On a class of stochastic partial differential equations with multiple invariant measures

In this work we investigate the long-time behavior, that is the existence and characterization of invariant measures as well as convergence of transition probabilities, for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. Contrary to the existing literature where typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, using a \textit{generalized dissipativity condition} combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we show that these results contain L\'evy driven Ornstein-Uhlenbeck processes, the Heath-Jarrow-Morton-Musiela equation as well as stochastic partial differential equations with delay as a particular case.


Introduction
Stochastic partial differential equations arise in the modelling of applications in mathematical physics (e.g. Navier-Stokes equations [21,17,9,35] or stochastic non-linear Schrödinger equations [4,13]), biology (e.g. catalytic branching processes [12,28]), and finance (e.g. forward prices [23,36,15]). While the construction of solutions to the underlying stochastic equations is an important mathematical issue, having applications in mind it is indispensable to also study their specific properties. Among them, an investigation of the long-time behavior of solutions, that is existence and uniqueness of invariant measures and convergence of transition probabilities, are often important and at the same time also challenging mathematical topics. In this work we investigate the long-time behavior of mild solutions to the stochastic partial differential equation of the form dX t = (AX t + F (X t ))dt + σ(X t )dW t + E γ(X t , ν) N (dt, dν), t ≥ 0 (1.1) on a separable Hilbert space H, where (A, D(A)) is the generator of a strongly continuous semigroup (S(t)) t≥0 on H, (W t ) t≥0 is a Q-Wiener process and N(dt, dν) denotes a compensated Poisson random measure. The precise conditions need to be imposed on these objects will be formulated in the subsequent sections.
In the literature the study on the existence and uniqueness of invariant measures often relies on different variants of a dissipativity condition. The simplest form of such a dissipativity condition is: There exists α > 0 such that Indeed, if (1.2) is satisfied, σ and γ are globally Lipschitz-continuous, and α is large enough, then there exists a unique invariant measure for the Markov process obtained from (1.1), see, e.g., [30,Section 16], [10,Chapter 11,Section 6], and [34]. Note that (1.2) is satisfied, if F is globally Lipschitz continuous and (A, D(A)) satisfies for some β > 0 large enough the inequality Ax, x H ≤ −β x 2 H , x ∈ D(A), i.e. (A, D(A)) is the generator of a strongly continuous semigroup satisfying S(t) L(H) ≤ e −βt . Here and below we denote by L(H) the space of bounded linear operators from H to H and by · L(H) its operator norm. For weaker variants of the dissipativity condition (e.g. cases where (1.2) only holds for x H , y H ≥ R for some R > 0), in general one can neither guarantee the existence nor uniqueness of an invariant measure. Hence, to treat such cases, additional arguments, e.g. coupling methods, are required. Such arguments have been applied to different stochastic partial differential equations on Hilbert spaces in [31,32,33] where existence and, in particular, uniqueness of invariant measures was studied. We also mention [8,22] for an extension of Harris-type theorems for Wasserstein distances, and [24,20] for extensions of coupling methods.
In contrast to the aforementioned methods and applications, several stochastic models exhibit phase transition phenomena where uniqueness of invariant measures fails to hold. For instance, the generator (A, D(A)) and drift F appearing in the Heath-Jarrow-Morton-Musiela equation do not satisfy (1.2), but instead F is globally Lipschitz continuous and the semigroup generated by (A, D(A)) satisfies S(t)x − P x H ≤ e −αt x − P x H for some projection operator P . Based on this property it was shown in [36,34] that the Heath-Jarrow-Morton-Musiela equation has infinitely many invariant measures parametrized by the initial state of the process, see also Section 6. Another example is related to stochastic Volterra equations as studied, e.g., in [6]. There, using a representation of stochastic Volterra equations via SPDEs and combined with some arguments originated from the study of the Heath-Jarrow-Morton-Musiela equation, the authors studied existence of limiting distributions allowing, in particular, that these distributions depend on the initial state of the process.
