Hypocoercivity for non-linear infinite-dimensional degenerate stochastic differential equations

The aim of this article is to construct solutions to second order in time stochastic partial differential equations and to show hypocoercivity of the corresponding transition semigroups. More generally, we analyze non-linear infinite-dimensional degenerate stochastic differential equations in terms of their infinitesimal generators. In the first part of this article we use resolvent methods developed by Beznea, Boboc and R\"ockner to construct diffusion processes with infinite lifetime and explicit invariant measures. The processes provide weak solutions to infinite-dimensional Langevin dynamics. The second part deals with a general abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser. In order to treat stochastic (partial) differential equations, Grothaus and Stilgenbauer translated these concepts to the Kolmogorov backwards setting taking domain issues into account. To apply these concepts in the context of infinite-dimensional Langevin dynamics we use an essential m-dissipativity result for infinite-dimensional Ornstein-Uhlenbeck operators, perturbed by the gradient of a potential. We allow unbounded diffusion operators as coefficients and apply corresponding regularity estimates. Furthermore, essential m-dissipativity of a non-sectorial Kolmogorov backward operator associated to the dynamic is needed. Poincar\'e inequalities for measures with densities w.r.t. infinite-dimensional non-degenerate Gaussian measures are studied. Deriving a stochastic representation of the semigroup generated by the Kolmogorov backward operator as the transition semigroup of a diffusion process enables us to show an $L^2$-exponential ergodicity result for the weak solution. Finally, we apply our results to explicit infinite-dimensional degenerate diffusion equations.


Introduction
Let (U , (·, ·) U ) and (V , (·, ·) V ) be two real separable Hilbert spaces. We equip W = U × V with the inner product (·, ·) W defined by Denote by B(U ) and B(V ) the Borel σ -algebra on U and V , on which we consider centered non-degenerate Gaussian measures μ 1 and μ 2 , respectively. The measures are uniquely determined by their covariance operators Q 1 ∈ L(U ) and Q 2 ∈ L(V ), respectively. Furthermore, we fix bounded linear operators K 12 ∈ L(U , V ), K 21 ∈ L(V , U ), a symmetric positive bounded linear operator K 22 ∈ L(V ) and a measurable potential : U → (−∞, ∞]. We assume that is bounded from below and U e − dμ 1 = 1. For such we consider the probability measure μ 1 = e − μ 1 on (U , B(U )) and the product measure on (W , B(W )). In [16] we analyzed the infinite-dimensional Langevin operator (L , FC ∞ b (B W )) in L 2 (W , μ , R) defined by where S and A applied to f ∈ FC ∞ b (B W ) are given via Here FC ∞ b (B W ) denotes the space of finitely based smooth (infinitely often differentiable with bounded derivatives) cylinder functions on W and D 1 , D 2 and D 2 2 the first and second order Fréchet derivatives in the first and second component, respectively, see Remark 2.1.
An application of the Itô-formula shows that (L , FC ∞ b (B W )) corresponds to the infinite-dimensional non-linear stochastic differential equation, heuristically given by which is càdlàg. The transition semigroup of the process X extended to L 2 (W , μ , R) coincides with the strongly continuous contraction semigroup associated to (L , FC ∞ b (B W )). The application of this abstract existence result requires that K 22 is of trace class, compare Assumption 3.8. Similar to [7,Theorem 6.4.2.], we show that (X t ) t≥0 has infinite lifetime and weakly continuous paths. In Theorem 4.7 we discuss the existence of weak solutions. We construct a cylindrical Brownian motion (W t ) t≥0 with values in V s.t. P μ -a.s. it holds for all ϑ ∈ D(Q −1 Above, (X t ) t≥0 = ((U t , V t )) t≥0 and P μ = W P w dμ (w).
Above, (T t ) t≥0 is the strongly continuous contraction semigroup generated by the closure of (L , FC ∞ b (B W )). Up to our knowledge this is the first time, this abstract hypocoercivity method is applied to show hypocoercivity of an infinite-dimensional degenerate stochastic differential equation. We would like to stress that the hypocoercivity result for the strongly continuous contraction semigroup (T t ) t≥0 also applies, if K 22 is not a trace class operator. The latter is needed to construct the stochastic process X . At the end of Sect. 6 we describe, assuming K 22 is a trace class operator, the connection between the transition semigroup of X and (T t ) t≥0 . Moreover, we provide an L 2 -exponential ergodicity result for the weak solution of the infinite-dimensional Langevin equation. In the last section, we present explicit examples to which the results from the previous sections are applicable. We would like to emphasize that our results can treat equations, beyond the framework of degenerate semi-linear stochastic partial differential equations, as considered in [28]. In particular, we are able to analyze degenerate stochastic reaction-diffusion equations. The main results obtained in this article are summarized in the following list: • We construct a μ -standard right process X with infinite lifetime and weakly continuous paths, providing a weak solution, via the martingale problem, to the infinite-dimension Langevin equation, see Theorems 4.3 and 4.7.
• Hypocoercivity, i.e. exponential convergence to equilibrium of the strongly continuous semigroup generated by the infinite-dimensional Langevin operator (L , FC ∞ b (B W )) in the Hilbert space L 2 (W , μ , R) with explicitly computable rate of convergence is proved in Theorem 6.10.
• We derive a stochastic representation of the semigroup (T t ) t≥0 in terms of the transition semigroup of the μ -standard right process X . Corollary 6.12 contains an L 2 -exponential ergodicity result, corresponding to the hypocoercivity of the semigroup generated by the infinite-dimensional Langevin operator (L , FC ∞ b (B W )).

