Moreover, denoting ux
1
and ux
2
by u
1
and u
2
respectively, an argument similar to the one leading to 19 yields the estimate
ψT ≤ E|x
1
− x
2
|
2
+ 2η − α Z
T
ψt d t + 146 Z
T
ht ψt d t,
where ψt := E sup
s≤t
|u
1
s − u
2
s|
2
. By Gronwall’s inequality we get ku
1
− u
2
k
2 2,
α
≤ e
2 η−α+146|h|
L1
|x
1
− x
2
|
2 L
2
, which proves that x 7→ ux is Lipschitz from L
2
to H
2, α
T , hence also from L
2
to H
2
T by the equivalence of the norms k · k
2, α
. Remark 13. As briefly mentioned in the introduction, one may prove under suitable assumptions
global well posedness for stochastic evolution equations obtained by adding to the right-hand side of 17 a term of the type Bt, ut dW t, where W is a cylindrical Wiener process on L
2
D, and B satisfies a Lipschitz condition analogous to the one satisfied by G in Theorem 12. An inspection of
our proof reveals that all is needed is a maximal estimate of the type 5 for stochastic convolutions driven by Wiener processes. To this purpose one may use, for instance, [10, Thm. 2.13]. Let us
also remark that many sophisticated estimates exist for stochastic convolutions driven by Wiener processes. Full details on well posedness as well as existence and uniqueness of invariant measures
for stochastic evolution equations of quasidissipative type driven by both multiplicative Poisson and Wiener noise will be given in a forthcoming article.
3 Invariant measures and Ergodicity
Throughout this section we shall additionally assume that G : Z × H → H is a deterministic Z ⊗ B H-measurable function satisfying the Lipschitz assumption
Z
Z
|Gz, u − Gz, v|
2
mdz ≤ K|u − v|
2
, for some K
0. The latter assumption guarantees that the evolution equation is well-posed by The- orem 12. Moreover, it is easy to see that the solution is Markovian, hence it generates a semigroup
via the usual formula P
t
ϕx := Eϕut, x, ϕ ∈ B
b
H. Here B
b
H stands for the set of bounded Borel functions from H to R.
3.1 Strongly dissipative case
Throughout this subsection we shall assume that there exist β
and ω
1
K such that 2〈A
β
u − A
β
v, u − v〉 + 2〈 f
λ
u − f
λ
v, u − v〉 − 2η|u − v|
2
≥ ω
1
|u − v|
2
20 for all
β ∈]0, β [, λ ∈]0, β
[, and for all u, v ∈ H. This is enough to guarantee existence and unique- ness of an ergodic invariant measure for P
t
, with exponentially fast convergence to equilibrium.
1546
Proposition 14. Under hypothesis 20 there exists a unique invariant measure ν for P
t
, which satisfies the following properties:
i Z
|x|
2
νd x ∞; ii let
ϕ ∈ ˙ C
0,1
H, R and λ ∈ M
1
H. Then one has Z
H
P
t
ϕx λ d x −
Z
H
ϕ νd y ≤ [ϕ]
1
e
−ω
1
t
Z
H×H
|x − y| λ d x νd y
Following a classical procedure see e.g. [13, 31, 32], let us consider the stochastic equation dut + Aut + f u d t =
ηut d t + Z
Z
Gz, ut− d ¯ µ
1
d t, dz, us = x,
21 where s ∈] − ∞, t[, ¯
µ
1
= µ
1
− Leb ⊗ m, and µ
1
t, B = µt, B,
t ≥ 0, µ
−t, B, t
0, for all B ∈ Z , with
µ an independent copy of
µ. The filtration ¯ F
t t∈R
on which µ
1
is considered can be constructed as follows:
¯ F
t
:= \
s t
¯ F
s
, ¯
F
s
:= σ {µ
1
[r
1
, r
2
], B : −∞ r
1
≤ r
2
≤ s, B ∈ Z }, N ,
where N stands for the null sets of the probability space Ω, F , P. We shall denote the value at time t ≥ s of the solution of 21 by ut; s, x.
For the proof of Proposition 14 we need the following lemma.
