Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise

We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations with a polynomially growing quasi-monotone nonlinearity and multiplicative Poisson noise. We also study existence and uniqueness of invariant measures for the associated semigroup in the Markovian case. A key role is played by a new maximal inequality for stochastic convolutions in $L_p$ spaces.


Introduction
The purpose of this paper is to obtain existence and uniqueness of solutions, as well as existence and uniqueness of invariant measures, for a class of semilinear stochastic partial differential equations driven by a discontinuous multiplicative noise. In particular, we consider the mild formulation of an equation of the type on L 2 (D), with D a bounded domain of n . Here −A is the generator of a strongly continuous semigroup of contractions, F is a nonlinear function satisfying monotonicity and polynomial growth conditions, andμ is a compensated Poisson measure. Precise assumptions on the data of the problem are given in Section 2 below. We would like to note that, under appropriate assumptions on the coefficients, all results of this paper continue to hold if we add a stochastic term of the type B(u(t)) dW (t) to the right hand side of (1), where W is a cylindrical Wiener process on L 2 (D) (see Remark 13 below). For simplicity we concentrate on the jump part of the noise. Similarly, all results of the paper still hold with minimal modifications if we allow the functions F and G to depend also on time and to be random.
While several classes of semilinear stochastic PDEs driven by Wiener noise, also with rather general nonlinearity F , have been extensively studied (see e.g. [9,11,12] and references therein), a corresponding body of results for equations driven by jump noise seems to be missing. Let us mention, however, several notable exceptions: existence of local mild solutions for equations with locally Lipschitz nonlinearities has been established in [20] (cf. also [26]); stochastic PDEs with monotone nonlinearities driven by general martingales have been investigated in [16] in a variational setting, following the approach of [21] (cf. also [3] for an ad hoc method); an analytic approach yielding weak solutions (in the probabilistic sense) for equations with singular drift and additive Lévy noise has been developed in [23]. The more recent monograph [31] deals also with semilinear SPDEs with monotone nonlinearity and additive Lévy noise, and contains a well-posedness result under a set of regularity assumptions on F and the stochastic convolution. In particular, continuity with respect to stronger norms (more precisely, in spaces continuously embedded into L 2 (D)) is assumed. We avoid such conditions, thus making our assumptions more transparent and much easier to verify.
Similarly, not many results are available about the asymptotic behavior of the solution to SPDEs with jump noise, while the literature for equations with continuous noise is quite rich (see the references mentioned above). In this work we show that under a suitably strong monotonicity assumption one obtains existence, uniqueness, and ergodicity of invariant measures, while a weaker monotonicity assumption is enough to obtain the existence of invariant measures.
Our main contributions could be summarized as follows: we provide a) a set of sufficient conditions for well-posedness in the mild sense for SPDEs of the form (1), which to the best of our knowledge is not contained nor can be derived from existing work; b) a new concept of generalized mild solution which allows us to treat equations with a noise coefficient G satisfying only natural integrability and continuity assumptions; c) existence of invariant measures without strong dissipativity assumptions on the coefficients of (1). It is probably worth commenting a little further on the first issue: it is in general not possible to find a triple V ⊂ H ⊂ V (see e.g. [16,21,32] for details) such that A+ F is defined from V to V and satisfies the usual continuity, accretivity and coercivity assumptions needed for the theory to work. For this reason, general semilinear SPDEs cannot be (always) treated in the variational setting. Moreover, the Nemitskii operator associated to F is in general not locally Lipschitz on L 2 (D), so one cannot hope to obtain global well-posedness invoking the local wellposedness results of [20], combined with a priori estimates. Finally, while the analytic approach of [23] could perhaps be adapted to our situation, it would cover only the case of additive noise, and solutions would be obtained only in the sense of the martingale problem.
The main tool employed in the existence theory is a Bichteler-Jacod-type inequality for stochastic convolutions on L p spaces, combined with monotonicity estimates. To obtain well-posedness for equations with general noise, also of multiplicative type, we need to relax the concept of solution we work with, in analogy to the deterministic case (see [4,7]). Finally, we prove existence of an invariant measure by an argument based on Krylov-Bogoliubov's theorem under weak dissipativity conditions. Existence and uniqueness of an invariant measure under strong dissipativity conditions is also obtained, adapting a classical method (see e.g. [13]).
The paper is organized as follows. In Section 2 all well-posedness results are stated and proved, and Section 3 contains the results on invariant measures. Finally, we prove in the Appendix an auxiliary result used in Section 2.
Let us conclude this section with a few words about notation. Generic constants will be denoted by N , and we shall use the shorthand notation a b to mean a ≤ N b. If the constant N depends on a parameter p, we shall also write N (p) and a p b. Given a function f : → , we shall denote its associated Nemitsky operator by the same symbol. Moreover, given an integer k, we shall write f k for the function ξ → f (ξ) k . For any topological space X we shall denote its Borel σ-field by (X ). We shall occasionally use standard abbreviations for stochastic integrals with respect to martingales and stochastic measures, so that H · X (t) : [19] for more details). Given two Banach spaces E and F , we shall denote the set of all