In this work we provide a general and unified approach for the study of multiple invariant measures and, moreover, we show that with dependence on the initial distribution the law of the mild solution of (1.1) is governed in the limit t → ∞ by one of the invariant measures. In particular, we show that the methods developed in [36,34,6] can be embedded as a special case of a general framework where one replaces (1.2) by a weaker dissipativity condition, which we call generalized dissipativity condition: (GDC) There exists a projection operator P 1 on the Hilbert space H and there exist constants α > 0, β ≥ 0 such that, for x, y ∈ D(A), one has: to the subspace x, y ∈ ker(P 1 ) ∩ D(A) shows that (GDC) contains the classical dissipativity condition on the subspace ker(P 1 ) ⊂ H. Contrary, restricting x, y to ran(P 1 ) ∩ D(A) does not yield the classical dissipativity condition but instead contains the additional term P 1 x − P 1 y 2 H which describes the influence of the non-dissipative part of the drift. Sufficient conditions and additional remarks, e.g., on this condition are collected in the end of Section 2 while particular examples are discussed in Sections 5 -7. Let us mention that (GDC) is satisfied if F is globally Lipschitz continuousmthe semigroup (S(t)) t≥0 is symmetric (i.e. S(t) * = S(t) for t ≥ 0), and uniformly convergent to P 1 (i.e. S(t) − P 1 L(H) −→ 0 as t → ∞).
Roughly speaking, we will show that under condition (GDC) and some restrictions on the projected coefficients P 1 F , P 1 σ, and P 1 γ, the Markov process obtained from (1.1) has for each initial data X 0 = x a limiting distribution π x depending only on P 1 x.
Moreover, the transition probabilities converge exponentially fast in the Wasserstein 2distance to this limiting distributions. In order to prove this result, we first decompose the state space H according to where I denotes the identity operator on H, and then investigate the components P 0 X t and P 1 X t separately. Based on an idea taken from [37], we construct, for each τ ≥ 0, a coupling of X t and X t+τ . This coupling will be then used to efficiently estimate the Wasserstein 2-distance for the solution started at two different points. This work is organized as follows. In Section 2 we state the precise conditions imposed on the coefficients of the SPDE (1.1), discuss some properties of the solution and then provide sufficient conditions for the generalized dissipativity condition (GDC). Based on condition (GDC) we derive in Section 3 an estimate on the trajectories of the process when started at two different initial points, i.e. we estimate the L 2 -norm of X x t − X y t when x = y. Based on this estimate, we state and prove our main results in Section 4. Examples are then discussed in the subsequent Sections 5 -7. In Section 5 we explicitly characterize the limiting distributions of the Lévy driven Ornstein-Uhlenbeck process with an operator (A, D(A)) generating an uniformly convergent semigroup. The Heath-Jarrow-Morton-Musiela equation is then considered in Section 6 for which we first show that the main results of Section 4 contain [36,34], and then extend these results by characterizing its limiting distributions more explicitly. Finally, we apply our results in Section 7 to an SPDE with delay.

Preliminaries
2.1. Framework and notation. Here and throughout this work, (Ω, F, (F t ) t∈R + , P) is a filtered probability space satisfying the usual conditions. Let U be a separable Hilbert space and W = (W t ) t∈R + be a Q-Wiener process with respect to (F t ) t∈R + on (Ω, F, (F t ) t∈R + , P), where Q : U → U is a non-negative, symmetric, trace class operator. Let E be a Polish space, E the Borel-σ-field on E, and µ a σ-finite measure on (E, E). Let N (dt, dν) be a (F t ) t≥0 -Poisson random measure with compensator dtµ(dν) and denote by N (dt, dν) = N (dt, dν) − dtµ(dν) the corresponding compensated Poisson random measure. Suppose that the random objects (W t ) t≥0 and N (dt, dν) are mutually independent.
