Essential m-dissipativity for Possibly Degenerate Generators of Infinite-dimensional Diffusion Processes

Abstract

First essential m-dissipativity of an infinite-dimensional Ornstein-Uhlenbeck operator N, perturbed by the gradient of a potential, on a domain $\mathcal {F}C_b^{\infty }$ of finitely based, smooth and bounded functions, is shown. Our considerations allow unbounded diffusion operators as coefficients. We derive corresponding second order regularity estimates for solutions f of the Kolmogorov equation $\alpha f-Nf=g$, $\alpha \in (0,\infty )$, generalizing some results of Da Prato and Lunardi. Second, we prove essential m-dissipativity for generators $(L_{\Phi },\mathcal {F}C_b^{\infty })$ of infinite-dimensional degenerate diffusion processes. We emphasize that the essential m-dissipativity of $(L_{\Phi },\mathcal {F}C_b^{\infty })$ is useful to apply general resolvent methods developed by Beznea, Boboc and Röckner, in order to construct martingale/weak solutions to infinite-dimensional non-linear degenerate stochastic differential equations. Furthermore, the essential m-dissipativity of $(L_{\Phi },\mathcal {F}C_b^{\infty })$ and $(N,\mathcal {F}C_b^{\infty })$, as well as the regularity estimates are essential to apply the general abstract Hilbert space hypocoercivity method from Dolbeault, Mouhot, Schmeiser and Grothaus, Stilgenbauer, respectively, to the corresponding diffusions.

1 Introduction

The classical Langevin dynamics (compare [23, Chapter 8.1])

$$\begin{aligned} \begin{aligned}&\mathrm {d}X_t=Y_t\mathrm {d}t\\&\mathrm {d}Y_t=(-Y_t-D\Phi (X_t) )\mathrm {d}t+\sqrt{2}\mathrm {d}W_t, \end{aligned} \end{aligned}$$

(1.1)

describes the evolution of the positions $X_t=(X_t^{(1)},...,X_t^{(n)})\in ({\mathbb {R}}^d)^n$ and velocities $Y_t=(Y_t^{(1)},...,Y_t^{(n)})\in ({\mathbb {R}}^d)^n$ of n particles in dimension d, where $(W_t)_{t\ge 0}$ is a standard Brownian Motion in $({\mathbb {R}}^d)^n$. I.e. the velocity of the particles is subjected to friction and a stochastic perturbation. The n-particle potential $\Phi :({\mathbb {R}}^d)^n\rightarrow {\mathbb {R}}$, with gradient $D\Phi$, affects the motion of the particles and can be used to model their interactions.

The equation has been studied under various aspects. In order to show exponential convergence to equilibrium of such type of non-coercive evolution equations, Cedric Villani developed the concepts of hypocoercivity, see [25]. Abstract hypocoercivity concepts for quantitative descriptions of convergence rates are introduced in [13]. These are translated to the corresponding stochastic equations, taking domain issues into account, in [15]. In [16] these concepts have been further generalized to the case where only a weak Poincaré inequality is needed. In this case one obtains (sub-)exponential convergence rates. Ergodicity and rate of convergence to equilibrium of the Langevin dynamics with singular potentials are elaborated e.g. in [3] and [17]. Recently, the dynamics and its hypocoercivity behavior is studied on abstract smooth manifolds, see [20]. The latter articles cited above have in common that they study the associated Kolmogorov backward operator. Applying Itô’s formula, the Kolmogorov backward operator associated to (1.1), also called Langevin operator, applied to $f\in C_0^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d)$ (w.l.o.g $n=1$) is given by

$$\begin{aligned} L_{\Phi }f=\Delta _2f-\langle x,D_2f\rangle -\langle D\Phi ,D_2f\rangle +\langle y,D_1f\rangle . \end{aligned}$$

Here, $C_0^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d)$ denotes the space of compactly supported smooth (infinitely often differentiable) functions from ${\mathbb {R}}^d\times {\mathbb {R}}^d$ to ${\mathbb {R}}$, x and y the projection to the spatial and the velocity component, respectively, $\langle \cdot ,\cdot \rangle$ the euclidean inner product, $\Delta _2$, $D_2$ the Laplacian and the gradient in the velocity component and $D_1$ the gradient in the spatial component. The key observation is the essential m-dissipativity of $(L_{\Phi },C_0^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d))$ defined in $L^2({\mathbb {R}}^d\times {\mathbb {R}}^d,\mu ^{\Phi },{\mathbb {R}})$, where

$$\begin{aligned} \mu ^{\Phi }=(2\pi )^{-\frac{d}{2}}e^{-\Phi (x)-\frac{1}{2}y^2}\lambda ^d\otimes \lambda ^d. \end{aligned}$$

We want to emphasize the degenerate structure of the Langevin equation, i.e. the noise only appears in the velocity component. The degeneracy of the equation corresponds to the fact that the Laplacian in the definition of $(L_{\Phi },C_0^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d))$ is degenerate, i.e. only acts in the velocity component. As the antisymmetric part of $(L_{\Phi },C_0^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d))$ in $L^2({\mathbb {R}}^d\times {\mathbb {R}}^d,\mu ^{\Phi },{\mathbb {R}})$ contains first order differential operators in the spatial component and the symmetric part only differential operators in the velocity component, the operator $(L_{\Phi },C_0^{\infty }({\mathbb {R}}^d\times {\mathbb {R}}^d))$ is non-sectorial.

In this article we address an infinite-dimensional generalization of the Langevin operator above. In order to do that let $(U,(\cdot ,\cdot )_U)$ and $(V,(\cdot ,\cdot )_V)$ be two real separable Hilbert spaces. Consider the real separable Hilbert space $W=U\times V$ with inner product $(\cdot ,\cdot )_W$ defined by

$$\begin{aligned} ((u_1,v_1),(u_2,v_2))_W=(u_1,u_2)_U+(v_1,v_2)_V,\quad (u_1,v_1),(u_2,v_2)\in W. \end{aligned}$$

Denote by $\mathcal {B}(U)$ and $\mathcal {B}(V)$ the Borel $\sigma$-algebra on U and V, on which we consider centered non-degenerate infinite-dimensional Gaussian measures $\mu _1$ and $\mu _2$, respectively. The measures are uniquely determined by their covariance operators $Q_1\in \mathcal {L}(U)$ and $Q_2\in \mathcal {L}(V)$. Furthermore, we consider bounded linear operators $K_{12}\in \mathcal {L}(U;V)$, $K_{21}\in \mathcal {L}(V;U)$ and a symmetric bounded linear operator $K_{22}\in \mathcal {L}(V)$. For a given measurable potential $\Phi :U\rightarrow (-\infty ,\infty ]$, which is bounded from below, we set $\rho _{\Phi }=\frac{1}{c_{\Phi }}e^{-\Phi }$, where $c_{\Phi }=\int _U e^{-\Phi }\mathrm {d}\mu _1$ and consider the measure $\mu _1^{\Phi }=\rho _{\Phi }\mu _1$ on $(U,\mathcal {B}(U))$. On $(W,\mathcal {B}(W))$ we introduce the product measure

$$\begin{aligned} \mu ^{\Phi }=\mu _1^{\Phi }\otimes \mu _2. \end{aligned}$$

The infinite-dimensional Langevin operator $(L_{\Phi },\mathcal {F}C_b^{\infty })$ is defined by

$$\begin{aligned} \mathcal {F}C_b^{\infty }\ni f \mapsto L_{\Phi }f=S_{\Phi }f-A_{\Phi }f\in L^2(\mu ^{\Phi }), \end{aligned}$$

where for $f\in \mathcal {F}C_b^{\infty }$, $S_{\Phi }f$ and $A_{\Phi }f$ are given by

$$\begin{aligned} S_{\Phi }f=&\mathrm {tr}[K_{22}D^2_2f]-(v,Q_2^{-1}K_{22}D_2f)_V ,\\ A_{\Phi }f=&(u,Q_1^{-1}K_{21}D_2f)_U+(D \Phi (u),K_{21}D_2f)_U-(v,Q_2^{-1}K_{12}D_1f)_V. \end{aligned}$$

Above, $\mathcal {F}C_b^{\infty }$ is a space of finitely based, smooth and bounded functions, see Definition 2.3 and 4.1, below. Furthermore, u and v denotes the projections of W to U and V, respectively. One of the major challenges in this article is to show essential m-dissipativity of the infinite-dimensional Langevin operator in $L^2(\mu ^{\Phi })$.