Well-posedness
Let (Ω, , ( t ) t≥0 , ) be a filtered probability space satisfying the usual conditions and E denote expectation with respect to . All stochastic elements will be defined on this stochastic basis, unless otherwise specified. The preditable σ-field will be denoted by . Let (Z, , m) be a measure space,μ a Poisson measure on [0, T ] × Z with compensator Leb ⊗ m, where Leb stands for Lebesgue measure. We shall set, for simplicity of notation, Z t = (0, t]×Z, for t ≥ 0, and L p (Z t ) := L p (Z t , Leb⊗ m). Let D be an open bounded subset of n with smooth boundary ∂ D, and set H = L 2 (D).
The norm and inner product in H are denoted by | · | and 〈·, ·〉, respectively, while the norm in L p (D), p ≥ 1, is denoted by | · | p . Given a Banach space E, we shall denote the set of all Evalued random variables ξ such that |ξ| p < ∞ by p (E). For compactness of notation, we also set p := p (L p (D)). Moreover, we denote the set of all adapted processes u : by p (T ) and p (T ), respectively. Note that ( p (T ), |[ · ]| p ) and ( p (T ), · p ) are Banach spaces. We shall also use the equivalent norms on p (T ) defined by and we shall denote ( p (T ), · p,α ) by p,α (T ).

