Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

0.27 MB·English

by  Giuseppe Da Prato

Singular Stochastic Equations on
Hilbert Spaces: Harnack Inequalities
for their Transition Semigroups
9
0
0
Giuseppe Da Prato ,
2 ∗
n Scuola Normale Superiore di Pisa, Italy
a
J

3 Michael R¨ockner
†
1
Faculty of Mathematics, University of Bielefeld, Germany

]
R and
P
Department of Mathematics and Statistics,
.
h
Purdue University, W. Lafayette, 47906, IN, U. S. A.
t
a
m
[ Feng-Yu Wang
‡

2
School of Math. Sci. and Lab. Math Com. Sys.
v
1 Beijing Normal University, 100875, China
6
0 and
2
. Department of Mathematics, Swansea University,
1
1 Singleton Park, SA2 8PP, Swansea, UK
8
0
:
v
i
X
Abstract
r
a
We consider stochastic equations in Hilbert spaces with singular
drift in the framework of [7]. We prove a Harnack inequality (in the
senseof [18])forits transition semigroup andexploitits consequences.
∗Supported in part by “Equazioni di Kolmogorov” from the Italian “Ministero della
Ricerca Scientifica e Tecnologica”
†Supported by the DFG through SFB-701 and IRTG 1132, by nsf-grant 0603742 as
well as by the BIBOS-ResearchCenter.
‡The corresponding author. Supported in part by WIMCS, Creative Research Group
Fund of the National Natural Science Foundation of China (No. 10721091) and the 973-
Project.
1
In particular, we prove regularizing and ultraboundedness properties
of the transition semigroup as well as that the corresponding Kol-
mogorov operator has at most one infinitesimally invariant measure
µ (satisfying some mild integrability conditions). Finally, we prove
existence of such a measure µ for non-continuous drifts.
2000 Mathematics Subject Classification AMS: 60H15, 35R15, 35J15,
Key words: Stochasticdifferential equations, Harnack inequality, monotone
coefficients, Yosida approximation, Kolmogorov operators.
1 Introduction, framework and main results
In this paper we continue our study of stochastic equations in Hilbert spaces
withsingulardriftthroughitsassociatedKolmogorovequationsstartedin[7].
The main aim is to prove a Harnack inequality for its transition semigroup
in the sense of [18] (see also [1, 16, 19] for further development) and exploit
its consequences. See also [14] for an improvement of the main results in
[16] concerning generalized Mehler semigroups. To describe our results more
precisely, let us first recall the framework from [7].
Consider the stochastic equation
dX(t) = (AX(t)+F(X(t)))dt+σdW(t)
(1.1)

X(0) = x H.