We also address essential m-dissipativity and regularity estimates in $L^2(\mu _1^{\Phi })$ for infinite-dimensional Ornstein-Uhlenbeck operators, perturbed by the gradient of a potential $\Phi$. Indeed we fix a possible unbounded linear operator (C, D(C)) in U and introduce the operator $(N,\mathcal {F}C_b^{\infty })$ in $L^2(\mu _1^{\Phi })$ defined by

$$\begin{aligned} \mathcal {F}C_b^{\infty }\ni f \mapsto Nf=&\mathrm {tr}[CD^2f]-(u,Q_1^{-1}CD f)_U-(D\Phi ,CDf)_U\in L^2(\mu _1^{\Phi }). \end{aligned}$$

In [2, 7] and [8] similar operators and corresponding regularity estimates are studied, but the results are restricted to bounded diffusion operators (C, D(C)) as coefficients.

The essential m-dissipativity of $(N,\mathcal {F}C_b^{\infty })$ in $L^2(\mu _1^{\Phi })$ is useful in various applications. E.g. to study stochastic quantization problems as in [8, Section 4] and to solve stochastic reaction diffusion equations as in [7, Section 5]. In addition, essential m-dissipativity and related regularity estimates of such operators, will be essential for our planed application of the general abstract hypocoercivity method from [15] to our infinite-dimensional setting. For this application it is needed to allow unbounded diffusion operators (C, D(C)) as coefficients in the definition of the perturbed Ornstein-Uhlenbeck operator $(N,\mathcal {F}C_b^{\infty })$.

The organization of this article is as follows. First, we fix notions and define several important spaces. Then properties of infinite-dimensional Gaussian measures are elaborated, especially the relation between finite and infinite-dimensional Gaussian measures in Lemma 2.1 and the integration by parts formula from Corollary  are focused. In Theorem 2.9 we use the integration by parts formula to describe Sobolev spaces w.r.t. infinite-dimensional Gaussian measures.

Sect. 3 introduces necessary conditions on (C, D(C)) (compare Hypothesis 3.1) to obtain essential m-dissipativity of an Ornstein-Uhlenbeck operator with diffusion operator (C, D(C)) as coefficient. In Theorem 3.6 we perturb this Ornstein-Uhlenbeck operator by the gradient of a potential $\Phi :U\rightarrow (-\infty ,\infty ]$, which is in $W^{1,2}(U,\mu _1,{\mathbb {R}})$ and bounded from below. If $D\Phi$ is strictly bounded by $\frac{1}{2\sqrt{\lambda _1}}$, where $\lambda _1$ is the biggest eigenvalue of $Q_1$, we obtain essential m-dissipativity of $(N,\mathcal {F}C_b^{\infty })$ in $L^2(\mu _1^{\Phi })$. Note that the restriction to such potentials is due to the possible unboundedness of (C, D(C)). In the second part of this section we imitate the strategy used in [7] to derive an infinite-dimensional second order regularity estimate for $f\in \mathcal {F}C_b^{\infty }$ in terms of $g=\alpha f-Nf$, $\alpha \in (0,\infty )$.

In Sect. 4, we deal with the essential m-dissipativity of $(L_{\Phi },\mathcal {F}C_b^{\infty })$ in $L^2(\mu ^{\Phi })$. First, we consider the case where the potential $\Phi =0$. We decompose our infinite-dimensional Langevin operator into countable finite-dimensional ones, to use arguments for finite-dimensional Langevin operators as described in [21] and [6]. We derive first order regularity estimates needed to add perturbations in terms of $\Phi$. I.e. we consider potentials $\Phi$ as in Theorem 4.11 and use the Neumann-Series theorem to obtain essential m-dissipativity of $(L_{\Phi },\mathcal {F}C_b^{\infty })$ in $L^2(\mu ^{\Phi })$. During the whole section we assume Hypothesis 4.3, which is the key to the decomposition described above.

Applications of the results, we derived in Sect. 3 and Sect. 4, are discussed in the last section. We propose an infinite-dimensional non-linear degenerate stochastic differential equation, see (5.1). With the results we achieved in this article and the resolvent methods from [1] we plan to solve it. Moreover, we elaborate, how the essential m-dissipativity of $(N,\mathcal {F}C_b^{\infty })$ can be used to show hypocoercivity of the semigroup $(T_t)_{t\ge 0}$ generated by the closure of $(L_{\Phi },\mathcal {F}C_b^{\infty })$ and how hypocoercivity of $(T_t)_{t\ge 0}$ is related to the long time behavior of the process solving (5.1). The main results obtained in this article are summarized in the following list:

• We prove essential m-dissipativity of perturbed Ornstein-Uhlenbeck operators $(N,\mathcal {F}C_b^{\infty })$ in $L^2(\mu _1^{\Phi })$. We allow possible unbounded diffusions (C, D(C)) as coefficients, see Theorem 3.6. There Hypothesis 3.1 is assumed and perturbations by the gradient of a potential $\Phi$, which is bounded from below and in $W^{1,2}(U,\mu _1,{\mathbb {R}})$, are considered. In addition, an appropriate bound for the gradient of $\Phi$, i.e. $\Vert D\Phi \Vert _{L^{\infty }(\mu _1)}<\frac{1}{2\sqrt{\lambda _1}}$, where $\lambda _1$ is the biggest eigenvalue of $Q_1$ (see Theorem 3.6), is needed.

• Considering potentials $\Phi \in W^{1,2}(U,\mu _1,{\mathbb {R}})$, which are convex, bounded from below and lower semicontinuous as in Hypothesis 3.9, we provide second order regularity estimate for $f\in \mathcal {F}C_b^{\infty }$ in terms of $g=\alpha f-Nf$, $\alpha \in (0,\infty )$. Indeed by Theorem 3.11 it holds

  $$\begin{aligned} \int _U \mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2\mathrm {d}\mu _1^{\Phi }\le 4\int _U g^2\mathrm {d}\mu _1^{\Phi }, \end{aligned}$$

  where (C, D(C)) is the possible unbounded diffusion coefficient in the definition of $(N,\mathcal {F}C_b^{\infty })$.

• Essential m-dissipativity of the infinite-dimensional Langevin operator $(L_{\Phi },\mathcal {F}C_b^{\infty })$ in $L^2(\mu ^{\Phi })$ is shown in Theorem 4.11. We consider potentials $\Phi$ in $W^{1,2}(U,\mu _1,{\mathbb {R}})$, which are bounded from below, with bounded gradient and assume Hypothesis 4.3 and 4.10.

2 Notations and Preliminaries

Let U and V be two real separable Hilbert spaces with inner products $(\cdot ,\cdot )_U$ and $(\cdot ,\cdot )_V$, respectively. The induced norms are denoted by $\Vert {\cdot }\Vert _U$ and $\Vert {\cdot }\Vert _V$. The set of all linear bounded operators from U to U and from U to V are denoted by $\mathcal {L}(U)$ and $\mathcal {L}(U;V)$. The adjoint of an operator $J\in \mathcal {L}(U;V)$ is denoted by $J^*$. By $\mathcal {L}^+(U)$ we shall denote the subset of $\mathcal {L}(U)$ consisting of all nonnegative symmetric operators. The subset of operators in $\mathcal {L}^+(U)$ of trace class is denoted by $\mathcal {L}^+_1(U)$ and the set of Hilbert-Schmidt operators by $\mathcal {L}_2(U)$.