In this work we investigate the long-time behavior of mild solutions to the stochastic partial differential equation where (A, D(A)) is the generator of a strongly continuous semigroup (S(t)) t≥0 on H, Here B(H) denotes the Borel-σ-algebra on H, and L 0 2 := L 0 2 (H) is the Hilbert space of all Hilbert-Schmidt operators from U 0 to H, where U 0 := Q 1/2 U is a separable Hilbert space endowed with the scalar product x, y 0 := Q −1/2 x, Q −1/2 y U = k∈N 1 λ k x, e k U e k , y U , ∀x, y ∈ U 0 , and Q −1/2 denotes the pseudoinverse of Q 1/2 . Here (e j ) j∈N denotes an orthogonal basis of eigenvectors of Q in U with corresponding eigenvalues (λ j ) j∈N . For comprehensive introductions to integration concepts in infinite dimensional settings we refer to [10] for the case of Q-Wiener processes and to [26] for compensated Poisson random measures as integrators. Throughout this work we suppose that the coefficients F, σ, γ are Lipschitz continuous. More precisely: (A1) There exist constants L F , L σ , L γ ≥ 0 such that for all x, y ∈ H Hence A − (β + √ L F ) is dissipative and thus by the Lumer-Phillips theorem the semigroup (S(t)) t≥0 generated by (A, D(A)) is quasi-contractive, i.e.
where all (stochastic) integrals are well-defined, see, e.g., [1], [26], and [16]. The obtained solution is a Markov process whose transition probabilities p t (x, dy) = P[X x t ∈ dy] are measurable with respect to x. Denote by (p t ) t≥0 its transition semigroup, i.e., for each bounded measurable function f : H −→ R, p t f is given by Using the continuous dependence on the initial condition, see (2.6), it can be shown In this work we investigate the the existence of invariant measures and convergence of the transition probabilities towards these measures for the Markov process (X x t ) t≥0 with particular focus on the cases where uniqueness of invariant measures fails to hold. By slight abuse of notation, we denote by p * t the adjoint operator to p t defined by Recall that a probability measure π on (H, B(H)) is called invariant measure for the semigroup (p t ) t≥0 if and only if p * t π = π holds for each t ≥ 0. Let P 2 (H) be the space of Borel probability measures ρ on (H, B(H)) with finite second moments. Recall that P 2 (H) is separable and complete when equipped with the Wasserstein-2-distance where H(ρ, ρ) denotes the set of all couplings of (ρ, ρ), i.e. Borel probability measures on H × H whose marginals are given by ρ and ρ, respectively, see [38, Section 6] for a general introduction to couplings and Wasserstein distances.

2.2.
Discussion of generalized dissipativity condition. In this section we briefly discuss the condition where λ 0 > 0 and λ 1 ≥ 0. Note that, if (2.9) and condition (2.1) are satisfied, then i.e. the generalized dissipativity condition (GDC) is satisfied for α = λ 0 − √ L F and for all x 0 ∈ H 0 and x 1 ∈ H 1 . Then (2.9) holds for P 1 being the orthogonal projection operator onto H 1 .
Proof. Let P 0 be the orthogonal projection operator onto H 0 . Since (S(t)) t≥0 leaves the closed subspace H 0 invariant, its restriction (S(t)| H 0 ) t≥0 onto H 0 is a strongly continuous semigroup of contractions on H 0 with generator (A 0 , D(A 0 )) being the H 0 part of A, that is Since H 0 is closed and S(t) leaves H 0 invariant, it follows that Ay = lim t→0 . Let P 1 the orthogonal projection operator onto H 1 . Arguing exactly in the same way shows that the restriction (S(t)| H 1 ) t≥0 is a strongly continuous semigroup of contractions on H 1 with generator (A 1 , D(A 1 )) given by Since (e λ 0 t S(t)| H 0 ) t≥0 is a strongly continuous semigroup of contractions on H 0 with generator A 0 + λ 0 I, and (e −λ 1 t S(t)| H 1 ) t≥0 is a strongly continuous semigroup of contractions on H 1 with generator A 1 − λ 1 I, we have by the Lumer-Phillips theorem (see [29,Theorem 4.3]) Hence we find that where the last equality follows from H 0 ⊥ H 1 . This proves the assertion.
