Communications in Partial Differential Equations, 31: 277–291, 2006
Copyright © Taylor & Francis Group, LLC
ISSN 0360-5302 print/1532-4133 online
DOI: 10.1080/03605300500357998
Strong Solutions of Stochastic Generalized Porous
Media Equations: Existence, Uniqueness,
and Ergodicity
G. DA PRATO1 , MICHAEL RÖCKNER2 ,
B. L. ROZOVSKII3 , AND FENG-YU WANG4
1
Scuola Normale Superiore, Pisa, Italy
Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany
3
Center of Applied Mathematics, USC, Los Angeles, California, USA
4
School of Mathematical Sciences, Beijing Normal University,
Beijing, China
2
Explicit conditions are presented for the existence, uniqueness, and ergodicity of the
strong solution to a class of generalized stochastic porous media equations. Our
estimate of the convergence rate is sharp according to the known optimal decay for
the solution of the classical (deterministic) porous medium equation.
Keywords Ergodicity; Stochastic porous medium equation; Strong solution.
AMS Subject Classification 76S05; 60H15.
1. Main Results
Let E m be a separable probability space and L L a negative definite
self-adjoint linear operator on L2 m having discrete spectrum with eigenvalues
0 > −1 ≥ −2 ≥ · · · → −
and L2 m-normalized eigenfunctions ei such that ei ∈ Lr+1 m for any i ≥ 1,
where r > 1 is a fixed number throughout this article. We assume that L−1 is
bounded in Lr+1 m, which is the case if L is a Dirichlet operator (cf. e.g., Ma
and Röckner, 1992) since in this case the interpolation theorem or simply Jensen’s
inequality implies etL r+1 ≤ e−1 t2/r+1 for all t ≥ 0 A classical example of L is
the Laplacian operator on a smooth bounded domain in a complete Riemannian
manifold with Dirichlet boundary conditions.
Accepted October 1, 2004
Address correspondence to Feng-Yu Wang, School of Mathematical Sciences, Beijing
Normal University, Beijing 100875, China; E-mail: [email protected]
277
278
Da Prato et al.
In this article we consider the stochastic differential equation
dXt = LXt + Xt dt + Q dWt (1.1)
where and are (nonlinear) continuous functions
on , and Q is a densely
qji ej i ≥ 1 such that qi2 =
defined linear operator on L2 m with Qei = j=1
2
j=1 qij satisfies
q =
qi2
i=1
i
< An appropriate Hilbert space H as state space for the solutions to (1.1) is given as
follows. Let
1
2
2
i mfei < H = f ∈ L m i=1
Define H to be its topological dual with inner product H . Identifying L2 m with
its dual, we get the continuous and dense embeddings
H 1 ⊂ L2 m ⊂ H
We denote the duality between H and H 1 by . Obviously, when restricted
to L2 m × H 1 this coincides with the natural inner product in L2 m, which we
therefore also denote by , and it is also clear that
f gH =
i−1 f ei g ei f g ∈ H
Furthermore, in (1.1) Wt = bti i∈ is a cylindrical Brownian motion on L2 m where
bti are independent one-dimensional Brownian motions on a complete probability
space P. Let t be the natural filtration of Wt . Then
QWt =
i=1
qij btj
ei t≥0
j=1
is a well-defined process taking values in H which is a martingale.
Recall that the classical porous medium equation reads
dXt = Xtm dt
on a domain in d , see e.g., Aronson (1986) and the references therein. So, we
may call (1.1) the generalized stochastic porous medium equation. Recently, the
existence and uniqueness of weak solutions as well as the existence of invariant
probability measures for the stochastic porous medium equation, i.e., (1.1) with
L = on a bounded domain in d with Dirichlet boundary conditions, were
proven in Da Prato and Röckner (2004,?), Bogachev et al. (2004), and Barbu et al.
(Preprint). In this paper we aim to prove the existence and uniqueness of strong
solutions of (1.1), in particular, describe the convergence rate of the solution, for
t → .