Suppose we have $J\in \mathcal {L}^+(U)$. If J is injective it is reasonable to talk about the inverse of $J:U\rightarrow J(U)$, which will be denoted by $J^{-1}$. Due to [22, Proposition 4.4.8.] there exists a unique operator $J^{\frac{1}{2}}\in \mathcal {L}^+(U)$ such that $(J^\frac{1}{2})^2=J$. If $J^{-1}$ exists, so does $(J^{\frac{1}{2}})^{-1}$, in this case we denote $(J^{\frac{1}{2}})^{-1}$ by $J^{-\frac{1}{2}}$. By $\mathcal {B}(U)$ we denote the Borel $\sigma$-algebra, i.e. the $\sigma$-algebra generated by the open sets in $(U,(\cdot ,\cdot )_U)$. The euclidean inner product and induced norm is denoted by $\langle \cdot ,\cdot \rangle$ and $|\cdot |$, respectively.

For a given measure space $(\Omega ,\mathcal {A},m)$ and a Banach space Y we denote by $L^p(\Omega ,m,Y)$, $p\in [0,\infty ]$ the Hilbert space of equivalence classes of $\mathcal {A}$-$\mathcal {B}(Y)$ measurable and p-integrable functions. The corresponding norm is denoted by $\Vert {\cdot }\Vert _{L^p(\Omega ,m,Y)}$. If $p=2$, the norm is induced by an inner product denoted by $(\cdot ,\cdot )_{L^2(\Omega ,m,Y)}$. In case ($\Omega ,\mathcal {A})$ is clear from the context and $Y={\mathbb {R}}^n$ for some $n\in {\mathbb {N}}$, we also write $L^2(m)$ instead of $L^2(\Omega ,m,{\mathbb {R}}^n)$. By $\lambda ^n$, $n\in {\mathbb {N}}$, we denote the Lebesgue measure on $({\mathbb {R}}^n,\mathcal {B}({\mathbb {R}}^n))$.

On the measurable space $(U,\mathcal {B}(U))$ we consider an infinite-dimensional non-degenerate Gaussian measure $\mu _1$ with covariance operator $Q_1\in \mathcal {L}^+_1(U)$. Since the measure is non-degenerate the operator $Q_1$ is injective and therefore positive ($(Q_1u,u)_U>0$ for all $u\in U$). For the definition and construction of these measures we refer to the first chapter of [10].

In the next lemma we discuss the important relation between finite and infinite-dimensional Gaussian measures. A proof can be found in [10, Corollary 1.19].

Lemma 2.1

Given $n\in {\mathbb {N}}$ and elements $l_1,...,l_n\in U$. The image measure $\mu _1^n$ of $\mu _1$ under the map

$$\begin{aligned} U \ni u \mapsto \big ((l_1,u)_U,...(l_n,u)_U \big )\in {\mathbb {R}}^n \end{aligned}$$

is the centered Gaussian measure on $({\mathbb {R}}^n,\mathcal {B}({\mathbb {R}}^n))$ with covariance matrix $Q_{1,n}=((Q_1l_i,l_j)_U)_{ij=1,...,n}$.

If $l_1,...,l_n$ is an orthonormal system of eigenvectors of $Q_1$ with corresponding eigenvalues $\lambda _1,...,\lambda _n$, the covariance matrix $Q_{1,n}$ of $\mu _1^n$ is given by the diagonal matrix $\mathrm {diag}(\lambda _1,...,\lambda _n)$.

During this article we have to perform explicit calculations of integrals with respect to Gaussian measures including monomials of order 2 and 4, therefore the following lemma is useful. To proof it, apply Lemma 2.1 and Isserlis formula from [18].

Lemma 2.2

For $l_1,l_2,l_3,l_4\in U$ it holds

$$\begin{aligned}&\int _U (u,l_1)_U(u,l_2)_U\mathrm {d}\mu _1(u)=(Q_1l_1,l_2)_U\quad \text {and}\\&\int _U (u,l_1)_U(u,l_2)_U(u,l_3)_U(u,l_4)_U\mathrm {d}\mu _1(u)\\&=(Q_1l_1,l_2)_U(Q_1l_3,l_4)_U+(Q_1l_1,l_3)_U(Q_1l_2,l_4)_U+(Q_1l_1,l_4)_U(Q_1l_2,l_3)_U. \end{aligned}$$

To cover more general situations we consider a measurable potential $\Phi :U\mapsto (-\infty ,\infty ]$, which is bounded from below. During the paper we will assume more or less restrictive assumptions on the potential. As in the introduction we set $\rho _{\Phi } =\frac{1}{c_{\Phi }}e^{-\Phi }$, where $c_{\Phi }=\int _U e^{-\Phi }\mathrm {d}\mu _1$. On $(U,\mathcal {B}(U))$ we consider the measure $\mu _1^{\Phi }$ defined by

$$\begin{aligned} \mu _1^{\Phi }=\rho _{\Phi }\mu _1. \end{aligned}$$

I.e. a measures having a density with respect to the infinite-dimensional Gaussian measure $\mu _1$. We fix an orthonormal basis $B_U=(d_i)_{i\in {\mathbb {N}}}$ of U.

Definition 2.3

For $n\in {\mathbb {N}}$, set $B_U^n=\mathrm {span}\lbrace d_1,...,d_n\rbrace$. The orthogonal projection from U to $B^n_U$ is denoted by $\overline{P}_n$ and the corresponding coordinate map by $P_n$, i.e. we have for all $u\in U$

$$\begin{aligned} \overline{P}_n(u)=\sum _{i=1}^n (u,d_i)_Ud_i\quad \text {and}\quad P_n(u)=\big ( (u,d_1)_U,...,(u,d_n)_U\big ). \end{aligned}$$

Let $C_b^{\infty }({\mathbb {R}}^n)$ be the space of all bounded smooth (infinitely often differentiable) real-valued functions on ${\mathbb {R}}^n$. The space of finitely based smooth and bounded functions on U, is defined by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_U )&=\lbrace U\ni u\mapsto \varphi (P_m(u))\in {\mathbb {R}}\mid m\in {\mathbb {N}}, \; \varphi \in C_b^{\infty }({\mathbb {R}}^{m})\rbrace . \end{aligned}$$

The subset of functions only depending on n-directions is defined correspondingly by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_U,n)&=\lbrace U\ni u\mapsto \varphi (P_n(u))\in {\mathbb {R}}\mid \varphi \in C_b^{\infty }({\mathbb {R}}^{n})\rbrace . \end{aligned}$$

For later use we define

$$\begin{aligned} L^2_{B_{U_n}}(U,\mu _1,{\mathbb {R}})=\lbrace U\ni u\mapsto f(P_n(u))\in {\mathbb {R}}\mid f\in L^2({\mathbb {R}}^n,\mu _1^n,{\mathbb {R}})\rbrace , \end{aligned}$$

where $\mu _1^n$ is the image measure of $\mu _1$ under $P_n$. Equipping $L^2_{B_{U_n}}(U,\mu _1,{\mathbb {R}})$ with the inner product from $L^2(U,\mu _1,{\mathbb {R}})$ we obtain another Hilbert space.

A very useful density result, proved in [7, Lemma 2.2], is stated in the next lemma.

Lemma 2.4

The function spaces $\mathcal {F}C_b^{\infty }(B_U)$ and $\mathcal {F}C_b^{\infty }(B_U,n)$ are dense in $L^2(U,\mu _1,{\mathbb {R}})$ and $L^2_{B_{U_n}}(U,\mu _1,{\mathbb {R}})$, respectively.