∈
Here His a real separable Hilbert space with inner product , and norm
h· ·i
, W = W(t), t 0, is a cylindrical Brownian motion on H defined on
| · | ≥
a stochastic basis (Ω,F,(F ) ,P) and the coefficients satisfy the following
t t≥0
hypotheses:
(H1) (A,D(A)) is the generator of a C -semigroup, T = etA, t 0, on H
0 t
≥
and for some ω R
∈
Ax,x ω x 2, x D(A). (1.2)
h i ≤ | | ∀ ∈
(H2) σ L(H) (the space of all bounded linear operators on H) such that
∈
σ is positive definite, self-adjoint and
2
∞
(i) (1+t−α) T σ 2 dt < for some α > 0, where denotes the
k t kHS ∞ k·kHS
Z0
norm on the space of all Hilbert–Schmidt operators on H.
(ii) σ−1 L(H).
∈
(H3) F : D(F) H 2H is an m-dissipative map, i.e.,
⊂ →
u v,x y 0, x,y D(F), u F(x), v F(y),
h − − i ≤ ∀ ∈ ∈ ∈
(“dissipativity”) and
Range (I F) := (x F(x)) = H.
− −
x∈D(F)
[
Furthermore, F (x) F(x), x D(F), is such that
0
∈ ∈
F (x) = min y .
0
| | y∈F(x)| |
Here we recall that for F as in (H3) we have that F(x) is closed, non
empty and convex.
The corresponding Kolmogorov operator is then given as follows: Let
E (H) denote the linear span of all real parts of functions of the form ϕ =
A
eihh,·i, h D(A∗), where A∗ denotes the adjoint operator of A, and define for
∈
any x D(F),
∈
1
L ϕ(x) = Tr (σ2D2ϕ(x))+ x,A∗Dϕ(x) + F (x),Dϕ(x) , ϕ E (H).
0 0 A
2 h i h i ∈
(1.3)
Additionally, we assume:
(H4) There exists a probability measure µ on H (equipped with its Borel
σ-algebra B(H)) such that
(i) µ(D(F)) = 1,
(ii) (1+ x 2)(1+ F (x) )µ(dx) < ,
0
| | | | ∞
ZH
(iii) L ϕdµ = 0 for all ϕ E (H).
0 A
∈
ZH
3
Remark 1.1 (i) A measure for which the last equality in (H4) (makes sense
and) holds is called infinitesimally invariant for (L ,E (H)).
0 A
(ii) Since ω in (1.2) is an arbitrary real number we can relax (H3) by allowing
that for some c (0, )
∈ ∞
u v,x y c x y 2, x,y D(F), u F(x), v F(y).
h − − i ≤ | − | ∀ ∈ ∈ ∈
We simply replace F by F c and A by A+c to reduce this case to (H3).
−
(iii) At this point we would like to stress that under the above assumptions
(H1)-(H4) (and (H5) below) because F is merely measurable and σ is not
0
Hilbert-Schmidt, it is unknown whether (1.1) has a strong solution.
(iv) Similarly as in [7] (see [7, Remark 4.4] in particular) we expect that
(H2)(ii) can be relaxed to the condition that σ = ( A)−γ for some γ
− ∈
[0,1/2]. However, some of the approximation arguments below become more
involved. So, for simplicity we assume (H2)(ii).
The following are the main results of [7] which we shall use below.
Theorem 1.2 (cf. [6, Theorem 2.3 and Corollary 2.5]) Assume (H1),
(H2)(i), (H3) and (H4). Then for any measure µ as in (H4) the op-
erator (L ,E (H)) is dissipative on L1(H,µ), hence closable. Its closure
0 A
(L ,D(L )) generates a C -semigroup Pµ, t 0, on L1(H,µ) which is
µ µ 0 t ≥
Markovian, i.e., Pµ1 = 1 and Pµf 0 for all nonnegative f L1(H,µ)
t t ≥ ∈
and all t > 0. Furthermore, µ is Pµ-invariant, i.e.,
t
Pµfdµ = fdµ, f L1(H,µ).
t ∀ ∈
ZH ZH
Below B (H),C (H)denotetheboundedBorel-measurable, continuous func-
b b
tions respectively from H into R and denotes the usual norm on L(H).
k·k
Theorem 1.3 (cf. [6, Proposition 5.7]) Assume (H1)–(H4) hold. Then
for any measure µ as in (H4) and H := supp µ (:=largest closed set of H
0
whose complement is a µ-zero set) there exists a semigroup pµ(x,dy), x H ,
t ∈ 0
t > 0, of kernels such that pµf is a µ-version of Pµf for all f B (H), t > 0,
t t ∈ b
where as usual
pµf(x) = f(y)pµ(x,dy), x H .
t t ∈ 0
ZH
4
Furthermore, for all f B (H), t > 0, x,y H
b 0
∈ ∈
e|ω|t
pµf(x) pµf(y) f σ−1 x y (1.4)
| t − t | ≤ √t 1 k k0k k| − |
