f ∈ Cb(H) and fn ⊂ Cb(H). We say that fn isπ-convergent to f and write fn

π→ f if

limn→∞

fn(x) = f (x), ∀ x ∈ H

supn∈N ‖fn‖0 <∞.


π-convergence was called bp-convergence in

E.B. Dynkin, Th. Probab. Appl.,1956.,

and

S.N. Ethier and T. Kurtz, Wiley 1986.

see also

B. Goldys and M. Kocan, JDE 2001, were a suitable topologycorresponding to π-convergence was constructed.


The infinitesimal generator of Rt

We define the infinitesimal generator L of Rt in Cb(H) asfollows. The domain D(L) of L is the set of all ϕ ∈ Cb(H) suchthat

There exists

limε→0

1ε

(Rεϕ(x)− ϕ(x)) =: Lϕ(x)

for all x ∈ H.

supε∈(0,1]

1ε‖Rεϕ− ϕ‖0 <∞.


We are going to study the resolvent set ρ(L) and the resolventR(λ,L) = (λ− L)−1 of L.

Results and proofs are straighforward generalizations of theclassical Hille–Yosida theorem (the difference being that Rt isnot strongly continuous), and so they will be only sketched.

For details see E. Priola, Studia Math., 1999.


Proposition 1

Let L be the infinitesimal generator of Rt in Cb(H). Then(i) (0,+∞) ⊂ ρ(L) and we have

R(λ,L)f (x) =

∫ +∞

0e−λtRt f (x)dt , f ∈ Cb(H), λ > 0, x ∈ H.

(Note that the integral below is only pointwise defined.)Moreover,

‖R(λ,L)f‖0 ≤1λ‖f‖0, λ > 0, f ∈ Cb(H).

(ii) If f ∈ Cb(H) and fn ⊂ Cb(H) is a sequence such thatfn

π→ f , we have R(λ,L)fnπ→ R(λ,L)f .


Proof

Let f ∈ Cb(H). Write for any λ > 0 and any x ∈ H

F (λ)f (x) =

∫ +∞

0e−λtRt f (x)dt .

It is easy to check that F (λ)f ∈ Cb(H), that

ddh

RhF (λ)f (x)|h=0 = λ

∫ +∞

0e−λsRsf (x)ds − f (x)

= λF (λ)f (x)− f (x),


and that the family of linear operators (∆h)h∈(0,1]

∆hF (λ)f :=1h

(RhF (λ)f − F (λ)f )

is equi-bounded in sup norm. This implies that F (λ)f ∈ D(L)and (λ− L)F (λ)f = f

Analogously one sees that if ϕ ∈ D(L) we haveF (λ)(λ− L)ϕ = ϕ.

This will achieve the proof that ρ(L) ⊃ (0,+∞).

Finally, (ii) is straightforward.


RemarkOne can make similar considerations by replacing space Cb(H)with Cb,n(H).

When some confusion may arise we will denote by

D(L,Cb,n(H))

the domain of the generator L of semigroup Rt acting inCb,n(H).

For details see DP, Birkhäuser, 2004.


Identification of L

It seems to be difficult to characterize the domain of L in Cb(H).

So, we shall try to find a subset D of Cb(H) with the followingproperties

L looks like an explicit differential operator on D .D is stable for Rt

For any ϕ ∈ D(L) there is a two-indices sequence(ϕn1,n2) ⊂ D , π- convergent to ϕ and such that (Lϕn1,n2) isπ- convergent to Lϕ.

We shall call such a set Y a core of L.


Exponential functions

We denote by EA(H) the linear span of the real parts offunctions

ϕh(x) := ei〈x ,h〉, x ∈ H

with h ∈ D(A∗).Recalling the Fourier tranform formula of a Gaussian measurewe see that

Rtϕh(x) = ei〈x ,etA∗h〉e−12 〈Qt h,h〉, h ∈ H.

Therefore EA(H) is stable for Rt .


Proposition 2

Let ϕ ∈ Cb(H), k = 0,1. Then there exists a double sequence(ϕm,n) ⊂ EA(H), π- convergent to ϕ in Cb(H).

Proof. The result is well known in finite dimensions (with anone index sequence).

In infinite dimensions we approach ϕ(x) by ϕ(Pnx) where (Pn)is a sequence of finite dimensional projectors convergent to theidentity.


Now we can easily see that

EA(H) ⊂ D(L,Cb,1(H))

and

Lϕ =12

Tr [BB∗D2ϕ] + 〈x ,A∗Dϕ〉, ∀ ϕ ∈ D(L,Cb,1(H)). (15)


Proposition 3

EA(H) is a core for (L,D(L,Cb,1(H))).

Proof. EA(H) is stable for Rt and π-dense in Cb,1(H). The proofis an arrangement of a classical result on C0-semigroups.

See DP-L. Tubaro, Czech. Math. Journal, 01, for details.


Remark

Obviously

EA(H) /∈ D(L,Cb(H)).