At this point it is worthwhile to mention that Onno van Gaans has investigated in [37] ergodicity for a class of Lévy driven stochastic partial differential equations where the semigroup (S(t)) t≥0 was supposed to be hyperbolic. Proposition 2.1 also covers this case, provided that the hyperbolic decomposition is orthogonal. The conditions of previous proposition are satisfied whenever (S(t)) t≥0 is a symmetric, uniformly convergent semigroup.
Remark 2.2. Suppose that (S(t)) t≥0 is a strongly continuous semigroup on H and there exists an orthogonal projection operator P on H and λ 0 > 0 such that Then the conditions of Proposition 2.1 are satisfied for H 0 = ker(P ) and H 1 = ran(P ) with λ 0 > 0 and λ 1 = 0. In particular, (S(t)) t≥0 is a semigroup of contractions.
The following example shows that (2.9) can also be satisfied for non-symmetric and non-convergent semigroups.

Key stability estimate
Define, for x, y ∈ D(A), the function Remark that under the additional assumption that (1.1) has a strong solution, the function is simply the generator L applied to the unbounded function z 2 H , see, e.g,. [2, equation (3.4)]). Since we work with mild solutions instead, all computations given below require to use additionally Yosida approximations for the mild solution of (1.1).
Below we first prove a Lyapunov-type estimate for L( · 2 H ) and then deduce from that by an application of the Itô-formula for mild solutions to (2.1) an estimate for the
Proof. Using first (A1) and then (GDC) we find that This proves the asserted inequality.
The following is our key stability estimate.

2)
and suppose that Then, for each x, y ∈ L 2 (Ω, F 0 , P; H) and all t ≥ 0, Proof. To simplify the notation, denote by (X t ) t≥0 the mild solution to (1.1) with initial condition x and by (Y t ) t≥0 the mild solution to (1.1) with initial condition y. Moreover, we write (X n t ) t≥0 and (Y n t ) t≥0 for the strong solutions to the corresponding Yosida -approximation systems By classical properties of the resolvent (see [29,Lemma 3.2]), one clearly has R n z → z as n → ∞ in H . Moreover, by properties of the Yosida approximation of mild solutions of SPDEs (compare e.g. with Appendix A2 in [26] or Section 2 in and hence there exists a subsequence (which is again denoted by n) such that X n t −→ X t and Y n t −→ Y t hold a.s. for each t ≥ 0. Following a method proposed in [2] we verify that sufficient conditions are satisfied to apply the generalized Itô-formula from Theorem A.2 to the function F (t, z) := e εt z 2 H , where ε = 2α − L σ − L γ is given by (3.2): Observe that, by condition (A1) and (3.3), one has Thus we can apply the generalized Itô-formula from Theorem A.2 and obtain (similar to (3.5) in [2]) where we used, for z, w ∈ D(A), the notation Taking expectations in (3.5) yields Below we prove that the right-hand-side tends to zero as n → ∞, which would imply the assertion of this theorem. To prove the desired convergence to zero we apply the generalized Lebesgue Theorem (see [26,Theorem 7.1.8]). For this reason we have to prove that holds a.s. for each s > 0 as n → ∞ and, moreover, there exists a constant C > 0 such that We start with the proof of (3.7). Denote F n s : For the first term I 1 we estimate Using that X n s → X s and Y n s → Y s as a.s. for some subsequence (also denoted by n), we easily find that the right-hand side tends to zero. The convergence of the second term follows from It remains to show the convergence of the third term. First, observe where the last inequality follows from condition (A1) combined with the inequality H . The first expression I 1 1 clearly tends to zero as n → ∞. For the second expression I 2 3 we use the inequality γ s (ν) − R n γ s (ν) 2 H ≤ 2(2 + λ) 2 γ s (ν) 2 H so that dominated convergence theorem is applicable, which shows that I 2 3 → 0 as n → ∞ a.s.. This proves (3.7). Concerning (3.8), we find that Hence the generalized Lebesgue Theorem is applicable, and thus the assertion of this theorem is proved.