Stochastic Generalized Porous Media Equations
279
To solve (1.1), we assume that there exist some constants c ≥ 0, ∈ such that
s
+ s
≤ c1 + s
r s − ts − t ≥ s − t
r+1
(1.2)
+ s − t s t ∈ 2
Since by the Cauchy-Schwarz inequality one has
r+1/2
2
23−r s − t
r+1
4
s
≤
sgns − t
r+1/2 sgnt
2
2
r + 1
r + 1
s
2
s
r−1/2
=
u
du ≤ s − t
u
r−1 du
t
t
(1.2) holds if 0 = 0 and there exists > 0 such that (cf. Bogachev et al., 2004)
+
r + 12
s
r−1 ≤ s ≤ 1 + s
r−1 4
s ∈ Next, assume that there exist < and ≤ such that
2
−m x − yL−1 x − y ≤ x − yr+1
r+1 + x − y2 x y ∈ Lr+1 m
(1.3)
where here and in the sequel, · p denotes the Lp -norm with respect to m for
any p ≥ 1 We note that since L−1 is bounded on Lr+1 m and r > 1, if there exist
constants c1 c2 ≥ 0 such that
s − t
≤ c1 s − t
r + c2 s − t
s t ∈ then
−1
2
−m x − yL−1 x − y ≤ c1 L−1 r+1 x − yr+1
r+1 + c2 1 x − y2 hence (1.3) holds for = c1 L−1 r+1 and = c2 1−1 Definition 1.1. Let dt = e−t dt. An H-valued continuous t -adapted process Xt
is called a solution to (1.1), if X ∈ Lr+1 + × × E × P × m such that
Xt ei = X0 ei +
t
0
m Xs Lei + Xs ei ds + qi Bti i ≥ 1 t > 0
(1.4)
j
where Bti = q1 j=1 qij bt (
= 0 if qi = 0) is an t -Brownian motion on (provided
i
it is nontrivial).
Remark 1.1. (i) We note that (1.4) indeed makes sense, since by the first inequality
in (1.2) we have
X
X ∈ Lr+1/r + × × E × P × m
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Da Prato et al.
(ii) We emphasize that for each solution of (1.4) there also exists a vectorvalued version of the equation. More precisely, the integral comes from an H-valued
random vector with a natural integrand which, however, takes values in a larger
Banach space . To describe this in detail, we need some preparations.
Consider the separable Banach space = Lr+1 m. Then we can obtain a
presentation of its dual space through the embeddings
⊂ H ≡ H ⊂ where H is identified with its dual through the Riesz-isomorphism. In other words,
is just the completion of H with respect to the norm
f = sup f gH gr+1 ≤1
f ∈ H
Since H is separable, so is . We note that this is different from the usual
representation of = Lr+1 m through the embedding
⊂ L2 m ≡ L2 m which, of course, gives Lr+1/r m as dual. But it is easy to identify the isomorphism
between Lr+1/r m and . Below denotes the duality between and .
Clearly, = H on × H.
Proposition 1.1. The linear operator
Lf = −
i mfei ei f ∈ L2 m
i=1
defines an isometry from Lr+1/r m to with dense domain. Its (unique) continuous
extension L to all of Lr+1/r m is an isometric isomorphism from Lr+1/r m onto such that
−Lf g
= mfg
for all f ∈ Lr+1/r m g ∈ Lr+1 m
(1.5)
Proof. Let f ∈ L2 m, N > n ≥ 1. Then
N
N
i mfei ei = sup m g mfei ei g ≤1 i=n
r+1
i=n
N
=
mfe
e
i i
i=n
r+1/r
r+1
2
r m
Since f = i=1 mfei ei with the series converging in L m, hence in L
(because r > 1), the first part of the assertion follows, and L is an isometry from
Lr+1/r m into . Now let T ∈ . Then there exists f ∈ Lr+1/r m such that for
all g ∈ Lr+1 m
T g
= mfg
= lim mfn g
n→
(1.6)
Stochastic Generalized Porous Media Equations
281
for some fn ∈ DL such that lim fn = f in Lr+1/r m. Hence for all g ∈ Lr+1 m
n→
T g
= lim mfn LL−1 g
n→
= lim mLfn L−1 g
(1.7)
n→
= lim −Lfn gH = − Lf g
n→
and the second assertion is proven. Since any f ∈ Lr+1/r m defines a T ∈ , the
last assertion follows from (1.6) and (1.7).