Remark 2.5

Given a Frechét differentiable function $f:U\rightarrow {\mathbb {R}}$. For $u\in U$ we denote by $Df(u)\in U$ the gradient of f in u. Analogously for a two times Frechét differentiable function $f:U\rightarrow {\mathbb {R}}$, we identify $D^2f(u)\in L(U)$ with the second order Frechét derivative in $u\in U$. For $i,j\in {\mathbb {N}}$ we denote by $\partial _if(u)=(Df(u),d_i)_U$ the partial derivative in the direction of $d_i$ and by $\partial _{ij}f(u)=(D^2f(u)d_i,d_j)_U$ the second order partial derivative in the direction of $d_i$ and $d_j$.

We continue this section with the important integration by parts formula for infinite-dimensional Gaussian measures, with and without densities. We assume that $B_U=(d_i)_{i\in {\mathbb {N}}}$ is an orthonormal basis of eigenvectors of $Q_1$ with corresponding eigenvalues $(\lambda _i)_{i\in {\mathbb {N}}}\subset (0,\infty )$. W.l.o.g. we assume that $(\lambda _i)_{i\in {\mathbb {N}}}$ is decreasing to zero.

Since $Q_1$ is injective, the inverse $Q_1^{-1}$ of $Q_1:U\rightarrow Q_1(U)$ exists. Obviously it holds

$$\begin{aligned} Q_1^{-1}d_i=\frac{1}{\lambda _i}d_i,\quad i\in {\mathbb {N}}, \end{aligned}$$

and therefore it is reasonable to define the operator $Q_1^{-\frac{1}{2}}$ on $\bigcup _{n\in {\mathbb {N}}}B_U^n$ determined by

$$\begin{aligned} Q_1^{-\frac{1}{2}}d_i=\frac{1}{\sqrt{\lambda _i}}d_i,\quad i\in {\mathbb {N}}. \end{aligned}$$

Theorem 2.6

For $f,g\in \mathcal {F}C_b^{\infty }(B_U)$ and $i\in {\mathbb {N}}$, it holds the integration by parts formula

$$\begin{aligned} \int _U \partial _ifg\mathrm {d}\mu _1=-\int _U f\partial _ig\mathrm {d}\mu _1+\int _U (u,Q^{-1}d_i)_U fg\mathrm {d}\mu _1. \end{aligned}$$

(2.1)

Proof

Apply Lemma 2.1, use the standard integration by parts formula and use Lemma 2.1 again. For a more detailed proof see [10, Lemma 10.1].

Fix a possible unbounded linear operator (C, D(C)) on U fulfilling the following hypothesis.

Hypothesis 2.7

• 1. (C, D(C)) is symmetric.

• 2. There is a strictly increasing sequence $(m_k)_{k\in {\mathbb {N}}}\subset {\mathbb {N}}$ such that for each $n\in {\mathbb {N}}$ with $n\le m_k$, it holds

  $$\begin{aligned} B_U^n\subset D(C)\quad C(B_U^n)\subset B_U^{m_k}. \end{aligned}$$

Remark 2.8

Note that for given $n\in {\mathbb {N}}$ and $f=\varphi (P_n(\cdot ))\in \mathcal {F}C_b^{\infty }(B_U)$ one has $Df=\sum _{i=1}^n\partial _i\varphi (P_n(\cdot ))d_i\in B_U^n$. Hence the hypothesis above ensure that expressions like CDf, $Q_1^{-\frac{1}{2}}CDf$ or $Q_1^{-1}CDf$ are well-defined.

Theorem 2.9

Assume that Hypothesis 2.7 hold. The operators

$$\begin{aligned} D:\mathcal {F}C_b^{\infty }(B_U)&\rightarrow L^2(U,\mu _1,U)\\ CD:\mathcal {F}C_b^{\infty }(B_U)&\rightarrow L^2(U,\mu _1,U)\\ Q_1^{-\frac{1}{2}}CD:\mathcal {F}C_b^{\infty }(B_U)&\rightarrow L^2(U,\mu _1,U)\\ (CD,CD^2):\mathcal {F}C_b^{\infty }(B_U)&\rightarrow L^2(U,\mu _1,U)\times L^2(U,\mu _1,\mathcal {L}_2(U)) \end{aligned}$$

are closable in $L^2(U,\mu _1,{\mathbb {R}})$.

Proof

As the proof of closability for the first three operators are essentially the same, we restrict ourself to the third and the last. Let $(f_n)_{n\in {\mathbb {N}}}\subset \mathcal {F}C_b^{\infty }(B_U)$ converge to 0 in $L^2(U,\mu _1,{\mathbb {R}})$ and be such that $Q_1^{-\frac{1}{2}}CDf_n\rightarrow F$ in $L^2(U,\mu _1,U)$ as $n\rightarrow \infty$. For $m\in {\mathbb {N}}$, there is some $m_k\in {\mathbb {N}}$ such that $B_U^m\subset B_U^{m_k}$. By symmetry and the invariance properties of (C, D(C)) and the fact that $(d_i)_{i\in {\mathbb {N}}}$ is an orthonormal basis of eigenvectors of $Q_1$ it holds

$$\begin{aligned} (Q_1^{-\frac{1}{2}}CDf_n,d_m)_U=\sum _{l=1}^{\infty } (Q_1^{-\frac{1}{2}}Cd_l,d_m)_U\partial _lf_n=\sum _{l=1}^{m_k} (Q_1^{-\frac{1}{2}}Cd_l,d_m)_U\partial _lf_n. \end{aligned}$$

For an arbitrary $g\in \mathcal {F}C_b^{\infty }(B_U)$ we obtain by the integration by parts formula

$$\begin{aligned}&\int _U (Q_1^{-\frac{1}{2}}CDf_n,d_m)_U g \mathrm {d}\mu _1\\ =&-\sum _{l=1}^{m_k}(Q_1^{-\frac{1}{2}}Cd_l,d_m)_U\int _U f_n(\partial _lg-(u,Q_1^{-1}d_l)_Ug)\mathrm {d}\mu _1. \end{aligned}$$

Observe that g and $\partial _lg-(u,Q_1^{-1}d_l)_Ug$ are in $L^2(U,\mu ,{\mathbb {R}})$ and therefore

$$\begin{aligned} \int _U (F,d_m)_U g \mathrm {d}\mu _1=0. \end{aligned}$$

By the density of $\mathcal {F}C_b^{\infty }(B_U)$ in $L^2(U,\mu _1,{\mathbb {R}})$ we conclude $(F,d_m)_U=0$ for all $m\in {\mathbb {N}}$, hence finally $F=0$. To show that the fourth operator is closable we proceed similarly. Indeed, let $(f_n)_{n\in {\mathbb {N}}}\subset \mathcal {F}C_b^{\infty }(B_U)$ converge to 0 in $L^2(U,\mu _1,{\mathbb {R}})$ and be such that $CDf_n\rightarrow F$ in $L^2(U,\mu _1,U)$ and $CD^2f_n\rightarrow A$ in $L^2(U,\mu _1,\mathcal {L}_2(U))$, as $n\rightarrow \infty$. As above $F=0$. Now for $l,m\in {\mathbb {N}}$ we find a corresponding $m_k$ with $l,m\le m_k$ such that

$$\begin{aligned} (CD^2f_nd_l,d_m)_U=\sum _{i=1}^{m_k} (Cd_l,d_i)_U\partial _{li}f_n. \end{aligned}$$

Hence for arbitrary $g\in \mathcal {F}C_b^{\infty }(B_U)$, we obtain by the integration by parts formula

$$\begin{aligned} \int _U(CD^2f_nd_l,d_m)_Ug\mathrm {d}\mu _1&=\sum _{i=1}^{m_k} (Cd_l,d_i)_U\int _U\partial _{li}f_ng\mathrm {d}\mu _1\\&=-\sum _{i=1}^{m_k} (Cd_l,d_i)_U\int _U \partial _if_n(\partial _lg-(u,Q_1^{-1}d_l)_Ug)\mathrm {d}\mu _1\\&=-\int _U (d_l,CDf_n)_U(\partial _lg-(u,Q_1^{-1}d_l)_Ug)\mathrm {d}\mu _1. \end{aligned}$$

Arguing as in the first part we observe $(Ad_l,d_m)_U=0$ in $L^2(U,\mu _1,{\mathbb {R}})$, implying $A=0$ in $L^2(U,\mu _1,\mathcal {L}_2(U))$.