Additive noise
Let us consider the equation where A is a linear maximal monotone operator on H; f : → is a continuous maximal monotone function satisfying the growth condition . Finally, η is just a constant and the corresponding term is added for convenience (see below). We shall assume throughout the paper that the semigroup generated by −A admits a unique extension to a strongly continuous semigroup of positive contractions on L 2d (D) and L d * (D), d * := 2d 2 . For simplicity of notation we shall not distinguish among the realizations of A and e −tA on different L p (D) spaces, if no confusion can arise. for a detailed proof). The latter class of operators includes also nonlocal operators such as, for instance, fractional powers of the Laplacian, and even more general pseudodifferential operators with negative-definite symbols -see e.g. [18] for more details and examples. Definition 2. Let x ∈ 2d . We say that u ∈ 2 (T ) is a mild solution of (2) if u(t) ∈ L 2d (D) -a.s. and -a.s. for all t ∈ [0, T ], and all integrals on the right-hand side exist.
Let us denote the class of processes G as above such that by p . Setting d * = 2d 2 , we shall see below that a sufficient condition for the existence of the integrals appearing in (3) is that G ∈ d * . This also explains the condition imposed on the sequence {G n } in the next definition.
Definition 3. Let x ∈ 2 . We say that u ∈ 2 (T ) is a generalized mild solution of (2) if there exist a sequence {x n } ⊂ 2d and a sequence {G n } ⊂ d * with x n → x in 2 and G n → G in 2 (L 2 (Z T )), such that u n → u in 2 (T ), where u n is the mild solution of (2) with x n and G n replacing x and G, respectively.
In order to establish well-posedness of the stochastic equation, we need the following maximal inequalities, that are extensions to a (specific) Banach space setting of the corresponding inequalities proved for Hilbert space valued processes in [27], with a completely different proof.
measurable function such that the expectation on the right-hand side of (4) below is finite. Then there exists a constant N = N (p, T ) such that where (p, T ) → N is continuous. Furthermore, let −A be the generator of a strongly continuous semigroup e −tA of positive contractions on E. Then one also has ds, (5) where N is the same constant as in (4).
Proof. We proceed in several steps.
Step 1. Let us assume that m(Z) < ∞ (this hypothesis will be removed in the next step). Note that, by Jensen's (or Hölder's) inequality and Fubini's theorem, one has therefore, since the right-hand side of (4) is finite, Fubini's theorem implies that Minkowski's inequality (see e.g. [24, Thm. 2.4]) implies that the second term on the right-hand side of the previous inequality is less than or equal to We have thus proved that Step 2. Let us turn to the general case m(Z) = ∞. Let {Z n } n∈ a sequence of subsets of Z such that ∪ n∈ Z n = Z, Z n ⊂ Z n+1 and m(Z n ) < ∞ for all n ∈ . By the Bichteler-Jacod inequality for real-valued integrands we have where m n (·) := m(· ∩ Z n ). Integrating both sides of this inequality with respect to ξ over D we obtain, using Fubini's theorem and Minkowski's inequality, we are left with Let us now prove that g n μ(T ) converges to g μ(T ) on D×Ω in Leb⊗ -measure as n → ∞. In fact, by the isometric formula for stochastic integrals with respect to compensated Poisson measures, we have which converges to zero as n → ∞ by the dominated convergence theorem. In fact, g n ↑ g a.e. on D × [0, T ] × Z, -a.s., and Finally, by Fatou's lemma, we have Step 3. Estimate (4) now follows immediately, by Doob's inequality, provided we can prove that g μ is an E-valued martingale. For this it suffices to prove that , the space of infinitely differentiable functions with compact support on D. In fact, we have, by the stochastic Fubini theorem, where the last term has s -conditional expectation equal to zero by well-known properties of Poisson measures. In order for the above computation to be rigorous, we need to show that the last stochastic integral is well defined: using Hölder's inequality and recalling that g ∈ p , we get Step 4. In order to extend the result to stochastic convolutions, we need a dilation theorem due to Fendler [15, Thm. 1]. In particular, there exist a measure space (Y, , n), a strongly continuous group of isometries T (t) onĒ := L p (Y, n), an isometric linear embedding j : Then we have, recalling that the operator norms of π and T (t) are less than or equal to one, where we have used again that T (t) is a unitary group and that the norms ofĒ and E are equal.
Remark 5. (i) The idea of using dilation theorems to extend results from stochastic integrals to stochastic convolutions has been introduced, to the best of our knowledge, in [17].
(ii) Since g μ is a martingale taking values in L p (D), it has a càdlàg modification, as it follows by a theorem of Brooks and Dinculeanu (see [8,Thm. 3]). Moreover, the stochastic convolution also admits a càdlàg modification by the dilation method, as in [17] or [31, p. 161].
We shall need to regularize the monotone nonlinearity f by its Yosida approximation f λ , λ > 0. In and f λ ∈Ċ 0,1 ( ) with Lipschitz constant bounded by 2/λ. For more details on maximal monotone operators and their approximations see e.g. [2,7]. Let us consider the regularized equation which admits a unique càdlàg mild solution u λ ∈ 2 (T ) because −A is the generator of a strongly continuous semigroup of contractions and f λ is Lipschitz (see e.g. [20,27,31]).
We shall now establish an a priori estimate for solutions of the regularized equations.
Proof. We proceed by the technique of "subtracting the stochastic convolution": set Then y λ is also a mild solution in L 2 (D) of the deterministic equation with random coefficients -a.s., where φ (t) := dφ(t)/d t. We are now going to prove that y λ is also a mild solution of (9) in L 2d (D). Settingf and rewriting (9) as y λ (t) + Ay λ (t) +f λ (t, y λ (t)) = 0, we conclude that (9) admits a unique mild solution in L 2d (D) by Proposition 17 below (see the Appendix).
Let y λβ be the strong solution in L 2d (D) of the equation which exists and is unique because the Yosida approximation A β is a bounded operator on L 2d (D). Let us recall that the duality map J : coincides with the Gâteaux derivative of φ → |φ| 2 2d /2. Therefore, multiplying (in the sense of the duality product of L 2d (D) and L 2d 2d−1 (D)) both sides of (10) by the function , thus the second term on the left hand side is positive because J is single-valued. Moreover, we have, omitting the dependence on t for simplicity of notation, as it follows by the monotonicity of f λ . Therefore we can write where we have used Hölder's and Young's inequalities with conjugate exponents 2d and 2d/(2d −1). A simple computation reveals immediately that there exists a constant N depending only on d and η such that . We thus arrive at the inequality and Gronwall's inequality yields hence also, thanks to (5) and the hypothesis that G ∈ d * , where the constant N does not depend on λ. Let us now prove that y λβ → y λ in 2 (T ) as β → 0: By well-known properties of the Yosida approximation we have therefore, by the dominated convergence theorem, the expectation of the left-hand side of the previous expression converges to zero as β → 0. Similarly, we also have where the integrand on the right-hand side converges to zero -a.s. for all s ≤ T . The last inequality also yields, recalling that y λ and G A belong to 2 (T ), hence, by the dominated convergence theorem, the expectation of the left-hand side of the previous expression converges to zero as β → 0. Finally, since A β generates a contraction semigroup, by definition off λ and the fact that f λ has Lipschitz constant bounded by 2/λ, we have using the above expressions, Gronwall's lemma, and letting β → 0, we obtain the claim. Therefore, by a lower semicontinuity argument, we get By definition of y λ we also infer that The a priori estimate just obtained for the solution of the regularized equation allows us to construct a mild solution of the original equation as a limit in 2 (T ), as the following proposition shows.