Since L−1 is bounded on Lr+1 m by our assumptions on L (which was not used
so far), we obtain the following consequence.
Corollary 1.2. Let L−1 Lr+1/r m → Lr+1/r m be the dual operator of L−1 Lr+1 m → Lr+1 m. Then the operator
J L L−1 Lr+1/r m → extends the natural inclusion L2 m ⊂ H ⊂ and for all f ∈ Lr+1/r m
= −mfL−1 g
Jf g
for all g ∈ Lr+1 m
(1.8)
Proof. If f ∈ L2 m, then L−1 f = L−1 f , hence Jf = f ∈ L2 m ⊂ H ⊂ . The
last assertion follows by (1.5).
Multiplied by i−1 , (1.4) by the above now reads
Xt ei = X0 ei +
t
0
LXs + JXs ei ds + QW ei By Remark 1.1, Proposition 1.1, and Corollary 1.2, the Bochner integrals
t
LXs + JXs ds
0
t ≥ 0
exist in . So, (1.4) can always be rewritten equivalently in vector form as an
equation in as
Xt = X0 +
t
0
LXs + JXs ds + QWt t ≥ 0
(1.9)
Note that by Definition 1.1, Xt ∈ H and also QWt ∈ H, hence the integral in (1.9) is
necessarily a continuous H-valued process.
Now we can state our main results.
Theorem 1.3. Assume 12 and 13 with ≥ and > . We have:
(1) For any 0 /H-measurable → H with 2H < , there exists a unique
solution X to (1.1) such that X0 = . Furthermore, there exists C > 0 such that
Xt 2H ≤ C1 + t−2/r−1 t > 0
(1.10)
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Da Prato et al.
In particular, for any x ∈ H there exists a unique solution Xt x to 11 with initial
value x, whose distributions form a continuous strong Markov process on H.
(2) For any two solutions X and Y of (1.1), we have for all t ≥ s ≥ 0
−2/r−1
r+1/2
Xt − Yt 2H ≤ Xs − Ys 1−r
t − s
H + r − 1 − 1
−2/r−1
r+1/2
≤ Xs − Ys 2H ∧ r − 1 − 1
t − s
(1.11)
Consequently, setting Pt Fx = FXt x for F H → , Borel measurable, so
that the expectation makes sense, we have that Pt t>0 is a Feller semigroup on
Cb H and, in addition, for Lipschitz continuous F
Pt Fx − Pt Fy
≤ F x − yH x y ∈ H
(1.12)
where F is the Lipschitz constant of F .
(3) Pt has a unique invariant probability measure and for some constant C > 0,
satisfies
sup Pt Fx − F ≤ CF t−1/r−1 t > 0
(1.13)
x∈H
for any Lipschitz continuous function F on H. Moreover, · r+1
r+1 < (4) If > then for any two solutions X and Y of (1.1), we have for all t ≥ s ≥ 0
Xt − Yt H ≤ Xs − Ys H e−−t−s
(1.14)
and there exists C > 0 such that
Xt − Yt H ≤ Ce−−t t ≥ 1
(1.15)
Consequently, for some constant C > 0,
sup Pt Fx − F ≤ CF e−−t t ≥ 1
(1.16)
x∈H
for any Lipschitz continuous function F on H.
Remark 1.2. (1) When Q = 0, the Dirac measure 0 is the unique invariant
measure. Thus, (1.13) with Fx = xH implies
sup Xt xH ≤ Ct−1/r−1 t > 0
x
This coincides with the optimal decay of the solution to the classical porous medium
equation obtained by Aronson and Peletier (1981, Theorem 2).
(2) In the case where = 0 and r = r + r m for ≥ 0 and m ≥ 3 odd,
and L = on a regular domain in d , in Barbu et al. (Preprint) and Da Prato
and Röckner (2004) much stronger integrability results for
2 measure
the invariant
have been proven, namely, if either m = 3 or > 0, then sign x
x
n < for
any n ≥ 1
Stochastic Generalized Porous Media Equations
283
(3) In the case where L = on a bounded smooth domain in d , the
existence of an invariant measure was proven in Bogachev et al. (2004) under
the conditions that 0 s
r−1 ≤ s ≤ C1 s
r−1 and s
≤ C + s
r for some
constants C 0 1 > 0 ∈ 0 40 1 r + 1−2 and all s ∈ Also in Bogachev
etal. (2004) stronger
integrability properties for have been proven, namely that
2 sign x x
< for all ∈ r + 1/2 r.