∧
and for all f Lip (H) (:= all bounded Lipschitz functions on H)
b
∈
pµf(x) pµf(y) e|ω|t f x y , t > 0, x,y H , (1.5)
| t − t | ≤ k kLip| − | ∀ ∈ 0
and
limpµf(x) = f(x), x H . (1.6)
t→0 t ∀ ∈ 0
(Here f , f denote the supremum, Lipschitz norm of f respectively.)
0 Lip
k k k k
Finally, µ is pµ-invariant.
t
Remark 1.4 (i) Both results above have been proved in [7] on L2(H,µ)
rather than on L1(H,µ), but the proofs for L1(H,µ) are entirely analogous.
(ii) In [7] we assume ω in (H1) to be negative, getting a stronger estimate
than (1.4) (cf. [7, (5.11)]). But the same proof as in [7] leads to (1.4) for
arbitrary ω R (cf. the proof of [7, Proposition 4.3] for t [0,1]). Then by
∈ ∈
virtue of the semigroup property and since pµ is Markov we get (1.4) for all
t
t > 0.
(iii) Theorem 1.3 holds in more general situations since (H2)(ii) can be
relaxed (cf. [7, Remark 4.4] and [5, Proposition 8.3.3]).
(iv) (1.4)above implies thatpµ, t > 0,isstrongly Feller, i.e., pµ(B (H))
t t b ⊂
C(H ) (=all continuous functions on H ). We shall prove below that under
0 0
the additional condition (H5) we even have pµ(Lp(H,µ)) C(H ) for all
t ⊂ 0
p > 1 and that µ in (H4) is unique. However, so far we have not been able
to prove that for this unique µ we have supp µ = H, though we conjecture
(cid:3)
that this is true.
For the results on Harnack inequalities, in this paper we need one more
condition.
(H5) (i) (1+ω A,D(A)) satisfies the weak sector condition (cf. e.g. [12]), i.e.,
−
there exists a constant K > 0 such that
(1+ω A)x,y K (1+ω A)x,x 1/2 (1+ω A)y,y 1/2, x,y D(A).
h − i ≤ h − i h − i ∀ ∈
(1.7)
5
(ii) There exists a sequence of A-invariant finite dimensional subspaces
H D(A) such that ∞ H is dense in H.
n ⊂ n=1 n
S
We note that if A is self-adjoint, then (H2) implies that A has a discrete
spectrum which in turn implies that (H5)(ii) holds.
Remark 1.5 Let (A,D(A)) satisfy (H1). Then the following is well known:
(i) (H5) (i) is equivalent to the fact that the semigroup generated by (1+
ω A,D(A)) on the complexification HC of H is a holomorphic contraction
−
semigroup on HC (cf. e.g. [12, Chapter I, Corollary 2.21]).
(ii) (H5) (i) is equivalent to (1+ω A,D(A)) being variational. Indeed,
−
let (E,D(E)) be the coercive closed form generated by (1 + ω A,D(A))
−
(cf. [12, Chapter I, Section 2]) and (E,D(E)) be its symmetric part. Then
define
e
V := D(E) with inner product E and V∗ to be its dual. (1.8)
Then e
V H V∗ (1.9)
⊂ ⊂
and 1+ω A : D(A) H has a natural unique continuous extension from
− →
V to V∗ satisfying all the required properties (cf. [12, Chapter I, Section 2,
in particular Remark 2.5]).
Now we can formulate the main result of this paper, namely the Harnack
inequality for pµ, t > 0.
t
Theorem 1.6 Suppose (H1) (H5) hold and let µ be any measure as in
−
(H4) and pµ(x,dy) as in Theorem 1.3 above. Let p (1, ). Then for all
t ∈ ∞
f B (H), f 0,
b
∈ ≥
pω x y 2
(pµf(x))p pµfp(y)exp σ−1 2 | − | , t > 0, x,y H .
t ≤ t k k (p 1)(1 e−2ωt) ∈ 0
(cid:20) − − (cid:21)
(1.10)
As consequences in the situation of Theorem 1.6 (i.e. assuming (H1)-(H5))
we obtain:
Corollary 1.7 For all t > 0 and p (1, )
∈ ∞
pµ(Lp(H,µ)) C(H ).
t ⊂ 0
6
Corollary 1.8 µ in (H4) is unique.
Because of this result below we write p (x,dy) instead of pµ(x,dy).
t t
Finally, we have
Corollary 1.9 (i) For every x H , p (x,dy) has a density ρ (x,y) with
0 t t
∈
respect to µ and
1
ρ (x, ) p/(p−1) , x H , p (1, ).
k t · kp ≤ exp σ−1 2 pω|x−y|2 µ(dy) ∈ 0 ∈ ∞
H −k k (1−e−2ωt)
R h i (1.11)
(ii) If µ(eλ|·|2) < for some λ > 2(ω 0)2 σ−1 2, then p is hyperbounded,