Proposition 7.
Assume that x ∈ 2d and G ∈ d * . Then equation (2) admits a unique càdlàg mild solution in 2 (T ) which satisfies the estimate Proof. Let u λ be the solution of the regularized equation (7), and u λβ be the strong solution of (7) with A replaced by A β studied in the proof of Lemma 6 (or see [29,Thm. 34.7]). Then u λβ − u µβ solves -a.s. the equation Note that we have thus also, by the monotonicity of A, Multiplying both sides by e −2ηt and integrating we get e −2ηs µ| f λ (u λβ (s))| 2 + λ| f µ (u µβ (s))| 2 ds.
Since u λβ → u λ in 2 (T ) as β → 0 (as shown in the proof of Lemma 6) and f λ is Lipschitz, we can pass to the limit as β → 0 in the previous equation, which then holds with u λβ and u µβ replaced by u λ and u µ , respectively. Taking supremum and expectation we thus arrive at Recalling where the constant N does not depend on λ, hence which shows that {u λ } is a Cauchy sequence in 2 (T ), and in particular there exists u ∈ 2 (T ) such that u λ → u in 2 (T ). Moreover, since u λ is càdlàg and the subset of càdlàg processes in 2 (T ) is closed, we infer that u is itself càdlàg.
Recalling that f λ (x) = f (J λ (x)), J λ x → x as λ → 0, thanks to the dominated convergence theorem and (12) we can pass to the limit as λ → 0 in the equation thus showing that u is a mild solution of (2).
The estimate for sup t≤T |u(t)| 2d 2d is an immediate consequence of (8). We shall now prove uniqueness. In order to simplify notation a little, we shall assume that f is η-accretive, i.e. that r → f (r) + ηr is accretive, and consequently we shall drop the first term on the right hand side of (2). This is of course completely equivalent to the original setting. Let {e k } k∈ ⊂ D(A * ) be an orthonormal basis of H and > 0. Denoting two solutions of (2) by u and v, we have for all k ∈ . Therefore, by Itô's formula, Summing over k and recalling that (I + A * ) −1 = (I + A) −1 * , we obtain Using the monotonicity of A and then letting tend to zero and we are left with which immediately implies that u = v by Gronwall's inequality.
Let us now prove Lipschitz continuity of the solution map. Set u 1 := u(x 1 ), u 2 := u(x 2 ), and denote the strong solution of (2) with A replaced by A β , f replaced by f λ , and initial condition x i , i = 1, 2, by u i λβ , i = 1, 2, respectively. Then we have, omitting the dependence on time for simplicity, -a.s. in the strong sense. Multiplying, in the sense of the scalar product of L 2 (D), both sides by u 1 λβ − u 2 λβ and taking into account the monotonicity of A and f , we get which implies, by Gronwall's inequality and obvious estimates, Since, as seen above, u i λβ → u i , i = 1, 2, in 2 (T ) as β → 0, λ → 0, we conclude by the dominated convergence theorem that u 1 − u 2 2 ≤ e ηT |x 1 − x 2 | 2 . Remark 8. We would like to emphasize that proving uniqueness treating mild solutions as if they were strong solutions, as is very often done in the literature, does not appear to have a clear justification, unless the nonlinearity is Lipschitz continuous. In fact, if u is a mild solution of (2) and u β is a mild (or even strong) solution of the equation obtained by replacing A with A β in (2), one would at least need to know that u β converges to the given solution u, which is not clear at all and essentially equivalent to what one wants to prove, namely uniqueness.
A general proof of uniqueness for mild solutions of stochastic evolution equations with dissipative nonlinear drift and multiplicative (Wiener and Poisson) noise is given in [28].
In order to establish well-posedness in the generalized mild sense, we need the following a priori estimates, which are based on Itô's formula for the square of the norm and regularizations. Lemma 9. Let x 1 , x 2 ∈ 2d , G 1 , G 2 ∈ d * , and u 1 , u 2 be mild solutions of (2) with x = x 1 , G = G 1 and x = x 2 , G = G 2 , respectively. Then one has (13) and Proof. Let u λ and u λβ be defined as in the proof of Proposition 7. Set w i (t) = e −ηt u i λβ (t). Itô's formula for the square of the norm in H yields where M is a local martingale. In particular, since A and f are monotone, we are left with In particular, taking expectations on both sides (if necessary, along a sequence {τ n } n∈ of localizing stopping times for the local martingale M , and then passing to the limit as n → ∞), we obtain where we have used the identity Recalling that u i λβ → u i λ , i = 1, 2, in 2 (T ) as β go to zero (see the proof of Lemma 6 or e.g. [27,Prop. 3.11]), we get that the above estimate holds true for u 1 λ , u 2 λ replacing u 1 λβ , u 2 λβ , respectively. Finally, since mild solutions are obtained as limits in 2 (T ) of regularized solutions for λ → 0, (13) follows.
By (15) and (16) we get sup t≤T e −2ηt |u 1 Note that where X i := G i * μ, i = 1, 2. Thanks to Davis' and Young's inequalities we can write Therefore we have hence, passing to the limit as β and λ go to zero, we obtain (14).
Proof. Let us choose a sequence {x n } ⊂ 2d such that x n → x in 2 , and a sequence {G n } ⊂ d * such that G n → G in 2 (L 2 (Z T )) (e.g. by a cut-off procedure 1 ). By Proposition 7 the stochastic equation admits a unique mild solution u n . Then (14) yields In particular {u n } is a Cauchy sequence in 2 (T ), whose limit u ∈ 2 (T ) is a generalized mild solution of (2). Since u n is càdlàg for each n by Proposition 7, u is also càdlàg.
i.e. the solution map is Lipschitz, which in turn implies uniqueness of the generalized mild solution.
Remark 11. One could also prove well-posedness in 2 (T ), simply using estimate (13) instead of (14). In this case one can also get explicit estimates for the Lipschitz constant of the solution map. On the other hand, one cannot conclude that a solution in 2 (T ) is càdlàg, as the subset of càdlàg processes is not closed in 2 (T ).