Notations
Let U and V be two real separable Hilbert spaces with inner products (·, ·) U and (·, ·) V , respectively. In particular there exist a orthonormal basis B U = (d i ) i∈N and B V = (e i ) i∈N of U and V , respectively. Let W = U × V be the real separable Hilbert space, equipped with the canonical inner product considered in the introduction.
The set of all linear bounded operators from U to U and from U to V are denoted by L(U ) and L(U , V ), respectively. The adjoint of an operator K ∈ L(U , V ) is denoted by K * ∈ L(V , U ). By L + (U ) we shall denote the subset of L(U ) consisting of all positive symmetric operators. The subset in L + (U ) of trace class operators is denoted by . For a given measure space ( , A, m) we denote by L p ( , m, Y ), p ∈ (0, ∞], the Banach space of equivalence classes of A-B(Y ) measurable and p-integrable functions. The corresponding norm is denoted by · L p ( ,m,Y ) . If p = 2 the norm is induced by an inner product denoted by (·, ·) L 2 ( ,m,Y ) . In case ( , A) and and Y is clear from the context we also write L 2 (m) instead of L 2 ( , m, R n ). By λ we denote the Lebesgue measure on (R, B(R)).

Remark 2.1 Given a Fréchet differentiable function
In this article we identify the Fréchet derivative with the gradient. Analogously, for a two times Fréchet differentiable function f : U → R, we identify the second order Fréchet derivative the second order partial derivative in the direction of d i and d j . In particular we have Concerning derivatives of sufficient smooth functions f : Analogously we define D 2 1 f , D 2 2 f as well as ∂ i j,1 f and ∂ i j,2 f .
The orthogonal projection from U to B n U = span{d 1 , ..., d n } is denoted by P n and the corresponding coordinate map by P n , i.e. we have for all u ∈ U , To avoid an overload of notation we use the same notation for the projection to B n V = span{e 1 , ..., e n }. The spaces of finitely based smooth (infinitely often differentiable with bounded derivatives) cylinder functions w.r.t. B U and B W are denoted by , respectively. As in [16] these spaces are defined by denote the spaces of finitely based smooth cylinder functions only dependent on the first n directions defined by denote the space of infinitively differentiable bounded functions from R m and R m × R m to R, respectively. By C ∞ 0 (R n ) we denote the space of compactly supported smooth functions from R n to R.

Assumptions and preliminaries
Let U , V and W as above. We fix two centered non-degenerate Gaussian measure μ 1 and μ 2 with covariance operators Q 1 ∈ L + 1 (U ) and Q 2 ∈ L + 1 (V ) on (U , B(U )) and (V , B(V )), respectively. Let B U = (d i ) i∈N and B V = (e i ) i∈N be the orthonormal basis of eigenvectors with corresponding eigenvalues (λ i ) i∈N and (ν i ) i∈N of Q 1 and Q 2 , respectively. W.l.o.g. we assume that (λ i ) i∈N and (ν i ) i∈N are decreasing to zero. On (W , B(W )) we define the measure μ by In [16] it is shown, that the function spaces FC ∞ b (B U ) and FC ∞ b (B W ) are dense in L 2 (U , μ 1 , R) and L 2 (W , μ, R), respectively.