Note that condition (3.3) is used to guarantee that the Itô-formula A.2 for Hilbert space valued jump diffusions can be applied for (x, t) → e tε x 2 H . The assertion of Proposition 3.2 is also true when ε ≤ 0, but will be only applied for the case when ε > 0.
4. Convergence to limiting distribution 4.1. The Case of Vanishing Coefficients. It follows from Proposition 3.2 that, for ε > 0, one has an estimate on the L 2 -norm of the difference X x t − X y t . Such an estimate alone does neither imply the existence nor uniqueness of an invariant distribution. However, if the coefficients F, σ, γ vanish at H 1 , then we may characterize the limiting distributions in L 2 . 2) are satisfied, that (S(t)) t≥0 leaves H 0 := ran(I − P 1 ) invariant, and that ran(P 1 ) ⊂ ker(A). Moreover, assume that Fix x ∈ L 2 (Ω, F 0 , P; H) and suppose that holds for this fixed choice of x. Then In particular, let ρ be the law of x ∈ L 2 (Ω, F 0 , P; H) and ρ 1 be the law of P 1 x, respectively. Then ρ 1 is an invariant measure.
Proof. Fix x ∈ L 2 (Ω, F 0 , P; H) with property (4.2) and set P 0 = I − P 1 . Since (S(t)) t≥0 leaves H 0 invariant we find that S(t)P 0 = P 0 S(t) and hence we obtain P 1 S(t)P 0 = P 1 P 0 S(t) = 0. Moreover, using (4.1) we find that P 1 X x t = P 1 S(t)x = P 1 S(t)P 0 x + P 1 S(t)P 1 x = P 1 x where we have that S(t)P 1 = P 1 due to ran(P 1 ) ⊂ ker(A). From this we conclude that where we have set F (y) := P 0 F (P 1 x+y), σ(y) := P 0 σ(P 1 x+y) and γ(y, ν) := P 0 γ(P 1 x+ y, ν) for all y ∈ H 0 and ν ∈ E. Since these coefficients share the same Lipschitz estimates as F, σ and γ, we can apply Proposition 3.2 to the process (P 0 X x t ) t≥0 obtained from the above auxiliary SPDE, we obtain where we have used that P 0 X 0 t = 0 due to (4.2). This theorem can be applied, for instance, to the Heath-Jarrow-Morton-Musiela equation, see Section 6. Below we discuss two simple examples showing that, in general, conditions ran(P 1 ) ⊂ ker(A) and (4.1) cannot be omitted. (a) Let X t = (X 1 t , X 2 t ) ∈ H = R 2 be given by Then (GDC) holds with P 1 being the projection onto the second coordinate. Hence (4.1) holds, while ran(P 1 ) ⊂ ker(A) is not satisfied. Since Y 2 t = e t Y 2 0 + t 0 e t−s dW 2 s it is clear that Y 2 t does not have a limiting distribution. Hence also Y t cannot have a limiting distribution.

4.2.
The General Case. In Theorem 4.1 we have assumed (4.1), (4.2), and that (S(t)) t≥0 leaves H 0 invariant. Below we continue with the more general case. Namely, for the projection operator P 1 given by condition (GDC) we decompose the process X x t according to X x t = P 0 X x t + P 1 X x t , with P 0 = I − P 1 and suppose that: (A2) The process P 1 X x t is deterministic of the form Our next condition imposes a control on this solution: (A3) For each x ∈ H 1 = ran(P 1 ) there exists X x ∞ ∈ H 1 and constants C(x) > 0, δ(x) ∈ (0, |ε|) such that Note that, if P 1 F (P 1 ·) = 0 then condition (A3) reduces to a condition on the limiting behavior of the semigroup (S(t)) t≥0 when restricted to H 1 = ran(P 1 ). In such a case condition (A3) is, for instance, satisfied if H 1 ⊂ ker(A). The following is our main result for this case.