By Theorem 2.9 it is reasonable to define $W^{1,2}(U,\mu _1,{\mathbb {R}})$, $W^{1,2}_{C}(U,\mu _1,{\mathbb {R}})$, $W^{1,2}_{Q_1^{-\frac{1}{2}}C}(U,\mu _1,{\mathbb {R}})$ and $W^{2,2}_{C}(U,\mu _1,{\mathbb {R}})$ as the domain of the closures of D, CD, $Q_1^{-\frac{1}{2}}CD$ and $(CD,CD^2)$ in $L^2(U,\mu _1,{\mathbb {R}})$, respectively. We still denote the closures of the differential operators from the theorem above by D, CD, $Q_1^{-\frac{1}{2}}CD$ and $(CD,CD^2)$. By [10, Proposition 10.6] every bounded function $f:U\rightarrow {\mathbb {R}}$ with bounded Frechét derivative is in $W^{1,2}(U,\mu _1,{\mathbb {R}})$ and the classical gradient of f coincides with Df in $L^2(U,\mu _1,{\mathbb {R}})$.

Remark 2.10

Adapting the proof of [11, Lemma 9.2.5] one can show that the integration by parts formula from Theorem 2.6 also holds in the case that $f,g\in W^{1,2}(U,\mu _1,{\mathbb {R}})$.

Invoking the remark above the following integration by parts formula for the measure $\mu _1^{\Phi }$ is valid.

Corollary 2.11

Assume the potential $\Phi :U\rightarrow (-\infty ,\infty ]$ is bounded from below and in $W^{1,2}(U,\mu _1,{\mathbb {R}})$. For $f,g\in \mathcal {F}C_b^{\infty }(B_U)$ and $i\in {\mathbb {N}}$, it holds the integration by parts formula

$$\begin{aligned} \int _U \partial _ifg\mathrm {d}\mu _1^{\Phi }=&-\int _U f\partial _ig\mathrm {d}\mu _1^{\Phi }+\int _U (u,Q^{-1}d_i)_U fg\mathrm {d}\mu _1^{\Phi }+\int _U \partial _i\Phi fg\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

3 Perturbed Ornstein-Uhlenbeck Operators and Corresponding Regularity Estimates

This section is devoted to an infinite-dimensional Ornstein-Uhlenbeck operator, perturbed by the gradient of the potential $\Phi$. As already mentioned in the introduction, such operators naturally occur during the application of the abstract Hilbert space hypocoercivity method. Also an infinite-dimensional regularity estimate is derived in the second part of this section. The proof of such estimates is motivated by the results from [7], where Giuseppe Da Prato and Alessandra Lunardi investigated Sobolev regularity for a class of second order elliptic partial differential equations in infinite-dimensions. As before $\mu _1$ is a centered non-degenerate Gaussian measure on the real separable Hilbert space U with covariance operator $Q_1\in \mathcal {L}_1^+(U)$. We fix an orthonormal basis of eigenvectors $B_U=(d_i)_{i\in {\mathbb {N}}}$ of $Q_1$. W.l.o.g. the corresponding sequence of eigenvalues $(\lambda _i)_{i\in {\mathbb {N}}}$ decreases to zero. We start with the operator $(N_0,\mathcal {F}C_b^{\infty }(B_U))$, defined in $L^2(U,\mu _1,{\mathbb {R}})$ by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_U)\ni f\mapsto N_0f=\mathrm {tr}[CD^2f]-(u,Q^{-1}CD f)_U\in L^2(U,\mu _1,{\mathbb {R}}), \end{aligned}$$

where we assume that (C, D(C)) is a possible unbounded linear operator on U. Since we allow such unbounded diffusions as coefficients, we cannot use general results from [7] or [11, Section 10]. Assuming Hypothesis 3.1 below, ensures that the expressions $\mathrm {tr}[CD^2f]$ and $Q_1^{-1}CD f$ are reasonable.

At this point we have to mention that the operator $N_0$ is well-defined in the sense that two representatives of the same equivalence class yield the same output. To see this, note that the measure $\mu _1$ has full topological support, i.e. the smallest closed measurable set with full measure is U. The proof of this statement can be found in [24], it relies on the fact that we assumed that the Hilbert space U is separable.

During this section we permanently assume the following hypothesis.

Hypothesis 3.1

• 1. (C, D(C)) is symmetric and positive.

• 2. There is a strictly increasing sequence $(m_k)_{k\in {\mathbb {N}}}\subset {\mathbb {N}}$ such that for each $n\in {\mathbb {N}}$ with $n\le m_k$, it holds

  $$\begin{aligned} B_U^n\subset D(C)\quad C(B_U^n)\subset B_U^{m_k}. \end{aligned}$$

For $n\in {\mathbb {N}}$, set $n^*=\min _{k\in {\mathbb {N}}}\lbrace m_k\vert n\le m_k\rbrace$.

Note that Hypothesis 3.1 adds positivity of (C, D(C)) to Hypothesis 2.7. This additional assumption is important to achieve essential m-dissipativity of $(N_0,\mathcal {F}C_b^{\infty }(B_U))$, which is proved in the theorem below.

Theorem 3.2

The operator $(N_0,\mathcal {F}C_b^{\infty }(B_U))$ is

• 1. dissipative in $L^2(U,\mu _1,{\mathbb {R}})$, with

  $$\begin{aligned} (N_0f,g)_{L^2(U,\mu _1,{\mathbb {R}})}=\int _U -(CDf,Dg)_U\mathrm {d}\mu _1, \end{aligned}$$

  for all $f,g\in \mathcal {F}C_b^{\infty }(B_U)$ and fulfills

• 2. the dense range condition $\overline{(Id-N_0)(\mathcal {F}C_b^{\infty }(B_U))}=L^2(U,\mu _1,{\mathbb {R}})$,

i.e. is essentially m-dissipative in $L^2(U,\mu _1,{\mathbb {R}})$. The resolvent in $\alpha \in (0,\infty )$ of the closure $(N_0,D(N_0))$ is denoted by $R(\alpha ,N_0)$.

Proof

The first item of the statement follows by the integration by parts formula from Theorem 2.6 together with the invariance properties of (C, D(C)). For the second statement we fix $n\in {\mathbb {N}}$, $f=\varphi (P_n(\cdot ))\in \mathcal {F}C_b^{\infty }(B_U,n)$ and $n^*\in {\mathbb {N}}$, according to Hypothesis 3.1. Extending $\varphi \in C_b^{\infty }({\mathbb {R}}^n)$ canonically to a function $\tilde{\varphi }\in C_b^{\infty }({\mathbb {R}}^{n^*})$ we can calculate

$$\begin{aligned}&N_0f(u)\\&=\sum _{i,j=1}^n \partial _{ij}\varphi (P_n(u)) (Cd_i,d_j)_U-\sum _{i=1}^n\sum _{j=1}^{n^*}(u,d_j)_U(d_j,Q_1^{-1}Cd_i)_U\partial _i\varphi (P_n(u))\\&=\sum _{i,j=1}^{n^*} \partial _{ij}\tilde{\varphi }(P_{n^*}(u)) (Cd_i,d_j)_U-\sum _{i,j=1}^{n^*}(u,d_j)_U\frac{1}{\lambda _j}(d_j,Cd_i)_U\partial _i\tilde{\varphi }(P_{n^*}(u)). \end{aligned}$$