Remark 3.1
The domain of the closure of We fix a measurable potential : U → (−∞, ∞], which is bounded from below. Moreover, during the whole article we assume U e − dμ 1 = 1 and ∈ W 1,2 (U , μ 1 , R). For such potentials , we consider the measure μ 1 = e − μ 1 on (U , B(U )). On (W , B(W )) we define the measure μ by Another well known approximation result providing a particular sequence of smooth functions in R n is stated in the next lemma. Such sequences are needed several times during the article.
where we define In particular, 0 ≤ ϕ m ≤ 1 and ϕ m = 1 on B m (0) for all m ∈ N. Moreover, ϕ m → 1 pointwisely on R n and Dϕ m , During the article we have to calculate certain Gaussian integrals as written down in the lemma below. The proof uses Isserlis formula from [22] and can be found in [16]. As already described in the introduction we define the infinite-dimensional Langevin operator (L , Throughout the whole article we assume the following: There is a strictly increasing sequence (m k ) k∈N ⊂ N such that for each n ∈ N with n ≤ m k , it holds (iii) There is a constant c K ∈ (0, ∞) such that (iv) D 2 L ∞ (μ 1 ) < ∞. Remark 3.5 Let n ∈ N and f = ϕ(P n (·), P n (·)) ∈ FC ∞ b (B W ) be given. One can show that D 2 f = n i=1 ∂ i,2 ϕ(P n (·), P n (·))e i ∈ B n V and D 1 f ∈ B n U . Hence the invariance properties stated in Assumption 3.4 ensure that expressions like Q −1 In particular, the measure μ is invariant for (L , FC ∞ b (B W )).  (ii) There exists ρ ∈ L 1 (μ ), such that for all n ∈ N, the function ρ n defined by and (ρ n ) n∈N converges to ρ in L 1 (μ ).
Below we introduce an infinite-dimensional Ornstein-Uhlenbeck operator perturbed by the gradient of the potential . This is necessary, since we plan to use the general abstract hypocoercivity framework, described by Grothaus and Stilgenabauer. Indeed, in Sect. 6 we see that these operators naturally appear as we show exponential convergence to equilibrium of the semigroup generated by (L , D(L )).
Moreover we define the operator (N , We want to highlight that the definition of the infinite-dimensional Ornstein Uhlenbeck operator above, allows potentially unbounded diffusion operators (C, D(C)) as coefficients. One can check that (C, D(C)) is symmetric and positive. Moreover, for each n ∈ N with n ≤ m k , it holds B n U ⊂ D(C) as well as C(B n U ) ⊂ B m k U . Assumption 3.10 below is central to apply the abstract hypocoercivity concept described in Sect. 5 and therefore to obtain exponential convergence to equilibrium of the semigroup generated by (L , D(L )). In particular, the assumption guarantees essential m-dissipativity of the infinite-dimensional Ornstein-Uhlenbeck operators (N , FC ∞ b (B U )) and corresponding regularity estimates, compare Theorem 3.11.
We denote the closure of (N , In applications, compare Sect. 7, the regularity estimates above can be used to check Item (v) from Assumption 3.10. In order to check the Poincaré type inequalities stated in Assumption 3.10 above the general Poincaré inequality from the proposition below is useful.
Proof The idea of the proof is to approximate by a sequence of convex and smooth functions. Afterwards, we apply the Poincaré inequality from [1, Proposition 4.5].
I.e., let α > 0 be given and denote by α the Moreau-Yosida approximation, defined by It is well known, see [12], that α is convex, differentiable with Lipschitz continuous , for all u ∈ U and lim α→0 α (u) = (u), for all u ∈ U . But to use the Poincaré inequality from [1, Proposition 4.5] α is not regular enough. Therefore we take β > 0 and define the function α,β by where (B, D(B)) is a self-adjoint negative definite operator such that B −1 is of trace class and μ β the infinite dimensional centered non-degenerate Gaussian measure with covariance operator 1 2 Note that α,β is defined in terms of the Mehler formula of an infinite-dimensional Ornstein-Uhlenbeck operator, see [8,Section 8.3].
Recalling the smoothing properties of such Ornstein-Uhlenbeck semigroups, one can check, compare [8, Section 11.6], that α,β is convex and has derivatives of all orders. Moreover, D α,β is Lipschitz continuous and has bounded derivatives of all orders. This is enough to verify that α,β fulfills Hypothesis 1.3 from [1]. Hence, by the Poincaré inequality from [1,Proposition 4.5] we obtain for all ( Since the derivative of α is Lipschitz continuous one can show that α has at most quadratic growth. Hence, we obtain lim β→0 α,β (u) = α (u), for all u ∈ U , by the theorem of dominated convergence. In particular, we get lim α→0 lim β→0 α,β (u) = I.e., taking the limits β → 0 and α → 0 in Inequality (2) yields the claim.
Note that the Poincaré inequality from the proposition above is valid without assuming that ∈ W 1,2 (U , μ 1 , R) and generalizes the one from [11, Theorem 12.3.8].
Indeed, since Q 1 is trace class, it is easier to estimate Nevertheless, the proof of Proposition 3.12 is inspired by the one from [11, Theorem 12.3.8].