Finally, we note that in this article the coefficient in front of the noise is
constant (i.e., the so-called additive noise). Under the usual Lipschitz assumptions,
however, properly reformulated versions of our results also hold for nonconstant
diffusion coefficients. Details on this will be contained in a forthcoming article.
2. Some Preliminaries
We shall make use of a finite-dimensional approximation argument to construct the
n
n
n
solution of (1.1). For any n ≥ 1, let rt = rt1 rtn solve the following SDE
n
(stochastic differential equation) on :
n
drti
=
qi dBti
− i m ei n
n
rtk ek
n
n
dt + m ei rtk ek dt
k=1
(2.1)
k=1
n
with r0i = X0 ei 1 ≤ i ≤ n, where X0 → H is a fixed 0 /H-measurable
map such that X0 2H < . Here and below for a topological space S we denote
its Borel -algebra by S. By Krylov (1999, Theorem 1.2) there exists a unique
solution to (2.1) for all t ≥ 0.
Lemma 2.1. Under the assumptions of Theorem 1.3, there exists a constant C > 0
n
n
independent of n and X0 such that Xt = ni=1 rti ei satisfies
0
T
n
m Xt r+1 dt ≤ C X0 2H + T T > 0
(2.2)
and
Xt 2H ≤ C1 + t−2/r−1 n
t > 0
(2.3)
s t ∈ (2.4)
Proof. By (1.2) we have
ss ≥ s0 + s
r+1 + s2 and by (1.3) we have
2
−m xL−1 x ≤ −0mL−1 x + xr+1
r+1 + x2 x ∈ Lr+1 m
Hence for all x ∈ spanei i ∈ −mxx − mxL−1 x
2
≤ 0
+ 0
1−1 x2 − − xr+1
r+1 − − x2 284
Da Prato et al.
Combining this with (2.1) and using Itô’s formula, we obtain
c n
c
1
n
n
dXt 2H ≤ dMt + 1 dt − 2 m Xt r+1 dt
2
2
2
(2.5)
n
for some local martingale Mt and
1 c2 > 0 independent of n. This
constants
cr+1/2
n
n
implies (2.2). Moreover, since m Xt r+1 ≥ 1
Xt r+1
H it follows from
(2.5) that
n
Xt 2H − Xsn 2H ≤ c1 t − s − c2
t
s
Xun 2H
r+1
2
du
0 ≤ s ≤ t
(2.6)
To prove (2.3), let h solve the equation
h t = −c2 htr+1/2 + c1 t ≥ 0
h0 = X0 2H + 4c1 /c2 2/r+1 (2.7)
Then it is easy to see that (2.6) implies
n
Xt 2H ≤ ht
t ≥ 0
(2.8)
n
Let t = ht − Xt 2H and
= inft ≥ 0 t ≤ 0
Suppose < , then by continuity ≤ 0 and by the mean-value theorem and (2.6),
(2.7), we obtain
t ≥ 0 − c2
t
0
r+1/2 h ur+1/2 − Xun 2H
du
≥ 4c1 /c2 2/r+1 − c
t
0
u du
0 ≤ t ≤ maxt∈0 tr−1/2 = c2 r+1
r−1/2 . By Gronwall’s lemma we arrive at
where c = c2 r+1
2
2
2/r+1 −c
≥ 4c1 /c2 e > 0. This contradiction proves (2.8).
To estimate ht let
= inf t ≥ 0 htr+1/2 ≤ 2c1 /c2 Since h0r+1/2 ≥ 4c1 /c2 > 2c1 /c2 ≥ t0 for some t0 > 0 independent of n. Indeed,
we may define t0 as above with h replaced by the solution to (2.7) with initial
condition h0 = 4c1 /c2 2/r+1 By (2.7) we have
h t ≤ −
c2
htr+1/2 2
0 ≤ t ≤ Therefore, for some constant c > 0 independent of n,
ht ≤ ct−2/r−1 0 ≤ t ≤ (2.9)
Stochastic Generalized Porous Media Equations
285
Clearly, h t ≤ 0 for all t ≥ 0, since by an elementary consideration we have
h ≥ c1 /c2 2/r+1 , consequently
−2/r−1
ht ≤ h ≤ c−2/r−1 ≤ ct0
t > Therefore, (2.3) holds.