Hence if we set

$$\begin{aligned} C_{n^*}=((Cd_i,d_j)_U)_{ij=1}^{n^*}\quad \text {and}\quad Q_{1,{n^*}}=((Q_1d_i,d_j)_U)_{ij=1}^{n^*}=\mathrm {diag}(\lambda _1,...,\lambda _{n^*}), \end{aligned}$$

we obtain

$$\begin{aligned} N_0f(u)=\mathrm {tr}[C_{n^*}D^2\tilde{\varphi }(P_{n^*}(u))]-\langle P_{n^*}(u),Q_{1,{n^*}}^{-1}C_{n^*}D\tilde{\varphi }(P_{n^*}(u))\rangle . \end{aligned}$$

It is therefore natural to consider the operator $(N_{0,{n^*}},C_b^{\infty }({\mathbb {R}}^{n^*}))$ defined by

$$\begin{aligned} C_b^{\infty }({\mathbb {R}}^{n^*})\ni \varphi \mapsto&N_{0,{n^*}}\varphi =\mathrm {tr}[C_{n^*}D^2\varphi ]- \langle \cdot ,Q^{-1}_{1,{n^*}}C_{n^*}D\varphi \rangle \in L^2({\mathbb {R}}^{n^*},\mu _1^{n^*},{\mathbb {R}}). \end{aligned}$$

As the matrix $C_{n^*}$ is symmetric and positive and $-C_{n^*}Q_{1,{n^*}}^{-1}$ has only negative eigenvalues we can use the argumentation of [21] (compare also Proposition 4.7 below) to obtain that $(N_{0,{n^*}},C_b^{\infty }({\mathbb {R}}^{n^*}))$ is essentially m-dissipative in $L^2({\mathbb {R}}^{n^*},\mu _1^{n^*},{\mathbb {R}})$, hence $(Id-N_{0,{n^*}})(C_b^{\infty }({\mathbb {R}}^{n^*}))$ is dense in $L^2({\mathbb {R}}^{n^*},\mu _1^{n^*},{\mathbb {R}})$. Given $\varepsilon >0$ and $h=g(P_{n^*}(\cdot ))\in L^2_{B_{U_{n^*}}}(U,\mu _1,{\mathbb {R}})$. As $(Id-N_{0,{n^*}})(C_b^{\infty }({\mathbb {R}}^{n^*}))$ is dense in $L^2({\mathbb {R}}^{n^*},\mu _1^{n^*},{\mathbb {R}})$ we find $\varphi \in C_b^{\infty }({\mathbb {R}}^{n^*})$ such that

$$\begin{aligned} \Vert (Id-N_{0,{n^*}})\varphi - g\Vert _{L^2({\mathbb {R}}^{n^*},\mu _1^{n^*},{\mathbb {R}})}<\varepsilon . \end{aligned}$$

Using Lemma 2.1 we obtain

$$\begin{aligned}&\Vert (Id-N_0)\varphi (P_{n^*}(\cdot ))-h\Vert _{L^2_{B_{U_{n^*}}} (U,\mu _1,{\mathbb {R}})}\\=&\Vert (Id-N_{0,{n^*}})\varphi - g\Vert _{L^2({\mathbb {R}}^n,\mu _1^{n^*},{\mathbb {R}})}<\varepsilon . \end{aligned}$$

In other words $(Id-N_0)(\mathcal {F}C_b^{\infty }(B_U,{n^*}))$ is dense in $L^2_{B_{U_{n^*}}}(U,\mu _1,{\mathbb {R}})$. Finally we use the result above to show that $(Id-N_0)(\mathcal {F}C_b^{\infty }(B_U))$ is dense in $L^2(U,\mu _1,{\mathbb {R}})$. Indeed let $\varepsilon >0$ and take an element $h\in L^2(U,\mu _1,{\mathbb {R}})$. By Lemma 2.4 we find $n\in {\mathbb {N}}$, w.l.o.g. $n=n^*$, and $g\in \mathcal {F}C_b^{\infty }(B_U,{n^*})\subset L^2_{B_{U_{n^*}}}(U,\mu _1,{\mathbb {R}})$ such that

$$\begin{aligned} \Vert h-g\Vert _{L^2(U,\mu _1,{\mathbb {R}})}< \frac{\varepsilon }{2}. \end{aligned}$$

As $\overline{(Id-N_0)(\mathcal {F}C_b^{\infty }(B_U,{n^*}))}=L^2_{B_{U_{n^*}}}(U,\mu _1,{\mathbb {R}})$ we find $f\in \mathcal {F}C_b^{\infty }(B_U,{n^*})$ such that

$$\begin{aligned} \Vert (Id-N_0)f-g\Vert _{L^2(U,\mu _1,{\mathbb {R}})}< \frac{\varepsilon }{2}. \end{aligned}$$

Hence the triangle inequality yields

$$\begin{aligned} \Vert (Id-N_0)f-h\Vert _{L^2(U,\mu _1,{\mathbb {R}})}< \varepsilon . \end{aligned}$$

Invoking the famous Lumer-Phillips theorem we obtain that the operator $(N_0,\mathcal {F}C_b^{\infty }(B_U))$ is essentially m-dissipative in $L^2(U,\mu _1,{\mathbb {R}})$. $\square$

Before we go ahead and perturb $(N_0,\mathcal {F}C_b^{\infty }(B_U))$ we need an $L^2(\mu _1)$ regularity estimate for the first and second order derivatives of a function $f\in \mathcal {F}C_b^{\infty }(B_U)$ in terms of $g\in L^2(U,\mu _1,{\mathbb {R}})$, where

$$\begin{aligned} \alpha f-N_0f=g, \end{aligned}$$

(3.1)

for a given $\alpha \in (0,\infty )$.

Theorem 3.3

Suppose we have $f\in \mathcal {F}C_b^{\infty }(B_U)$ and $g=\alpha f-N_0f$, $\alpha \in (0,\infty )$, as in Equation (3.1) above. It holds $g\in W^{1,2}(U,\mu _1,{\mathbb {R}})$ and the identities

$$\begin{aligned}&\int _U\alpha f^2+(CD f,D f)_U\mathrm {d}\mu _1=\int _Ugf\mathrm {d}\mu _1\\&\int _U\alpha (CD f,D f)_U+\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1 =\int _U (D g,CD f)_U\mathrm {d}\mu _1, \end{aligned}$$

are valid. In particular it holds

$$\begin{aligned} \int _U\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1=\int _U (N_0f)^2\mathrm {d}\mu _1. \end{aligned}$$

Proof

Due to the definition of $N_0$ and the fact that F is in $\mathcal {F}C_b^{\infty }(B_U)$ it is easy to see that g is infinitely often differentiable. As Dg is in $L^2(U,\mu _1,U)$ and has at most linear growth (compare equation (3.2)), an approximation argument shows that g is in $W^{1,2}(U,\mu _1,{\mathbb {R}})$.