Theorem 4.3. Suppose that condition (GDC) holds for some projection operator P 1 , that conditions (A1) -(A3), (3.3) and (3.2) are satisfied. Then the following assertions hold:
(a) For each x ∈ H there exists an invariant measure π δx ∈ P 2 (H) for the Markov semigroup (p t ) t≥0 and a constant K(α, β, ε, h) > 0 such that (b) Suppose, in addition to the conditions of (A3), that there are constants δ and C, such that Then, for each ρ ∈ P 2 (H), there exists an invariant measure π ρ ∈ P 2 (H) for the Markov semigroup (p t ) t≥0 and a constant K(α, β, ε) > 0 such that The proof of this theorem relies on the key stability estimate formulated in Proposition 3.2 and is given at the end of this section. So far we have only shown the existence of invariant measures parametrized by the initial state of the process. However, under the given conditions it can also be shown that π δx as well as π ρ depend only on the H 1 part of x or ρ, respectively.

4.3.
Construction of a coupling. Let x ∈ H and let (X x t ) t≥0 be the unique mild solution to (2.7). Below we construct for given τ ≥ 0 a coupling for the law of (X x t , X x t+τ ). Let (Y x,τ t ) t≥0 be the unique mild solution to the SPDE ) is a Q-Wiener process and a Poisson random measure with respect to the filtration (F τ s ) s≥0 defined by F τ s = F s+τ .

Proof. (a) Since (2.7) has a unique solution it follows from the Yamada-Watanabe
Theorem (see [25]) that also uniqueness in law holds for this equation. Since the driving noises N τ and W τ in (4.5) have the same law as N and W from (2.7), it follows that the unique solution to (4.5) has the same law as the solution to (2.7). This proves the assertion.
(b) Set X x,τ t := X x t+τ , then by direct computation we find that where in the last equality we have used, for appropriate integrands Φ(s, ν) and Ψ(s), that Hence (X x,τ t ) t≥0 also solves (4.5) with F τ 0 = F τ and initial condition X x,τ 0 = X x τ . Consequently, the assertion follows from Proposition 3.2 applied to X x,τ t and Y x,τ t .

Proof of Theorem 4.3.
Proof of Theorem 4.3. Fix x ∈ H and recall that p t (x, ·) denotes the transition probabilities of the Markov process obtained from (2.7). Below we prove that (p t (x, ·)) t≥0 ⊂ P 2 (H) is a Cauchy sequence with respect to the Wasserstein distance W 2 . Fix t, τ ≥ 0. We treat the cases τ ∈ (0, 1] and τ > 1 separately. Case 0 < τ ≤ 1: Then using the coupling lemma 4.5.(b) yields The first term I 1 can be estimated by To estimate the second term I 2 we first observe that by condition (A2) we have P 1 Y x,τ s = P 1 X x s and hence by condition (A3) one has for each s ≥ 0 that This readily yields Inserting this into the definition of I 2 gives Case τ > 1: Fix some N ∈ N with τ < N < 2τ and define a sequence of numbers (a n ) n=0,...,N by a n := τ N n, n = 0, . . . , N.
Hence using first the convexity of the Wasserstein distance and then (4.6) we find that For the second term we first use (A2) so that P 1 X x s = P 1 X P 1 x s , P 1 X y s = P 1 X P 1 y s and hence we find for each T > 0 a constant C(T ) > 0 such that for t ∈ [0, T ] Let us choose a particular coupling G as follows: By disintegration we write ρ(dx) = ρ( . Then G is, for A, B ∈ B(H), given by where G is a probability measure on H 2 1 given, for . For this particular choice of G we find that and hence I 2 = 0. This proves (4.8) and completes the proof.