According to (2.2) in Lemma 2.1, X n is bounded in Lr+1 + × × E ×
P × m, where dt = e−t dt Thus, there exists a subsequence nk → and a
process X such that X nk → X weakly in Lr+1 + × × E × P × m. To prove
that this limit provides a solution of (1.1), we shall make use of Theorem 3.2 in
Chapter 1 of Krylov and Rozovskii (1981). We state this result in detail for the
reader’s convenience specialized to our situation.
Theorem 2.2 (Krylov and Rozovskii, 1981, Theorem I.3.2). Consider three maps
v + × → , ṽ + × → , h + × → H such that:
(i) v is + ⊗ /-measurable and vt = vt · is t /-measurable for all
t ≥ 0;
(ii) ṽ is + ⊗ / -measurable and ṽt = ṽt · is t / -measurable.
T
Moreover, 0 ṽt dt < P-a.s. for all T > 0;
(iii) h is an H-valued t -adapted continuous local semi-martingale.
Set
h̃t =
t
0
ṽs ds + ht If h̃t = vt for × P-a.e. t , then h̃t is an H-valued continuous t -adapted
process satisfying the following Itô formula for the square of the norm:
h̃t 2H = h̃02H + 2
t
0
ṽs vs ds + 2
t
0
h̃s dhs H + ht (2.10)
where h denotes the quadratic variation process of h.
3. Proof of the Existence
n n By Lemma 2.1 and (1.2), Xt and Xt are bounded in
Lr+1/r + × × E × P × m, where dt = e−t dt Hence there exist a
subsequence nk → and processes U V ∈ Lr+1/r + × × E × P × m such
that
a)
X nk → U X nk → V
weakly in Lr+1/r + × × E × P × m (3.1)
Moreover, by Lemma 2.1, we may also assume that
X nk → X
weakly in Lr+1 + × × E × P × m
(3.2)
E.g. by Diestel (1975, Chap. 3, § 7) we may also assume that the Cesaro means of the
sequences in (3.1) converge strongly in Lr+1/r + × × E × P × m so the limits
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Da Prato et al.
have + ⊗ ⊗ -measurable versions. Furthermore, as continuous processes
the approximants are all progressively measurable as Lr+1/r m-valued processes,
hence so are their limits. In particular, these are adapted. The same holds for the
sequence in (3.2), respectively its limit with r + 1/r replaced by r + 1. Below we
always consider versions of U , V , X with all these measurability properties and
denote them by the same symbols. Since for t ≥ 0
t n m Xsnk Lei + m ei Xsnk ds + qi Bti m Xt k ei = X0 ei +
0
1 ≤ i ≤ nk it follows from (3.1) and (3.2) that, for any real-valued bounded measurable
process ,
T t
T
t mX t ei dt = t X0 ei + mUs Lei + mVs ei ds + qi Bti dt
0
0
0
for all T > 0. Thus,
mX t ei = X0 ei +
b)
t
0
mUs Lei + Vs ei ds + qi Bti for × P-a.e. t i ≥ 1
(3.3)
To apply Theorem 2.2, let
ṽs = LUs + JVs T
By (1.2), Lemma 2.1, Proposition 1.1, and Corollary 1.2, we have 0 v̄s ds < for any T > 0. So, we see that in Theorem 2.2 conditions (i), (ii) with v = X and also
(iii) with h = QW are satisfied and by Proposition 1.1 and Corollary 1.2, (3.3) with
ei replaced by i−1 ei implies
X t ei = X0 ei +
t
0
ṽs ei ds + QWt ei i ≥ 1 for × P-a.e. t .
Hence defining
Xt = X0 +
t
0
ṽs ds + QWt t ≥ 0
(3.4)
we see that
X=X
× P-a.e.