Weak solutions
In this section we construct a solution to the infinite-dimensional Langevin equation where the involved operators were specified in Assumption 3.4, above. As mentioned in the introduction, this equation corresponds to the infinite-dimensional Langevin . First we construct a martingale solution, compare Proposition 4.4 below. Second we show that the martingale solution also provides a weak solution.
to W and the space of continuous bounded functions from W to R, respectively. The potential theoretic notions we need in this article are from [4] and [23]. During the whole section we assume Assumption 3.4. Therefore, we know by Theorem 3.6 we know that the closure (L ,

Remark 4.1
Using that μ is invariant for L and [15, Lemma 1.9, App. B] we know that the semigroup (T t ) t≥0 generated by (L , D(L )) is Markovian, i.e. positive preserving and T t 1 = 1 for all t ≥ 0. Note that the same holds true for the strongly continuous contraction resolvent (R(α, L )) α>0 corresponding to (L , D(L )).
For k, n ∈ N and (u, v) ∈ W we set: Obviously, for all k ∈ N the sets F k are T -compact and increasing. Below we show that the sequence (F k ) k∈N provides a T -compact μ -nest, if Assumption 4.2 is valid. The idea of the proof follows the lines of [4, Proposition 5.5].

Lemma 4.2 Assume Assumption 3.8 holds true. Then the sequence of functions
is non-negative and converges to a non-negative function h in L 1 (μ ). Moreover, the increasing sequence of T -compact sets Note that we used Lemma 3.3 in the calculation above. Since we assume that K 22 is of trace class, we observe that (h n ) n∈N is a non-negative Cauchy-sequence in L 1 (μ ) and therefore convergent to a non-negative function h ∈ L 1 (μ ). Using a suitable selection of cut-off functions we obtain N n ∈ D(L ) for all n ∈ N with Similar to [4,Proposition 5.5] one can show that the closure of (L , FC ∞ b (B W )) in L 1 (μ ) also generates a strongly continuous contraction semigroup with corresponding resolvent denoted by (R 1 (α, L )) α>0 . Note that this resolvent coincides with (R(α, L )) α>0 on L 2 (μ ). Applying R 1 (1, L ) on both sides of the inequality above results in Note that the inequality should be read μ -a.e. and makes use of the Markovianity of the resolvent. As (N n ) n∈N , (g n ) n∈N and (h n ) n∈N converge to N , g and h in L 1 (μ ) we obtain μ -a.e.
Having the T -compact μ -nest at hand the application of [4, Theorem 1.1] is possible.