Ornstein-Uhlenbeck process on Hilbert space
Let H be a separable Hilbert space and let (Z t ) t≥0 be a H-valued Lévy process on a stochastic basis (Ω, F, (F t ) t≥0 , P) with the usual conditions. Then, following [3], it has characteristic exponent Ψ of Lévy-Khinchine form, i.e. E e i u,Zt H = e tΨ(u) , u ∈ H, t > 0, with Ψ given by where b ∈ H denotes the drift, Q denotes the covariance operator being a positive, symmetric, trace-class operator on H, and µ is a Lévy measure on H. Let (S(t)) t≥0 be a strongly continuous semigroup on H. The Ornstein-Uhlenbeck process driven by (Z t ) t≥0 is the unique mild solution to D(A)) denotes the generator of (S(t)) t≥0 , i.e. (X x t ) t≥0 satisfies The characteristic function of (X x t ) t≥0 is given by For additional properties, references and related results we refer to the review article [3] where also the existence, uniqueness and properties of invariant measures are discussed. Following these results, the Ornstein-Uhlenbeck process has a unique invariant measure provided that (S(t)) t≥0 is uniformly exponentially stable, that is and the Lévy measure µ satisfies a log-integrability condition for its big jumps Below we show that for a uniformly convergent semigroup (S(t)) t≥0 the corresponding Ornstein-Uhlenbeck process may admit multiple invariant measures parameterized by the range of the limiting projection operator of the semigroup.
Theorem 5.1. Suppose that (S(t)) t≥0 is uniformly exponentially convergent, i.e. there exists a projection operator P on H and constants M ≥ 1, α > 0 such that Suppose that the Lévy process satisfies the following conditions: The covariance operator Q satisfies P Qu = 0 for all u ∈ H.
In particular, the set of all limiting distributions for the Ornstein-Uhlenbeck process (X x t ) t≥0 is given by {δ x * µ ∞ | x ∈ ker(P )}, where µ ∞ denotes the law of X 0 ∞ . Proof. We first prove the existence of a constant C > 0 such that To do so we estimate We find by (5.2) that S(r)x H ≤ M e −αr x H for all x ∈ ker(P ) and hence For the second term I 2 we use ran(Q) ⊂ ker(P ) so that where C > 0 is a generic constant. Proceeding similarly for the last term, we obtain where we have used, for a = u H e −αr , b = z H , the elementary inequalities min{1, ab} ≤ C log(1 + ab) see [18, appendix]. Combining the estimates for I 1 , I 2 , I 3 , I 4 we conclude that ( Since, in view of (5.3), u −→ ∞ 0 Ψ(S(r) * u)dr is continuous at u = 0, the assertion follows from Lévy's continuity theorem combined with the particular form of (5.4).
Below we briefly discuss an application of this result to a stochastic perturbation of the Kolmogorov equation associated with a symmetric Markov semigroup. Let X be a Polish space and η a Borel probability measure on X. Let (A, D(A)) be the generator of a symmetric Markov semigroup (S(t)) t≥0 on H := L 2 (X, η). Then there exists, for each f ∈ D(A), a unique solution to the Kolmogorov equation (see, e.g., [29]) Below we consider an additive stochastic perturbation of this equation in the sense of Itô, i.e. the stochastic partial differential equation where (Z t ) t≥0 is an L 2 (X, η)-valued Lévy process with characteristic function Ψ. Let (v(t); f )) t≥0 be the unique mild solution to this equation.