Multiplicative noise
Let us consider the stochastic evolution equation with initial condition measurable function, and we denote its associated Nemitski operator, which is a mapping from We have the following well-posedness result for (17) in the generalized mild sense.

Theorem 12.
Assume that x ∈ 2 and G satisfies the Lipschitz condition Then (17) admits a unique generalized solution u ∈ 2 (T ). Moreover, the solution map is Lipschitz from 2 to 2 (T ).

Proof.
For v ∈ 2 (T ) and càdlàg, consider the equation Since (s, z) → G(s, z, v(s−)) satisfies the hypotheses of Proposition 10, (18) admits a unique generalized mild solution belonging to 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 2,α (T ) for a suitable choice of α > 0. Setting 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, The previous inequality in turn implies where we have set X i := G(·, ·, v i − ) μ and we have used the identities An application of Davis' and Young's inequalities, as in the proof of Lemma 9, yields 2 sup We have thus arrived at the estimate (19) Setting = 1/12 and φ(t) = sup s≤t e −2αs |u 1 (s) − u 2 (s)| 2 , we can write, by the hypothesis on G, hence, by Gronwall's inequality, Choosing α large enough, we obtain that there exists a constant N = N (T ) < 1 such that Banach's fixed point theorem then implies that F admits a unique fixed point in 2,α (T ), which is the (unique) generalized solution of (17), recalling that the norms · 2,α , α ≥ 0, are all equivalent. Since the fixed point of F can also be obtained as a limit of càdlàg processes in 2 (T ), by the well-known method of Picard's iterations, we also infer that the generalized mild solution is càdlàg.
Moreover, denoting u(x 1 ) and u(x 2 ) by u 1 and u 2 respectively, an argument similar to the one leading to (19) yields the estimate where ψ(t) := sup s≤t |u 1 (s) − u 2 (s)| 2 . By Gronwall's inequality we get which proves that x → u(x) is Lipschitz from 2 to 2,α (T ), hence also from 2 to 2 (T ) by the equivalence of the norms · 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 B(t, u(t)) 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 (quasi)dissipative type driven by both multiplicative Poisson and Wiener noise will be given in a forthcoming article.