Theorem 4.3 Assume that Assumption 3.8 holds true. Then there exists a countable
A is a core for (L , D(L )) and 2. A separates the points of W .
Let T 0 be the topology on W generated by A. Then there exists a μ -standard right process (see [4,Appendix B.]) with the state space W (endowed with the topology T 0 ) whose transition semigroup denoted by and identified with its extension to L 2 (μ ). Denote by P μ the probability measure on ( , F) defined for A ∈ F by Then it holds (i) The process has infinite lifetime P μ -a.e..
(ii) The process is T -continuous, P μ -a.e.. (iii) Every element from D(L ) has a μ -quasi continuous version (with respect to the topology T 0 ).
Using a suitable sequence of cut-off functions provided by Lemma 3.2 we can approximate f by a function g = ψ(P n (·), We obtain convergence of the sequence (ψ n m (P n (·), P n (·))) m∈N to g w.r.t. the (L , D(L )) graph-norm. We can conclude that A n is dense in FC ∞ b (B W , n) w.r.t. the (L , D(L )) graph-norm. In particular, the smallest Q-algebra A containing n∈N {ϕ(P n (·), P n (·)) | ϕ ∈ A n }, is a core for (L , D(L )). Moreover, A is a subset of D(L ) ∩ C b (W ) and by construction countable. It is easy to see that A also separates the points of W .
The existence of an ] there is a μ -standard right process with state space W (endowed with the topology T 0 ) whose transition semigroup ( p t ) t>0 extended to L 2 (μ ) coincides with (T t ) t>0 . Additionally by [4, Theorem 1.1] we know that Item (iii) is fulfilled and that (X t ) t≥0 is càdlàg w.r.t. the weak topology T , P μa.e.. Adapting the arguments from [4, Proposition 5.6.] we obtain T -continuity of (X t ) t≥0 . Infinite lifetime P μ -a.e. follows as in [ is the μ -standard right process provided there. Given a non-negative g 0 ∈ L 2 (μ ) with W g 0 dμ = 1. Denote by ν the measure with μ -density g 0 and define the probability measure P ν on ( , F) by Then X solves the martingale problem for (L , D(L )) under P ν . I.e., for all f ∈ D(L ) the process (M defines an (F t ) t≥0 martingale w.r.t. P ν .
If in addition f 2 ∈ D(L ) and L f ∈ L 4 (μ ) then the process (N also defines an (F t ) t≥0 martingale w.r.t. P ν . To construct weak solutions we calculate how the operator (L , D(L )) acts on a certain class of functions.
Then it holds f i , g i , f i f j , g i g j ∈ D(L ), L f i , L g i ∈ L 4 (μ ) and Proof Use suitable cut-off functions provided by Lemma 3.2 and note that D is bounded by Assumption 3.4.
Using the process X = ( , F, Proof Given i, j ∈ N and t ∈ [0, ∞). By the calculations in Lemma 4.6 and Equation (6) we obtain Hence the quadratic variation of M [g i ] and M [g j ] at time t is given by Using that K 22 is trace class we can argue as in [26, Proposition 2.1.10] to show that the series above converges in L 2 ( , F, P ν , C([0, ∞), W )) and defines a 2K 22 Brownian motion. Furthermore, one can show the existence of a cylindrical Brownian motion for all t ∈ [0, ∞). Hence, we obtain by the calculations in Lemma 4.6 and Equation (5), P μ -a.s. for t ∈ [0, ∞) . If we multiply the equation above with (θ, e i ) V and sum over all i ∈ N we get P μ -a.s.
In the last equation we used the dominated convergence theorem. It is applicable since for θ ∈ D(Q −1 Moreover, (V s ) s∈[0,t] and (U s ) s∈[0,t] are bounded P μ -a.s., since they are Tcontinuous P μ -a.s.. The quadratic variation of M [ f i ] at time t ∈ [0, ∞) is equal to zero, by the calculations in Lemma 4.6 and Equation (6). Therefore we get P μa.s. for t ∈ [0, ∞) Using similar arguments as above we obtain P μ -a.s. for ϑ ∈ D(Q −1 2 K 12 ) and for t ∈ [0, ∞)

The Hilbert space hypocoercivity method
This section is devoted to the abstract Hilbert space hypocoercivity method presented in [17]. It is a fundamental tool to establish hypocoercivity of infinite-dimensional Langevin dynamics. The method originated by J. Dolbeault, C. Mouhot and C. Schmeiser in [13], where an algebraic hypocoercivity method, i.e. excluding domain issues, for linear kinetic equations is developed. Below, W always denotes a real Hilbert space with scalar product (·, ·) and induced norm · . All considered operators are assumed to be linear and defined on linear subspaces of W .

Hypocoercivity data (D).
(D1) The Hilbert space. Let (W , F, μ) be a probability space. Set H = L 2 (W , μ, R) and equip it with the usual standard scalar product (·, ·) and denote by · the induced norm. In applications, in particular for the Langevin dynamics later on, the upcoming lemma is a useful tool to verify (A4). Next we formulate the hypocoercivity theorem. The proof can be found in [17], nevertheless we describe in detail how to explicitly compute the constants determining the speed of convergence, compare also [18, Theorem 1.1].

Theorem 5.2 Assume that (D) and (A1)-(A4) hold. Then for each θ
where (T t ) t≥0 denotes the C 0 -semigroup introduced in (D2). The constant θ 2 is explicitly computable in terms of m , M , c 1 and c 2 and given as 1 4 Proof In view of [17, Theorem 2.18] we first choose δ > 0 such that and then ε > 0 small enough such that as well. This particular choice ensures that Now chose κ > 0 smaller or equal than the minimum above. Again, by [17, Theorem 2.18] we obtain T t g − (g, 1) ≤ κ 1 e −κ 2 t g − (g, 1) for all t ≥ 0, To explicitly compute θ 1 and θ 2 as promised in the assertion, we have to specify δ > 0, ε > 0 and the corresponding κ > 0. We use the strategy from [ Moreover, we define In particular κ = v 1+v m r M ,c 1 +s M s M is a valid choice. The convergence rate in terms of κ 1 and κ 2 is given by Therefore, choosing θ 1 = 1 + v and θ 2 = 1 2 κ yields the claimed rate of convergence.