t
∞ ∧ k k
i.e. p < for some t > 0.
t L2(H,µ)→L4(H,µ)
k k ∞
Corollary 1.10 For simplicity, let σ = I and instead of (H1) assume that
more strongly (A,D(A)) is self-adjoint satisfying (1.2). We furthermore as-
sume that F L2(H,µ).
0
| | ∈
(i) There exists M B(H ), M D(F), µ(M) = 1 such that for every
0
∈ ⊂
x M equation (1.1) has a pointwise unique continuous strong solution (in
∈
the mild sense see (4.11) below), such that X(t) M for all t 0 P-a.s..
∈ ≥
(ii) Suppose there exists Φ C([0, )) positive and strictly increasing such
∈ ∞
that lim s−1Φ(s) = and
s→∞
∞
∞ dr
Ψ(s) := < , s > 0. (1.12)
Φ(r) ∞ ∀
Zs
If there exists a constant c > 0 such that
F (x) F (y),x y c Φ( x y 2), x,y D(F), (1.13)
0 0
h − − i ≤ − | − | ∀ ∈
then p is ultrabounded with
t
λ(1+Ψ−1(t/4))
kptkL2(H,µ)→L∞(H,µ) ≤ exp (1 e−ωt/2)2 , t > 0,
h − i
holding for some constant λ > 0.
Remark 1.11 We emphasize that since the nonlinear part F of our Kol-
0
mogorov operator is in general not continuous, it was quite surprising for us
that in this infinite dimensional case nevertheless the generated semigroup
P maps L1- functions to continuous ones as stated in Corollary 1.7.
t
7
The proof that Corollary 1.9 follows from Theorem 1.6 is completely
standard. So, we will omit the proofs and instead refer to [16], [19].
Corollary 1.7 is new and follows whenever a semigroup p satisfies the
t
Harnack inequality (see Proposition 4.1 below).
Corollary1.8isnew. Since(1.10)impliesirreducibilityofpµ andCorollary
t
1.7 implies that it is strongly Feller, a well known theorem due to Doob
immediately implies that µ is the unique invariant measure for pµ, t > 0. pµ,
t t
however, depends on µ, so Corollary 1.8 is a stronger statement. Corollary
1.10 is also new.
Theorem 1.6 as well as Corollaries 1.7, 1.8 and 1.10 will be proved in
Section 4. In Section 3 we first prove Theorem 1.6 in case F is Lipschitz,
0
and in Section 2 we prepare the tools that allow us to reduce the general case
to the Lipschitz case. In Section 5 we prove two results (see Theorems 5.2
and 5.4) on the existence of a measure satisfying (H4) under some additional
conditions and present an application to an example where F is not contin-
0
uous. For a discussion of a number of other explicit examples satisfying our
conditions see [7, Section 9].
2 Reduction to regular F
0
Let F be as in (H3). As in [7] we may consider the Yosida approximation of
F, i.e., for any α > 0 we set
1
F (x) := (J (x) x), x H, (2.1)
α α
α − ∈
where for x H
∈
J (x) := (I αF)−1(x), α > 0,
α
−
and I(x) := x. Then each F is single valued, dissipative and it is well known
α
that
limF (x) = F (x), x D(F), (2.2)
α 0
α→0 ∀ ∈
F (x) F (x) , x D(F). (2.3)
α 0
| | ≤ | | ∀ ∈
Moreover, F is Lipschitz continuous, so F is B(H)-measurable. Since F
α 0 α
is not differentiable in general, as in [7] we introduce a further regularization
by setting
F (x) := eβBF (eβBx+y)N (dy), α,β > 0, (2.4)
α,β α 1 B−1(e2βB−1)
2
ZH
8
where B : D(B) H H is a self-adjoint, negative definite linear operator
⊂ →
such that B−1 is of trace class and as usual for a trace class operator Q the
measure N is just the standard centered Gaussian measure with covariance
Q
given by Q.
F is dissipative, of class C∞, has bounded derivatives of all the orders
α,β
and F F pointwise as β 0.
α,β α
→ →
Furthermore, for α > 0
F (x)
α,β
c := sup | | : x H, β (0,1] < . (2.5)
α
1+ x ∈ ∈ ∞
(cid:26) | | (cid:27)
We refer to [10, Theorem 9.19] for details.
Now we consider the following regularized stochastic equation
dX (t) = (AX (t)+F (X (t)))dt+σdW(t)
α,β α,β α,β α,β
(2.6)