(3.5)
Therefore, by Theorem 2.2, X is an H-valued continuous t -adapted process
and (2.10) holds with X replacing h̃. Therefore, to prove that X solves (1.1), by
Proposition 1.1 and Corollary 1.2, it suffices to show that
mei Vs − i Us = mei X s − i X s This will be proven by the following two steps.
i ≥ 1 for × P-a.e. s (3.6)
Stochastic Generalized Porous Media Equations
+ → + bounded, Borel measurable with
c) We claim that for any
compact support,
nk 2
H
lim inf
k→
tXt
0
287
dt ≥
tX t 2H dt
0
(3.7)
Since X nk → X weakly in L2 + × × E × P × m, by Fatou’s lemma we have
tX t 2H dt =
0
i−1 i=1
=
0
2
tm ei X t dt
k→
1
2
i−1 lim i=1
≤
0
i−1 lim inf k→
i=1
n tm ei Xt k mei X t dt
0
n 2
tm ei Xt k dt
2
1
i−1 tm ei X dt
2 i=1
0
1 1
n tXt k 2H dt + tX t 2H dt
≤ lim inf 2 k→
2 0
0
+
Since X ∈ L2 + × × E × P × m so that
(3.7) immediately.
d)
0
tX t 2H dt < this implies
By (2.10) in Theorem 2.2, (1.5), and (1.8), we have
Xt 2H = X0 2H − 2
t
0
mX s Us + mL−1 X s Vs ds +
i−1 qi2 t
(3.8)
i=1
On the other hand, by Itô’s formula,
n
Xt 2H = X0 2H − 2
t
0
n
m Xsn Xsn + L−1 Xsn Xsn ds +
i−1 qi2 t
i=1
(3.9)
Then for any ∈ Lr+1 + × × E × P × m, we obtain from (1.2), (1.3), and
(3.9) that
m Xsnk − s Xsnk − s 0
+ L−1 Xsnk − s Xsnk − s ds
t = 2 m Xsnk Xsnk + L−1 Xsnk Xsnk ds
0 ≤ Ik t = 2
t
0
m s Xsnk − s + Xsnk s 0
+ L−1 Xsnk s + L−1 s Xsnk − s ds
− 2
t
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Da Prato et al.
nk 2
H
= −Xt
+ X0 2H +
nk
i−1 qi2 t
i=1
m s Xsnk − s + Xsnk s 0
+ L−1 Xsnk s + L−1 s Xsnk − s ds
− 2
t
(3.10)
Since X nk → U and X nk → V weakly in Lr+1/r + × × E × P × m
and X nk → X weakly in Lr+1 + × × E × P × m, for any + → + ,
bounded, Borel-measurable with compact support, we obtain from (3.10) and
(3.7) that
0≤
tdt −X t 2H +X0 2H + i−1 qi2 t
0
−2
i=1
t
0
m s Us −s +X s s +L−1 X s s +L−1 s Vs −s ds Combining this with (3.8), we arrive at
0≤
t
tdt
0
0
m X s − s Us − s + L−1 X s − s Vs − s ds
(3.11)
By first taking s = X s − ˜ s ei for given > 0 and ˜ ∈ L + × × P, then
dividing by and letting → 0, we obtain from the continuity of and the
dominated convergence theorem, which is valid due to (1.2), (1.3), and because
X ∈ Lr+1 0 × × E × P × m, that
t
tdt
0
0
˜ s m ei Us − X s − i−1 Vs − X s ds ≥ 0
Since
+ → + bounded, Borel-measurable with compact support, and
˜ ∈ L + × × P are arbitrary, this implies (3.6).
4. Proof of the Other Assertions
a) Proofs of the uniqueness, (1.10), (1.11), and (1.13). (1.10) is an immediate
consequence of (2.3) and (3.7), since X is P-a.e. continuous in t. Let X and Y be
solutions of (1.1). Then by (1.9), Z = X − Y solves the equation
Zt = Z0 +
t
0
LXt − Yt + JXt − Yt dt
t ≥ 0
Thus, by Theorem 2.2 with h = 0, (1.5), (1.8), and finally by (1.2), (1.3), we obtain
m Xs − Ys Xs − Ys 0
+ L−1 Xs − Ys Xs − Ys ds ≤ 0
Zt 2H = Z0 2H − 2
t
(4.1)
Stochastic Generalized Porous Media Equations
289
If X0 = Y0 , this implies Zt = 0 for all t ≥ 0. By a slight modification of a standard
argument one obtains as usual that the uniqueness also implies the stated Markov
property and hence the semigroup property of Pt . The strong Markov property then
follows from the Feller property of Pt proven below, since all solutions of (1.1) have
continuous sample paths in H.