To show the first equation, we multiply (3.1) with f and integrate over U with respect to $\mu _1$. An application of the integration by parts formula from Theorem 2.6 results in

$$\begin{aligned} \int _U \alpha f^2+(CD f,Df)_U\mathrm {d}\mu _1= \int _U g f\mathrm {d}\mu _1. \end{aligned}$$

To show the second equation we differentiate (3.1) with respect to the k-th direction yielding

$$\begin{aligned} \alpha \partial _k f-N_0\partial _kf+(d_k,Q_1^{-1}CD f)_U=\partial _k g \end{aligned}$$

(3.2)

Now we multiply the equation above with $\partial _l f(d_k,Cd_l)_U$. In order to structure the arguments we treat the resulting terms separately. If we sum over all indices’s a direct calculation shows that the first and third term on the left hand side of the equation above is equal to $\alpha (CD f,D f)_U$ and $(CD f,Q_1^{-1}CD f)_U$, respectively. The right hand side of the equation is then equal to $(D g,CDf)_U$. We also get

$$\begin{aligned} \sum _{k,l=1}^{\infty }(d_k,Cd_l)_U\int _U -N_0\partial _k f \partial _l f\mathrm {d}\mu _1&=\sum _{k,l=1}^{\infty }(d_k,Cd_l)_U\int _U (CD \partial _k f, D \partial _l f)_U\mathrm {d}\mu _1\\&=\int _U \mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1, \end{aligned}$$

and therefore the second equation from the statement is shown. Rearranging the terms of the equation we just derived yields

$$\begin{aligned} \int _U\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1=\int _U (D(-N_0f),CDf)_U\mathrm {d}\mu _1. \end{aligned}$$

Now it holds

$$\begin{aligned} \int _U (D(-N_0f),CDf)_U\mathrm {d}\mu _1&=\sum _{k,l=1}^{\infty }(d_k,Cd_l)_U \int _U \partial _k (-N_0f)\partial _lf\mathrm {d}\mu _1\\&=\sum _{k,l=1}^{\infty }(d_k,Cd_l)_U\int _U N_0f(\partial _{kl}f-(u,Q_1^{-1}d_k)_U\partial _lf)\mathrm {d}\mu _1\\&=\int _U (N_0f)^2\mathrm {d}\mu _1, \end{aligned}$$

where we used that $N_0f\in W^{1,2}(U,\mu _1,{\mathbb {R}})$ and Remark 2.10.

Note that the infinite sums in the calculations above are actually finite ones. $\square$

Remark 3.4

Given an arbitrary $f\in \mathcal {F}C_b^{\infty }(B_U)$. There is some $n\in {\mathbb {N}}$, s.t. $Df\in B_U^n$. In particular $CDf\in B_U^n$, by Hypothesis 3.1. Therefore

$$\begin{aligned} \frac{1}{\lambda _1}\Vert CDf\Vert ^2_U\le \Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U. \end{aligned}$$

(3.3)

Hence by the last equality in Theorem 3.3 we achieve

$$\begin{aligned} \begin{aligned} \int _U\frac{1}{\lambda _1}\Vert CDf\Vert ^2_U+\mathrm {tr} [(CD^2f)^2]\mathrm {d}\mu _1&\le \int _U\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U +\mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1\\&=\int _U (N_0f)^2\mathrm {d}\mu _1. \end{aligned} \end{aligned}$$

(3.4)

Corollary 3.5

It holds

$$\begin{aligned} D(N_0)\subset W^{1,2}_{Q_1^{-\frac{1}{2}}C}(U,\mu _1,{\mathbb {R}})\cap W^{2,2}_{C}(U,\mu _1,{\mathbb {R}}) \end{aligned}$$

and for all $f\in D(N_0)$ we have the inequality

$$\begin{aligned} \begin{aligned} \int _U\frac{1}{\lambda _1}\Vert CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2] \mathrm {d}\mu _1&\le \int _U\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1\\&=\int _U (N_0f)^2\mathrm {d}\mu _1. \end{aligned} \end{aligned}$$

(3.5)

Proof

Given $f\in D(N_0)$. We find a sequence $(f_n)_{n\in {\mathbb {N}}}\subset \mathcal {F}C_b^{\infty }(B_U)$ such that $f_n\rightarrow f$ and $N_0f_n\rightarrow N_0f$ in $L^2(U,\mu _1,{\mathbb {R}})$ as $n\rightarrow \infty$. By (In)equality (3.4) $(f_n)_{n\in {\mathbb {N}}}$ is a Cauchy-sequence in $W^{1,2}_{Q_1^{-\frac{1}{2}}C}(U,\mu _1,{\mathbb {R}})\cap W^{2,2}_{C}(U,\mu _1,{\mathbb {R}})$. Hence $f\in W^{1,2}_{Q_1^{-\frac{1}{2}}C}(U,\mu _1,{\mathbb {R}})\cap W^{2,2}_{C}(U,\mu _1,{\mathbb {R}})$ and the (in)equality of the statement is shown. $\square$

Using (In)equality (3.5) and Neumann-Series theorem we are able to deal with perturbations of $(N_0,\mathcal {F}C_b^{\infty }(B_U))$ as described in the following theorem.

Theorem 3.6

Assume that $\Phi$ is in $W^{1,2}(U,\mu _1,{\mathbb {R}})$, bounded from below and with $\Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}< \frac{1}{4\lambda _1}$. The operator $(N,\mathcal {F}C_b^{\infty }(B_U))$ defined by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_U)\ni f\mapsto Nf=&\mathrm {tr}[CD^2f]-(u,Q_1^{-1}CD f)_U\\&-(D\Phi ,CDf)_U\in L^2(U,\mu _1^{\Phi },{\mathbb {R}}), \end{aligned}$$

• 1. fulfills

  $$\begin{aligned} (Nf,g)_{L^2(U,\mu _1^{\Phi },{\mathbb {R}})}=\int _U -(CDf,Dg)_U\mathrm {d}\mu _1^{\Phi }, \end{aligned}$$

  for all $f,g\in \mathcal {F}C_b^{\infty }(B_U)$, in particular is dissipative in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$. Furthermore we have

• 2. the dense range condition $\overline{(Id-N)(\mathcal {F}C_b^{\infty }(B_U))}=L^2(U,\mu _1^{\Phi },{\mathbb {R}})$.

In particular $(N,\mathcal {F}C_b^{\infty }(B_U))$ is essentially m-dissipative in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$. The resolvent in $\alpha \in (0,\infty )$ of the closure (N, D(N)) is denoted by $R(\alpha ,N)$.

Proof

The first item of the statement follows by the integration by parts formula from Corollary 2.11 together with the invariance properties of the involved operators. For $f\in L^2(U,\mu _1,{\mathbb {R}})$ set

$$\begin{aligned} Tf=-(D\Phi ,CDR(1,N_0)f)_U. \end{aligned}$$

Since $D(N_0)\subset W^{1,2}_{Q_1^{-\frac{1}{2}}C}(U,\mu _1,{\mathbb {R}}) \subset W^{1,2}_{C}(U,\mu _1,{\mathbb {R}})$ the definition above is reasonable. Using the Cauchy-Schwarz inequality, Inequality (3.5) and the assumption on $\Phi$ we observe

$$\begin{aligned} \Vert Tf\Vert _{L^2(\mu _1)}^2&=\int _U (D\Phi ,CDR(1,N_0)f)_U^2\mathrm {d}\mu _1\\&\le \Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\int _U \Vert CDR(1,N_0)f\Vert ^2_U\mathrm {d}\mu _1\\&<\frac{1}{4}\int _U (N_0R(1,N_0)f)^2\mathrm {d}\mu _1\\&=\frac{1}{4}\int _U (f-R(1,N_0)f)^2\mathrm {d}\mu _1\le \Vert f\Vert _{L^2(\mu _1)}^2. \end{aligned}$$

Therefore the linear operator $T:L^2(U,\mu _1,{\mathbb {R}})\rightarrow L^2(U,\mu _1,{\mathbb {R}})$ is well-defined with operator norm less than one. Hence by the Neumann-Series theorem we obtain that $(Id-T)^{-1}$ exists in $\mathcal {L}(L^2(U,\mu _1,{\mathbb {R}}))$. In particular for a given $g\in L^2(U,\mu _1,{\mathbb {R}})$ we find $f\in L^2(U,\mu _1,{\mathbb {R}})$ with $f-Tf=g$ in $L^2(U,\mu _1,{\mathbb {R}})$. Since $(N_0,D(N_0))$ is m-dissipative, there is $h\in D(N_0)$ with $(Id-N_0)h=f$. This yields

$$\begin{aligned} (Id-N_0)h+(D\Phi ,CDh)_U=f+(D\Phi ,CDR(1,N_0)f)_U=f-Tf=g. \end{aligned}$$