X (0) = x H.
 α,β
∈
It is well known that (2.6) has a unique mild solution X (t,x), t 0. Its
α,β
≥
associated transition semigroup is given by
Pα,βf(x) = E[f(X (t,x))], t > 0, x H,
t α,β ∈
for any f B (H). Here E denotes expectation with respect to P.
b
∈
Proposition 2.1 Assume (H1) (H4). Then there exists a K -set K H
σ
− ⊂
such that µ(K) = 1 and for all f B (H), T > 0 there exist subsequences
b
∈
(α ), (β ) 0 such that for all x K
n n
→ ∈
lim lim Pαn,βmf(x) = pµf(x) weakly in L2(0,T;dt). (2.7)
• •
n→∞m→∞
Proof. This follows immediately from the proof of [7, Proposition 5.7]. (A
closer look at the proof even shows that (2.7) holds for all x H = supp
0
µ.) (cid:3) ∈
As we shall see in Section 4, the proof of Theorem 1.6 follows from Propo-
sition 2.1 if we can prove the corresponding Harnack inequality for each Pα,β.
t
Hence in the next section we confine ourselves to the case when F is dissi-
0
pative and Lipschitz.
3 The Lipschitz case
Inthissectionweassumethat(H1)-(H3)and(H5)holdandthatF in(H3)is
0
in addition Lipschitz continuous. The aimof this section is to prove Theorem
9
1.6 for such special F (see Proposition 3.1 below). We shall do this by finite
0
dimensional (Galerkin) approximations, since for the approximating finite
dimensional processes we can apply the usual coupling argument.
We first note that since F is Lipschitz (1.1) has a unique mild solution
0
X(t,x), t 0, for every initial condition x H (cf.[10]) and we denote the
≥ ∈
corresponding transition semigroup by P , t > 0, i.e.
t
P f(x) := E[f(X(t,x))], t > 0, x X,
t
∈
where f B (H).
b
∈
Now we need to consider an appropriate Galerkin approximation. To this
endlete D(A), k N,beorthonormalsuchthatH = linear span e ,...,
k n 1
∈ ∈ {
e , n N. Hence e : k N is an orthonormal basis of (H, , ). Let
n k
} ∈ { ∈ } h· ·i
π : H H be the orthogonal projection with respect to (H, , ). So, we
n n
→ h· ·i
can define
A := π A (= A by (H5)(ii)) (3.1)
n n |Hn |Hn
and, furthermore
F := π F , σ := π σ .
n n 0|Hn n n |Hn
Obviously, σ : H H is a self-adjoint, positive definite linear operator
n n n
→
on H . Furthermore, σ is bijective, since it is one-to-one. To see the latter,
n n
one simply picks an orthonormal basis eσ,...,eσ of H with respect to the
{ 1 n} n
inner product , defined by x,y := σx,y . Then if x H is such that
σ σ n
h· ·i h i h i ∈
σ x = π σx = 0, it follows that
n n
x,eσ = σx,eσ = 0, 1 i n.
h iiσ h ii ∀ ≤ ≤
But x = n x,eσ eσ, hence x = 0.
i=1h iiσ i
Now fix n N and on H consider the stochastic equation
n
P ∈
dX (t) = (A X (t)+F (X (t)))dt+σ dW (t)
n n n n n n n
(3.2)

X (0) = x H ,
 n n
∈
whereW (t) = π W(t) = n e ,W(t) e .
n n i=1h k i k
(3.2) has a unique strong solution X (t,x), t 0, for every initial con-
n
dition x H which is paPthwise continuous P-a.s.≥. Consider the associated
n
∈
transition semigroup defined as before by
Pnf(x) = E[f(X (t,x))], t > 0, x H , (3.3)
t n ∈ n
where f B (H ).
b n
∈
Below we shall prove the following:
10