Similarly to (4.1), we have for 0 ≤ s ≤ t that
Xt − Yt 2H − Xs − Ys 2H ≤ −2
t
s
− Xu − Yu 22 + − Xu − Yu r+1
r+1 du
(4.2)
r+1/2
· r+1
Noting that ≥ > , · 22 ≥ 1 · 2H , · r+1
H , and that for
r+1 ≥ 1
r+1/2
> 0 the function ht = + Xs − Ys H 1−r + r − 1 − 1
t − s−2/r−1 ,
t ≥ s, solves for s ≥ 0 fixed the equation
ht = −2 − 1
r+1/2 r+1/2
ht
t ≥ s≥ 0
h0 = Xs − Ys H + 2 (4.3)
due to the same comparison argument as in the proof of Lemma 2.1, it follows that
Xt − Yt 2H ≤ ht ∀t ≥ s. Letting → 0, this implies (1.11) and the Feller property
of Pt . By (1.10) it follows that Pt F x < for all Lipschitz continuous F H → .
Now (1.12) is obvious by (1.11).
b)
Proof of (3). Let 0 be the Dirac measure at 0 ∈ H. Set
n =
1 n
P dt
n 0 0 t
n ≥ 1
We intend to show the tightness of n . Then by the Feller property of Pt , the
weak limit of a subsequence provides an invariant probability measure of Pt . By
Lemma 2.1 and the weak convergence of X nk to X, we have
H
m
x
2 n dx =
1 n
m
Xt 0
2 dt ≤ C
n 0
for some constant C > 0 and all n ≥ 1 where we set m
x
2 = if x L2 m.
Since the function x → m
x
2 is compact, that is, x ∈ H m
x
2 ≤ r is relatively
compact in H for any r ≥ 0, we conclude that n is tight.
Next, let be an invariant probability measure. For any bounded Lipschiz
function F on H, (1.11) implies that there exists C > 0 such that
Pt Fx − F ≤
H
FXt x − FXt y
dy ≤ CF t−1/r−1
for all x ∈ H t > 0. Thus, (1.13) holds and hence Pt has a unique invariant measure.
Let be the invariant probability measure of Pt . It remains to show that
· r+1
r+1 < .
By (1.10), since is Pt -invariant, we have
H
x2H dx =
2
X1 x
H dx ≤ 2C < (4.4)
290
Da Prato et al.
Next, since X is the weak limit of X nk in Lr+1 0 × × E × P × m, by
Hölder’s inequality we have
2
1
m
Xt x
r+1 dt = lim
2
k→ 1
≤ lim inf
n m Xt k Xt r sgnXt dt
k→
2
1
n m
Xt k r+1 dt
1/r+1
2
1
r/r+1
m
Xt r+1
dt
Therefore, by (2.2), for some C1 > 0
2
1
m
Xt x
r+1 dt ≤ C1 1 + x2H x ∈ H
Then by (4.4), we obtain
· r+1
r+1 =
c)
2
dx
H
1
m
Xt x
r+1 dt < Exponential ergodicity, i.e., proof of (4).
If > then (4.2) implies
Xt − Yt 2H ≤ Xs − Ys 2H e−2−t−s t ≥ s ≥ 0
So, (1.14) holds. Combining this with (1.11), we arrive at
Xt+1 − Yt+1 2H ≤ X1 − Y1 2H e−2−t ≤ Ce−2−t t ≥ 0
This implies (1.15) and (1.16) immediately.
Acknowledgments
Supported in part by the DFG through the Forschergruppe “Spectral Analysis,
Asymptotic Distributions and Stochastic Dynamics,” the BiBoS Research Centre,
NNSFC (10025105, 10121101), RFDP (20040027009).
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