Invariant measures and Ergodicity
Throughout this section we shall additionally assume that G : Z × H → H is a (deterministic) ⊗ (H)-measurable function satisfying the Lipschitz assumption for some K > 0. The latter assumption guarantees that the evolution equation is well-posed by Theorem 12. Moreover, it is easy to see that the solution is Markovian, hence it generates a semigroup via the usual formula P t ϕ(x) := ϕ(u(t, x)), ϕ ∈ B b (H). Here B b (H) stands for the set of bounded Borel functions from H to .

Strongly dissipative case
Throughout this subsection we shall assume that there exist β 0 and ω 1 > K such that for all β ∈]0, β 0 [, λ ∈]0, β 0 [, and for all u, v ∈ H. This is enough to guarantee existence and uniqueness of an ergodic invariant measure for P t , with exponentially fast convergence to equilibrium.
Note that we have, by Young's inequality, for any > 0, By (20) we thus have, for β < β 0 , λ < λ 0 ∧ β 0 , Taking expectations in (23), applying the above inequality, and passing to the limit as β → 0, λ → 0, and n → ∞, yields Note that, similarly as before, we have Setting we are left with We can now choose such that ω 2 > 0. Gronwall's inequality then yields Set u 1 (t) := u(t; s 1 , x), u 2 (t) := u(t; s 2 , x) and w(t) = u 1 (t) − u 2 (t), with s 2 < s 1 . Then w satisfies the equation with initial condition w(s 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 and hence, by Gronwall's inequality, Estimate (24) therefore implies that there exists a constant N such that which converges to zero as s 1 → −∞. We have thus proved that {u(0; s, x)} s≤0 is a Cauchy net in 2 , hence there exists ζ = ζ(x) ∈ 2 such that u(0; s, x) → ζ in 2 as s → −∞. Let us show that ζ does not depend on x. In fact, let x, y ∈ 2 and set u 1 (t) = u(t; s, x), u 2 (t) = u(t; 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, which implies ζ(x) = 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 ∈ 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 where in the last step we have used the estimate (26).

Weakly dissipative case
In this subsection we replace the strong dissipativity condition (20) with a super-linearity assumption on the nonlinearity f , and we prove existence of an invariant measure by an argument based on Krylov-Bogoliubov's theorem.

Theorem 16. Assume that (i) f satisfies the super-linearity condition
is compactly embedded into L 2 (D).
Then there exists an invariant measure for the transition semigroup associated to the generalized mild solution of (17).
Proof. Let u, u n λβ , Γ, and Γ n be defined as in the proof of Lemma 15. Then an application of Itô's formula yields the estimate we obtain, taking into account the monotonicity of A β , By assumption (i) and Jensen's inequality, we have Passing to the limit as β → 0, λ → 0, n → ∞, recalling the definition of Γ and Γ n , and taking into account the Lipschitz continuity of G, shows that y(t) := |u(t)| 2 satisfies the differential inequality (in its integral formulation, to be more precise) y ≤ a y − b y 1+α + c, y(0) = |x| 2 , for some positive constants a and c. By simple ODE techniques one obtains that y(t) is bounded for all t, i.e. |u(t)| 2 ≤ C for all t ≥ 0, for some positive constant C.
As we have seen above, the right-hand side of the inequality converges, as β → 0, λ → 0, n → ∞, to where N is a constant that does not depend on λ, β, and n, and we have used the fact that |u(t)| 2 is bounded for all t ≥ 0. In analogy to an earlier argument, setting z β := (I + βA) −1 z, z ∈ H, we have 〈A β z, z〉 = 〈Az β , z β + z − z β 〉 = 〈Az β , z β 〉 + β|A β z| 2 .
In particular, setting v n λβ := (I + βA) −1 u n λβ ∈ D( ), we obtain where B c R stands for the complement in of the closed ball of radius R in . Since balls in are compact sets of L 2 (D), we infer that (ν n ) n≥1 is tight, and Krylov-Bogoliubov's theorem guarantees the existence of an invariant measure.

A Auxiliary results
The following proposition is a slight modification of [30, Thm. 6.1.2] and it is used in the proof of Lemma 6. Here [0, T ] ⊂ and E is a separable Banach space.
As a second step, we show that the map