Corollary 5.2. Suppose that the semigroup generated by (A, D(A)) on L 2 (X, η) satisfies (5.2) with the projection operator and assume that the Lévy process (Z t ) t≥0 satisfies the conditions (i) -(iii) of Theorem is a random variable whose characteristic function is given by

The Heath-Jarrow-Mortion-Musiela equation
The Heath-Jarrow-Morton-Musiela equation (HJMM-equation) describes the term structure of interest rates in terms of its forward rate dynamics modelled, for β > 0 fixed, on the space of forward curves H β = {h : R + → R : h is absolutely continuous and h β < ∞} , (6.1) Such space was first motivated and introduced by Filipovic [14]. Note that h(∞) := lim x→∞ h(x) exists, whenever h ∈ H β . It is called the long rate of the forward curve h. The HJMM-equation on H β is given by where (W t ) t≥0 is a Q-Wiener process and N (dt, dν) is a compensated Poisson random measure on E with compensator dtµ(dν) as defined in Section 2 for H := H β , and (i) A is the infinitesimal generator of the shift semigroup (S(t)) t∈R + on H β , that is The special form of the drift stems from mathematical finance and is sufficient for the absence of arbitrage opportunities. We denote the space of all forward rates with long rate equal to zero by H 0 β = {h ∈ H β : h(∞) = 0}. For the construction of a unique mild solution to (6.2) the following conditions have been introduced in [11]: ν ∈ E and t ≥ 0. (B3) There is an M ≥ 0 such that, for all h ∈ H β , and some β ′ > β has the weak derivative given by (B5) There are constants L σ , L γ > 0 such that, for all h 1 , h 2 ∈ H β , we have The following is the basic existence and uniqueness result for the Heath-Jarrow-Morton-Musiela equation (6.2).
This constant can be choosen as Moreover, for each initial condition h ∈ L 2 (Ω, F 0 , P; H β ) there is a unique adapted, cádlág mild solution (r t ) t≥0 to (6.2).
Proof. This result can be found essentially in [11], where the bound on L F is an immediate result from its derivation.
Using the space of all functions with zero long rate we obtain the decomposition where h(∞) ∈ R is identified with a constant function. Denote by the corresponding projections onto H 0 β and R, respectively. Such a decomposition of H β was first used in [36] to study invariant measures for the HJMM-equation driven by a Q-Wiener process. An extension to the Lévy driven HJMM-equation was then obtained in [34]. The next theorem shows that the results of Section 4 contain the HJMM-equation as a particular case.
Proof. Observe that the assertion is an immediate consequence of Theorem 4.3 and Corollary 4.4. Below we briefly verify the assumptions given in these statements. Condition (A1) follows from (B1), (B5), and (6.3) while the growth condition (3.3) is satisfied by (B3) and the fact that · β ≤ · β ′ for β < β ′ . It is not difficult to see that and that (S(t)) t≥0 leaves H 0 β as well as R ⊂ H β invariant. Hence Remark 2.2 yields that It follows from the considerations in Section 2 (see (2.10)) that (GDC) is satisfied for α = β 2 − √ L F . Consequently, ε = β − 2 √ L F − L σ − L γ and (3.2) holds due to (6.5). Since the coefficients map into H 0 β and S(t)P 1 h = h(∞) = P 1 h, conditions (A2), (A3) and (4.4) are trivially satisfied. The particular form of the estimate (6.6) follows from the proof of Theorem 4.3.
We close this section by applying our results for the particular example of coefficients as introduced in [34]. and since min(a, b 1 ) − min(a, b 2 ) ≤ |b 1 − b 2 | for a, b 1 , b 2 ∈ R + , we also have β . This shows that the Lipschitz condition for σ holds, in particular, for L σ = 1. Consequently, by taking γ ≡ 0, the conditions (B1) -(B5) are satisfied with L σ = 1 and L γ = 0 and M = 1 β for the Lipschitz and growth constants. By (6.4) we get for all β ′ > β. Choosing β ≥ 3 and β ′ > β large enough such that L F < 1, we find that where a and σ are as before and (γ(t, ν)) t≥0 is a predictable, H-valued stochastic process for each ν ∈ E such that For this purpose we first introduce the class of quasi-sublinear functions. for all x, y ≥ 0.