Hypocoercivity of the langevin dynamics
The aim of this section is to show exponential convergence to equilibrium in our extended hypocoercivity framework. Hypocoercivity refers to the semigroup generated by the infinite-dimensional Langevin operator. At this point we emphasize that the calculations and arguments to verify the data conditions (D1)-(D7) and the hypocoercivity Assumptions (A1)-(A4) below are similar to the associated dual statement in the Fokker plank setting in [13] and including domain issues in [17,18]. However, the infinite-dimensionality of our problem results in more challenging calculations and the need of advanced arguments. Consequently we take a closer look at the verification of (D1)-(D7) and (A1)-(A4), again. Throughout this section we assume Assumption 3.4. Hence by Theorem 3.6 the data conditions (D1)-(D4), (D6) and (D7) are already fulfilled. To check the rest we have to specify the orthogonal projection P S .
where the integration is understood w.r.t. the second variable. An application of Fubini's theorem and the fact that (V , B(V ), μ 2 ) is a probability space shows that P S is a well-defined orthogonal projection on L 2 (μ ) satisfying In the definition above we canonically embed L 2 (μ 1 ) into L 2 (μ ). Using that μ is a probability measure one can check that the map P : L 2 (μ ) → L 2 (μ ) given as is an orthogonal projection with It is important to mention that for each f ∈ FC ∞ b (B W ), the functions P S f and P f admit a unique version in The next lemma includes the proof of the data condition (D5) and tells us more about the structure of the infinite-dimensional Langevin operator. Recall the operators defined in Definition 3.9. To avoid to heavy notation we assume w.l.o.g. min k∈N {m k | n ≤ m k } = n for all n ∈ N.

Lemma 6.2 The data condition (D5) holds true, i.e. P(L 2 (μ )) ⊂ D(S), S P = 0 as well as P(FC
Proof Given P f ∈ P(L 2 (μ )). Choose a sequence ( . For all k ∈ N, it holds S P f k = 0, since P f k only depends on the first variable. Using the fact that (S, D(S)) is a closed operator we can conclude that P f ∈ D(S) and S P = 0. As and a sequence of cut-off functions (ϕ m ) m∈N ⊂ C ∞ 0 (R n ) provided by Lemma 3.2. Here, n ∈ N corresponds to the n such that f ∈ FC ∞ b (B W , n). Now we define ϕ n m (v) = ϕ m (P n (v)) and the sequence (g m ) m∈N by Hence, the lac of boundedness of A P f in the second variable is compensated with the sequence of cut-off functions and therefore (g m ) m∈N ⊂ FC ∞ b (B W ). Using the product rule we can calculate for all m ∈ N The theorem of dominated convergence, implies that (A g m ) m∈N converges to In order to show Item (ii) we have to calculate D 1 (A P f ) and D 2 (A P f ). Due to the structure of A P f , D 2 (A P f ) is easily derived to be −Q −1 2 K 12 D 1 f S . Using the fact that A P f only depends on the first n-directions in the first variable one can calculate for all (u, v) ∈ W To sum up we get Using Lemma 3.3 we obtain for all 1 ≤ i, j ≤ n. We can conclude Again, note that the proof above is based on [18,Lemma 3.4] and lifted to our infinite-dimensional setting. Without further tools we can directly show the algebraic relation stated in Sect. 5 Assumption (A1).
Since the Gaussian measure μ 2 is centered we have Hence we can conclude that P S A P f = 0. The fact that yields P A P f = 0 as desired.
The next checkpoint on our list is Assumption (A2), namely the microscopic coercivity. I.e., we address the symmetric part S of L and need an advanced dissipativity statement. In applications this can be checked via the Poincaré inequality for infinitedimensional Gaussian measures stated in Proposition 3.12. Moreover, the remark below can be useful.
Proof In view of the Poincaré inequality from Assumption 3.10 and the integration by parts formula from Theorem 3.6, we obtain for all The upcoming statements are devoted to prove Assumption (A3), i.e. the macroscopic coercivity. Recall the operator G from Sect. 5. Proposition 6.6 Assume Item (i) from Assumption 3.10 is valid. Then the linear oper- Proof For the moment we consider the operator (N ,

The integration by parts formula from Theorem 3.11 implies for all
This implies By Lemma 6.2 Item (iii) we obtain the representation of (G, FC ∞ b (B W )) promised in the assertion. Note that the calculation above also shows G f = N f S . Another application of the integration by parts formula from Theorem 3.11 shows that (G, FC ∞ b (B W )) is symmetric and dissipative in L 2 (μ ). As densely defined symmetric operators on a Hilbert space are essentially self-adjoint if and only if they are essentially mdissipative, it is left to show that We proof this by showing that implies g = 0. So suppose the statement (7) is true. Given n ∈ N and an arbitrary ele- and for all m ∈ N it holds as m → ∞ by the theorem of dominated convergence. Therefore Since n ∈ N was arbitrary it holds .
Proof Given f ∈ FC ∞ b (B W ). It holds by Lemma 3.3 and Item (iv) from Assumption 3.10 With the help of Lemma 5.1 we want verify Assumption (A4), which addresses the boundedness of the auxiliary operators. The central tool to apply the lemma is the elliptic a priori estimate of Dolbeaut, Mouhot and Schmeiser from [13, Sec. 2, Eq. (2.2), Lem 8]. Since Dolbeaut, Mouhot and Schmeiser only deals with finite dimensions in [13], we had to lift their result to our infinite-dimensional setting. Recall that this lifting is discussed in Sect. 3.