Lemma 15. There exists a random variable ζ ∈ L
2
such that u0; s, x → ζ in L
2
as s → −∞ for all x ∈ L
2
. Moreover, there exists a constant N such that E
|u0; s, x − x|
2
≤ e
−2ω
1
|s|
N 1 + E|x|
2
22 for all s
0. Proof. Let u be the generalized mild solution of 21. Define Γt, z := Gz, ut−, and let Γ
n
be an approximation of Γ, as in the proof of Proposition 10. Let us denote the strong solution of the
equation
dut + A
β
ut + f
λ
ut d t = ηut d t + Z
Z
Γ
n
t, z d ¯ µ
1
d t, dz, us = x,
by u
n λβ
. By Itô’s lemma we can write |u
n λβ
t|
2
+ 2 Z
t s
〈A
β
u
n λβ
r, u
n λβ
r〉 + 〈 f
λ
u
n λβ
r, u
n λβ
r〉 − η|u
n λβ
r|
2
d r = |x|
2
+ 2 Z
t s
Z
Z
〈Γ
n
r, z, u
n λβ
r〉 ¯ µ
1
d r, dz + Z
t s
Z
Z
|Γ
n
r, z|
2
µ
1
d r, dz. 23
1547
Note that we have, by Young’s inequality, for any ǫ 0,
−〈 f
λ
u, u〉 = −〈 f
λ
u − f 0, u − 0〉 − 〈 f
λ
0, u〉 ≤ −〈 f
λ
u − f 0, u − 0〉 + ǫ
2 |u|
2
+ 1
2 ǫ
| f
λ
0|
2
. Since f
λ
0 → f 0 as λ → 0, there exists δ 0, λ 0 such that
| f
λ
0|
2
≤ | f 0|
2
+ δ2 ∀λ λ
. By 20 we thus have, for
β β ,
λ λ ∧ β
, − 2〈A
β
u
n λβ
, u
n λβ
〉 − 2〈 f
λ
u
n λβ
, u
n λβ
〉 + 2η|u
n λβ
|
2
≤ −ω
1
|u
n λβ
|
2
+ ǫ|u
n λβ
|
2
+ ǫ
−1
| f 0|
2
+ δ. Taking expectations in 23, applying the above inequality, and passing to the limit as
β → 0, λ → 0, and n → ∞, yields
E |ut|
2
≤ E|x|
2
− ω
1
− ǫ Z
t s
E |ur|
2
d r + ǫ
−1
| f 0|
2
+ δ t − s
+ E Z
t s
Z
Z
|Γr, z|
2
mdz d r. Note that, similarly as before, we have
Z
Z
|Gu, z|
2
mdz ≤ 1 + ǫK|u|
2
+ 1 + ǫ
−1
Z
Z
|G0, z|
2
mdz for any u ∈ H, therefore, by definition of Γ,
E Z
t s
Z
Z
|Γr, z|
2
mdz d r ≤ 1 + ǫK
Z
t s
E |ur|
2
d r + 1 + ǫ
−1
G0, ·
2 L
2
Z,m
t − s. Setting
ω
2
:= ω
1
− K − ǫ1 + K, N :=
ǫ
−1
| f 0|
2
+ δ + 1 + ǫ
−1
G0, ·
2 L
2
Z,m
, we are left with
E |ut|
2
≤ E|x|
2
− ω
2
Z
t s
E |ur|
2
d r + N t − s. We can now choose
ǫ such that ω
2
0. Gronwall’s inequality then yields E
|ut|
2
® 1 + e
−ω
2
t+|s|
E |x|
2
. 24
Set u
1
t := ut; s
1
, x, u
2
t := ut; s
2
, x and wt = u
1
t − u
2
t, with s
2
s
1
. Then w satisfies the equation
d w + Aw d t + f u
1
− f u
2
d t = ηw d t + Gu
1
− Gu
2
d ¯ µ,
1548
with initial condition ws
1
= x − u
2
s
1
, in the generalized mild sense. By an argument completely similar to the above one, based on regularizations, Itô’s formula, and passage to the limit, we obtain
E |wt|
2
≤ E|x − u
2
s
1
|
2
− ω
1
− K Z
t s
1
E |wr|
2
d r, and hence, by Gronwall’s inequality,
E |u
1
0 − u
2
0|
2
= E|w0|
2
≤ e
−ω
1
−K|s
1
|
E |x − u
2
s
1
|
2
. Estimate 24 therefore implies that there exists a constant N such that
E |u
1
0 − u
2
0|
2
≤ e
−ω
1
−K|s
1
|
N 1 + E|x|
2
, 25
which converges to zero as s
1
→ −∞. We have thus proved that {u0; s, x}
s≤0
is a Cauchy net in L
2
, hence there exists ζ = ζx ∈ L
2
such that u0; s, x → ζ in L
2
as s → −∞. Let us show that ζ
does not depend on x. In fact, let x, y ∈ L
2
and set u
1
t = ut; s, x, u
2
t = ut; s, y. Yet another argument based on approximations, Itô’s formula for the square of the norm and the monotonicity
assumption 20 yields, in analogy to a previous computation, E
|u
1
0 − u
2
0|
2
≤ e
−ω
1
−K|s|
E |x − y|
2
, 26
which implies ζx = ζ y, whence the claim. Finally, 25 immediately yields 22.
Proof of Proposition 14. Let ν be the law of the random variable ζ constructed in the previous
lemma. Since ζ ∈ L
2
, i will follow immediately once we have proved that ν is invariant for
P
t
. The invariance and the uniqueness of ν is a well-known consequence of the previous lemma, see
e.g. [10]. Let us prove ii: we have
Z
H
P
t
ϕx λ d x −
Z
H
ϕ y νd y =
Z
H
Z
H
P
t
ϕx λ d x νd y −
Z
H
Z
H
P
t
ϕ y λ d x νd y
≤ Z
H×H
|P
t
ϕx − P
t
ϕ y| λ d x νd y
≤ [ϕ]
1
e
−ω
1
t
Z
H×H
|x − y| λ d x νd y,
where in the last step we have used the estimate 26.
3.2 Weakly dissipative case