One can see, however, that integrals of exponential functionsbelong to D(L,Cb(H)). More precisely, let IA(H) be the linearspan of real parts of all functions of the form

∫ a

0g(s)ei〈esAx ,h〉ds : a > 0, h ∈ D(A∗), g ∈ C1([0,a])).

then one can check that IA(H) is a core for D(L,Cb(H)).


Notice only that if

ϕ(x) =

∫ a

0ei〈esAx ,h〉ds,

we have

〈Dϕ(x), z〉 = i∫ a

0ei〈esAx ,h〉〈esAz,h〉ds,

so that

〈Dϕ(x),Ax〉 =

∫ a

0ei〈esAx ,h〉ds −

∫ a

0ei〈x ,h〉ds.


Hypoellipticity of L

Here we consider the operator L acting in Cb(H). We call Lhypoelliptic if

ϕ ∈ Bb(H), t > 0⇒ Rtϕ ∈ C∞b (H).

Theorem 4L is hypoelliptic if and only if

etA(H) ⊂ Q1/2t (H), ∀ t > 0. (16)


Comments on the condition etA(H) ⊂ Q1/2t (H)

Condition (16) is equivalent to null controllability (in any times)of the following deterministic system

X ′(t) = AX (t) + Bu(t), X (0) = x . (17)

System (17) is called null controllable in time T > 0 if for anyx ∈ H there exists u ∈ L2(0,T ; H) such that X (T ) = 0.


Moreover the minimal needed energy for driving X to 0 in timeT is precisely given by |ΛT x |2, where

ΛT := Q−1/2T eTA. (18)

See J. Zabczyk Institute of Mathematics, Polish Academy ofSciences, 1981

and

DP-J. Zabczyk, Cambridge 02.


The case when L is elliptic

We say that L is elliptic if (BB∗)−1 ∈ L(H).

In this case L is hypoelliptic, that is condition (16) is fulfilled.Take in fact for simplicity B = I and set

u(t) = − xT

etA, t ∈ [0,T ].

Then

X (T ) = eTAx − 1T

∫ T

0e(T−s)AesAds = 0.

So, system (17) is null controllable and

‖Λt‖ ≤1√t, ∀ t > 0.


Proof of Theorem 4

We shall only sketch the proof of the “if” part, for details see

DP–J.Zabczyk, Cambridge, 02.

So, we assume that

etA(H) ⊂ Q1/2t (H)

and take ϕ ∈ Bb(H). In view of the Cameron–Martin theoremwe have

NetAx ,Qt<< NQt


Setting now

ρ(x , y) :dNetAx ,Qt

dNQt

(y) = e−12 |Λt x |2+〈Λt x ,Q

−1/2t y〉,

we have

Rtϕ(x) =

∫H

e−12 |Λt x |2+〈Λt x ,Q

−1/2t y〉ϕ(y)NQt (dy).

Now the conclusion follows by straightforward computations.


Regularity results

We are here concerned with the elliptic Kolmogorov equation

λϕ− Lϕ = f , (19)

where λ > 0 and f ∈ Cb(H).

Since the resolvent set of L includes (0,+∞), equation (19) hasa unique solution ϕ ∈ D(L).

We are going to study the regularity of ϕ.


Regularity results

We need the following new assumption.

Hypothesis

There exists α ∈ [1/2,1) and cα > 0 such that

‖Λt‖ ≤ cαt−α, ∀ t ≥ 0. (20)

We have seen that this assumption is fulfilled when L is elliptic,that is when (BB∗)−1 ∈ L(H), with α = 1

2 .


Proposition 5Assume that (20) is fulfilled.

Let λ > 0, f ∈ Cb(H) and let ϕ be the solution of equation (19).Then ϕ ∈ C1

b(H) and there exists Kα > 0 such that

|Dϕ(x)| ≤ Kαλα−1 ‖ϕ‖0, ∀ x ∈ H. (21)

Consequently D(L) ⊂ C1b(H).


Proof

It follows by taking the Laplace transform in the formulae

〈DRtϕ(x),h〉 =

∫H〈Λth,Q

−1/2t y〉ϕ(etAx + y)NQt (dy),

|DRtϕ(x)| ≤ ‖Λt‖‖ϕ‖0, ∀t > 0.


The elliptic equation can be studied in f ∈ Cb,1(H)) which minormodifications.Let us notice, for further use the following result.

Proposition 6Assume that hypotheses (16) and (20) hold.For each ϕ ∈ D(L,C1,b(H)) there exists a two indices sequence(ϕm,n) ∈ EA(H) such that

(i) limm→∞

limn→∞

ϕm,n(x) = ϕ(x), limm→∞

limn→∞

Lϕm,n(x) = Lϕ(x) andlim

m→∞lim

n→∞Dϕm,n(x) = Dϕ(x) for all x ∈ H.

(ii) |ϕm,n(x)|+ |Lϕm,n(x)|+ |Dϕm,n(x)| ≤ ‖ϕ‖b,1(1 + |x |), for allx ∈ H.


Documents

Kolmogorov equations in Hilbert spaces 1 · Kolmogorov equations in Hilbert spaces 1 ... The ﬁrst goal would be to construct a general theory of second ... space of test functions