2.2 Multiplicative noise
Let us consider the stochastic evolution equation dut + Aut d t + f ut d t =
ηut d t + Z
Z
Gt, z, ut− ¯ µd t, dz
17 with initial condition u0 = x, where G : Ω × [0, T ] × Z × R × D → R is a P ⊗ Z ⊗ B R ⊗ B R
n
- measurable function, and we denote its associated Nemitski operator, which is a mapping from
Ω × [0, T ] × Z × H → H, again by G. We have the following well-posedness result for 17 in the generalized mild sense.
Theorem 12. Assume that x ∈ L
2
and G satisfies the Lipschitz condition E
Z
Z
|Gs, z, u − Gs, z, v|
2
mdz ds ≤ hs|u − v|
2
, where h ∈ L
1
[0, T ]. Then 17 admits a unique generalized solution u ∈ H
2
T . Moreover, the solution map is Lipschitz from L
2
to H
2
T . Proof. For v ∈ H
2
T and càdlàg, consider the equation dut + Aut d t + f ut d t =
ηut d t + Z
Z
Gs, z, vs− ¯ µds, dz,
u0 = x. 18
Since s, z 7→ Gs, z, vs− satisfies the hypotheses of Proposition 10, 18 admits a unique gen- eralized mild solution belonging to H
2
T . Let us denote the map associating v to u by F . We are going to prove that F is well-defined and is a contraction on H
2, α
T for a suitable choice of α 0. Setting u
i
= F v
i
, i = 1, 2, with v
1
, v
2
∈ H
2
T , we have du
1
− u
2
+ [Au
1
− u
2
+ f u
1
− f u
2
] d t = ηu
1
− u
2
d t + Z
Z
[G·, ·, v
1 −
− G·, ·, v
2 −
] d ¯ µ
in the mild sense, with obvious meaning of the slightly simplified notation. We are going to assume that u
1
and u
2
are strong solutions, without loss of generality: in fact, one otherwise approximate A, f and G with A
β
, f
λ
, and G
n
, respectively, and passes to the limit in equation 19 below, leaving the rest of argument unchanged. Setting w
i
t = e
−αt
u
i
t, i = 1, 2, we have, by an argument completely similar to the one used in the proof of Lemma 9,
|w
1
t − w
2
t|
2
≤ η − α Z
t
e
−2αs
|u
1
s − u
2
s|
2
ds + [w
1
− w
2
]t + 2
Z
Z
t
e
−2αs
u
1
s− − u
2
s−, Gs, z, v
1
s− − Gs, z, v
2
s− ¯ µds, dz
.
1544
The previous inequality in turn implies ku
1
− u
2
k
2 2,
α
≤ η − α Z
T
E sup
s≤t
e
−2αs
|u
1
s − u
2
s|
2
ds + 2E sup
t≤T
w
1 −
− w
2 −
· X
1
− X
2
+ E Z
T
Z
Z
e
−2αs
|Gs, z, v
1
s− − Gs, z, v
2
s−|
2
mdz ds, where we have set X
i
:= G·, ·, v
i −
⋆ ¯ µ and we have used the identities
E sup
t≤T
[w
1
− w
2
]t = E[w
1
− w
2
]T = E
Z
T
Z
Z
e
−2αs
|Gs, z, v
1
s − Gs, z, v
2
s|
2
mdz ds. An application of Davis’ and Young’s inequalities, as in the proof of Lemma 9, yields
2E sup
t≤T
w
1 −
− w
2 −
· X
1
− X
2
≤ 6ǫE sup
t≤T
|w
1
t − w
2
t|
2
+ 6ǫ
−1
E Z
T
Z
Z
e
−2αs
|Gs, z, v
1
s − Gs, z, v
2
s|
2
mdz ds, because [X
1
− X
2
] = [w
1
− w
2
]. We have thus arrived at the estimate 1 − 6ǫku
1
− u
2
k
2 2,
α
≤ η − α Z
T
E sup
s≤t
e
−2αs
|u
1
s − u
2
s|
2
d t + 1 + 6ǫ
−1
E Z
T
Z
Z
e
−2αs
|Gs, z, v
1
s − Gs, z, v
2
s|
2
mdz ds 19
Setting ǫ = 112 and φt = E sup
s≤t
e
−2αs
|u
1
s − u
2
s|
2
, we can write, by the hypothesis on G, φT ≤ 2η − α
Z
T
φt d t + 146|h|
L
1
kv
1
− v
2
k
2 2,
α
, hence, by Gronwall’s inequality,
ku
1
− u
2
k
2 2,
α
= φT ≤ 146|h|
1
e
2 η−αT
kv
1
− v
2
k
2 2,
α
. Choosing
α large enough, we obtain that there exists a constant N = N T 1 such that kF v
1
− F v
2
k
2, α
≤ N kv
1
− v
2
k
2, α
. Banach’s fixed point theorem then implies that F admits a unique fixed point in H
2, α
T , which is the unique generalized solution of 17, recalling that the norms k · k
2, α
, α ≥ 0, are all equivalent. Since the fixed point of F can also be obtained as a limit of càdlàg processes
in H
2
T , by the well-known method of Picard’s iterations, we also infer that the generalized mild solution is càdlàg.
1545
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
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