Proposition 6.8 Assume that Item (v) from Assumption 3.10 holds true with constant
i.e. the first inequality in (A4) holds true with c 1 = C 1 .
Proof To proof the statement we aim to use Lemma 5.1. Given f ∈ FC ∞ b (B W ) and set g = (I d − G) f . As shown in [17,Proposition 2.15 ] it holds where the last equality is due to Lemma 6.3. An approximation with cut-off functions, as in the proof of Lemma 6.2, shows that A P f ∈ D(S) with Using Lemma 3.3 and Assumption 3.10 we get Therefore the claim follows by Lemma 5.1.

Proposition 6.9
Assume that Item (i) and (ii) from Assumption 3.10 is satisfied. Then the operator (B A (I − P), FC ∞ b (B W )) is bounded and I.e., the second inequality in (A4) holds true with c 2 = 2 √ 2.
Proof As in the proposition above we want to apply Lemma 5.1. For f ∈ FC ∞ b (B W ) and g = (I − G) f it holds by [17,Proposition 2.15 ]

Moreover, by Lemma 3.3 it holds for all
Recall the definition of N from Proposition 6.6. Substituting the second equation into the first, one arrives at Since Pg = P f − PG f = P f − N P f , we get by the regularity estimates from Theorem 3.11 Hence the claim follows by Lemma 5.1 (ii).
In conclusion, we can apply Theorem 5.2 to get the promised hypocoercivity result, summarized in the next theorem. Theorem 6.10 Assume that Assumption 3.4 and 3.10 hold true. Then the operator The semigroup (T t ) t≥0 generated by the closure of (L , FC ∞ b (B W )) is hypocoercive. In particular, for each θ 1 ∈ (1, ∞) there is some θ 2 ∈ (0, ∞) such that The constant θ 2 determining the speed of convergence can be explicitly computed as Proof By the considerations above we can determine the constants from (A2)-(A4). I.e., it holds Applying Theorem 5.2 with these constants yields the claim.

Remark 6.11
In view of Remark 3.12 from [18] one can study the rate of convergence in terms of K 22 . Indeed, if we assume that K 22 = α Q 2 for some α ∈ (0, ∞) one can calculate that ω 1 = αλ 1 and C 1 = α are valid constants. Hence for small α we obtain a bad exponential convergence rate. This reflects the observation that for small α the equation becomes almost deterministic.
Since θ 2 converges to zero for α → ∞ the rate of exponential convergence gets arbitrary bad by considering large α. This is consistent with the calculations in [18,Remark 3.12], where a finite-dimensional Langevin equation is considered. Indeed in [18,Remark 3.12] the authors show that the exponential convergence rate in a large damping regime, which corresponds to large α, is of order 1 α and hence arbitrary bad.
We want to end this section with an L 2 -exponential ergodicity result for the weak solution provided by Theorem 4.7, which is shown in a manifold setting in [24, Corollary 5.2].
Then for all t ∈ (0, ∞) and g ∈ L 2 (μ ), T t g is a μ -version of p t g. Furthermore, it holds for all t ∈ (0, ∞). We call a weak solutions X with this property L 2 -exponentially ergodic, i.e. ergodic with a rate that corresponds to exponential convergence of the corresponding semigroup.
Proof The relation between T t g and p t g, for t > 0 and g ∈ L 2 (μ ), is part of Theorem 4.3. To show ergodicity, let t ∈ (0, ∞) and g ∈ L 2 (μ ) be given. For Note, to obtain the first equality we argued as in [24,Corollary 5.2]. Afterwards, we used the Cauchy-Schwarz inequality and the hypocoercivity of the semigroup. In the last line we computed the integral.
Remark 6.13 From Corollary 6.12 above we can follow, roughly speaking, that time average converges to space average in L 2 (P μ ) with rate t − 1 2 . If the spectrum of (L , D(L )) contains a negative eigenvalue −κ with corresponding eigenvector g, then this rate is optimal. Indeed, by a similar reasoning as in the calculation above we get for all t ∈ (0, ∞) Equality above holds, as the application of the Cauchy-Schwarz inequality is not necessary. Moreover, note that (g, 1) L 2 (μ ) = 1 −κ (L g, 1) L 2 (μ ) = 0.

Examples
We fix a real separable Hilbert space U and a self-adjoint operator Q ∈ L + (U ) with corresponding basis of eigenvectors (e i ) i∈N and a decreasing sequence eigenvalues of we consider Q 1 = Q α 1 and Q 2 = Q α 2 as covariance operators of two centered Gaussian measures μ 1 and μ 2 , respectively. Now we fix a potential ∈ W 1,2 (U , μ 1 , R), which is bounded from below, with bounded gradient and such that U e − dμ 1 = 1. For β 1 , β 2 ∈ [0, ∞) we set K 12 = Q β 1 and K 22 = Q β 2 . Since K 21 = K * 12 , also K 21 = Q β 1 . For this particular choice of operators the infinite-dimensional degener-ate stochastic differential equation (1) reads as follows The associated generator is given by Assuming Item (iv) from Assumption 3.4 is easily verified. Since by construction the other items from Assumption 3.4 are valid, Theorem 3.6 is applicable. I.e., we obtain essential m-dissipativity of (L , FC ∞ b (B W )) in L 2 (W , μ , R). To construct the corresponding diffusion process X with infinite lifetime, we have to verify the items from Assumption 3.8. In terms of α 1 , α 2 , β 1 and β 2 we need: to ensure that K 22 is of trace class. (ii) A function ρ ∈ L 1 (μ ), such that for all n ∈ N, the function ρ n defined by is in L 1 (μ ). Furthermore, for all (u, v) ∈ W we have to check and lim n→∞ ρ n = ρ in L 1 (μ ).
To show hypocoercivtiy, with explicitly computable constants, of the semigroup (T t ) t≥0 generated by (L , D(L )) we use the results from Sect. 6. It is easy to check that the data conditions and the algebraic relation P AP = 0 on D are fulfilled. Hence we are left to verify the microscopic/macroscopic coercivity and the boundedness of the auxiliary operators. The operators (C, D(C)) and (Q −1 1 C, D(Q −1 1 C)) on U are given by C = K 21 Q −1 2 K 12 = Q −α 2 +2β 1 and Q −1 1 C = Q −α 1 −α 2 +2β 1 , respectively. In order to obtain the microscopic coercivity, we use Remark 6.4. I.e., it is enough to assume that To get the macroscopic hypocoercivity we need Moreover, we have to find a constant ω 2 ∈ (0, ∞) s.t. for all f ∈ FC ∞ b (B U ) If : U → (−∞, ∞] is convex and lower semicontinuous we get in view of Proposition 3.12 for all f ∈ FC ∞ b (B U ) In particular, in terms of α 1 , α 2 , β 1 we need to get the macroscopic coercivity with ω 2 = λ 1 . To obtain boundedness of the auxiliary operators, we have to find a constant C 1 ∈ (0, ∞) such that for all f ∈ FC ∞ b (B W ) and g = (I − P A 2 P) f it holds By the regularity estimate from Theorem 3.11 we have I.e., in terms of α 1 , α 2 , β 1 , β 2 this translates to In summary, to obtain hypocoercivity we need the following three conditions β 2 − α 2 ≤ 0, 2β 1 − α 1 ≤ α 2 and 2β 1 − α 1 ≤ 2β 2 − α 2 .
Since the second inequality is implied by the first and the third, we only need In the corollary below we summarize the results we just derived.
(i) If (11) and either (12) or (13) hold true, then there exist a corresponding diffusion process X with infinite lifetime providing a weak solution to (9). (ii) If is convex and lower-semicontinuous, (14) and (15) are valid, then the strongly continuous contraction semigroup (T t ) t≥0 corresponding to (L , FC ∞ b (B W )) is hypocoercive, with explicitly computable constants.
If the assumptions from Item (i) and (ii) are valid, then X is an L 2 -exponentially ergodic weak solution.
Note the boundedness of φ implies that φ grows at most linear. Moreover, such potentials are lower semicontinuous by Fatou's lemma and bounded from below. Using [9, Proposition 5.2] we know that is in W 1,2 (U , μ 1 , R) with D (u) = φ • u for u ∈ L 2 ((0, 1), λ, R). In particular, we obtain As φ is convex the same holds true for . Now the degenerate stochastic reactiondiffusion equation reads as dU t = (− ) −β 1 +α 2 V t dt dV t = − (− ) −β 2 +α 2 V t + (− ) −β 1 +α 1 U t + (− ) −β 1 φ • U t dt Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.