If we can show that $D(N_0)\subset D(N)$ with $Nf=N_0f-(D\Phi ,CDf)_U$ for all $f\in D(N_0)$ the proof is finish by the Lumer-Philipps theorem. Indeed this implies

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_U)\subset L^2(U,\mu _1,{\mathbb {R}})\subset (Id-N)(D(N_0)) \subset (Id-N)(D(N)). \end{aligned}$$

i.e. the dense range condition, as $\mathcal {F}C_b^{\infty }(B_U)$ is dense in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$. So let $f\in D(N_0)$ be given. There is a sequence $(f_n)_{n\in {\mathbb {N}}}\subset \mathcal {F}C_b^{\infty }(B_U)$ s.t. $f_n\rightarrow f$ and $N_0f_n\rightarrow N_0f$ in $L^2(U,\mu _1,{\mathbb {R}})$. As $\rho _{\Phi }=\frac{1}{c_{\Phi }}e^{-\Phi }$ is bounded it is easy to see that $f_n\rightarrow f$ in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$. In view of the assumptions on $D\Phi$ and the Inequality (3.5) we can estimate

$$\begin{aligned}&\Vert N_0f-(D\Phi ,CDf)_U-Nf_n\Vert ^2_{L^2(\mu _1^{\Phi })}\\ \le&2\Vert N_0(f-f_n)\Vert ^2_{L^2(\mu _1^{\Phi })}+2\int _U (D\Phi ,CD(f-f_n))_U^2\mathrm {d}\mu _1^{\Phi }\\ \le&2\Vert \rho _{\Phi }\Vert _{L^{\infty }(\mu _1)}\big (\Vert N_0(f-f_n)\Vert ^2_{L^2(\mu _1)}+\Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\int _U \Vert CD(f-f_n)\Vert ^2_U\mathrm {d}\mu _1\big )\\ \le&2\Vert \rho _{\Phi }\Vert _{L^{\infty }(\mu _1)}\frac{5}{4}\Vert N_0(f-f_n)\Vert ^2_{L^2(\mu _1)}. \end{aligned}$$

I.e. $Nf_n\rightarrow N_0f-(D\Phi ,CDf)_U$ in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$. Since (N, D(N)) is closed by construction we obtain $D(N_0)\subset D(N)$ with $Nf=N_0f-(D\Phi ,CDf)_U$ for all $f\in D(N_0)$ as desired. $\square$

The following lines are devoted to derive a $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$ regularity estimate for the first and second order derivatives of a function $f\in \mathcal {F}C_b^{\infty }(B_U)$ in terms of $g\in L^2(U,\mu _1,{\mathbb {R}})$ related via

$$\begin{aligned} \alpha f-Nf=g, \end{aligned}$$

(3.6)

for some given $\alpha \in (0,\infty )$.

Theorem 3.7

Assume that $\Phi :U\rightarrow {\mathbb {R}}$ is Fréchet differentiable, bounded from below and $D\Phi :U\rightarrow U$ is Lipschitz continuous. If $f\in \mathcal {F}C_b^{\infty }(B_U)$, g and $\alpha \in (0,\infty )$ are as in Equation (3.6) it holds

$$\begin{aligned}&\int _U\alpha f^2+(CD f,D f)_U\mathrm {d}\mu _1^{\Phi }=\int _Ugf\mathrm {d}\mu _1^{\Phi }\\&\int _U\alpha (CD f,D f)_U+\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2]+(D^2\Phi CDf,CDf)_U\mathrm {d}\mu _1^{\Phi }\\&=\int _U (D g,CD f)_U\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

In particular we have

$$\begin{aligned}&\int _U \Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2]+(D^2\Phi CDf,CDf)_U\mathrm {d}\mu ^{\Phi } =\int _U (Nf)^2\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

Proof

The first equation follows by multiplying (3.6) with f, an integration over U with respect to $\mu _1^{\Phi }$ and an application of the first item in Theorem 3.6. To show the second equation we differentiate (3.6) with respect to the k-th direction resulting in

$$\begin{aligned} \alpha \partial _k f-N\partial _kf+(d_k,Q_1^{-1}CD f)_U+\sum _{i=1}^{\infty } (d_i,CDf)_U\partial _{ki}\Phi =\partial _k g. \end{aligned}$$

Note that the infinite sum in the line above is actually a finite one. Moreover $\partial _{ki}\Phi$ exists $\mu _1$-a.e., since the Lipschitz continuous function $\partial _i\Phi :U\rightarrow {\mathbb {R}}$ is Gateaux differentiable $\mu _1$-a.e. by [10, Proposition 10.11]. Now we multiply the equation above with $\partial _l f(d_k,Cd_l)_U$. If we sum over all indices’s a direct calculation shows that the first and third term as well as the right hand side is equal to $\alpha (CD f,D f)_U$, $\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U$ and $(D g,CDf)_U$, respectively. For the second term we calculate

$$\begin{aligned}&\sum _{k,l=1}^{\infty }(d_k,Cd_l)_U\int _U -N\partial _k f \partial _l f\mathrm {d}\mu _1^{\Phi }\\&\quad = \sum _{k,l=1}^{\infty }(d_k,Cd_l)_U\int _U (CD \partial _k f, D \partial _l f)_U\mathrm {d}\mu _1^{\Phi }=\int _U \mathrm {tr}[(CD^2f)^2]\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

Moreover we have

$$\begin{aligned} \sum _{k,l,i=1}^{\infty }(d_i,CDf)_U\partial _lf(d_k,Cd_l)_U \partial _{ki}\Phi&=\sum _{k,i=1}^{\infty } (d_i,CDf)_U(d_k,CDf)_U\partial _{ki}\Phi \\&=(D^2\Phi CDf,CDf)_U, \end{aligned}$$

from which we conclude the second equation. As in Theorem 3.3 we can rearrange the terms of the second equation to get

$$\begin{aligned}&\int _U\Vert Q_1^{-\frac{1}{2}}CDf\Vert ^2_U+\mathrm {tr}[(CD^2f)^2] +(D^2\Phi CDf,CDf)_U\mathrm {d}\mu _1^{\Phi }\\&=\int _U (D(-Nf),CDf)_U\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

Note that $N_0f,(D\Phi ,CDf)_U\in W^{1,2}(U,\mu _1,{\mathbb {R}})$, by [10, Proposition 10.11] and [10, Proposition 10.9] as well as $\partial _jf\rho _{\Phi }\in W^{1,2}(U,\mu _1,{\mathbb {R}})$, $j\in {\mathbb {N}}$. Using the integration by parts formula from Remark 2.10 we therefore get

$$\begin{aligned}&\int _U (D(-Nf),CDf)_U\mathrm {d}\mu _1^{\Phi }\\&\quad =\sum _{i,j=1}^{\infty }(Cd_i,d_j)_U\int _U \big (\partial _i(-N_0f)+ \partial _i(D\Phi ,CDf)_U\big )\partial _jf\rho _{\Phi }\mathrm {d}\mu _1\\&\quad =\sum _{i,j=1}^{\infty }(Cd_i,d_j)_U\int _U Nf\big (\partial _{i}(\partial _jf\rho _{\Phi })-(u,Q_1^{-1}d_i)_U\partial _jf\rho _{\Phi }\big )\mathrm {d}\mu _1\\&\quad =\int _U (Nf)^2\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

Note that the infinite sums in the calculations above are actually finite ones.$\square$

Remark 3.8

Suppose we are in the situation of Theorem 3.7. Using the Cauchy-Schwarz inequality and the first equation in Theorem 3.7 we obtain

$$\begin{aligned} \int _U \alpha f^2+(CDf,Df)_U\mathrm {d}\mu _1^{\Phi }&\le \Vert g\Vert _{L^2(\mu _1^{\Phi })} \Vert f\Vert _{L^2(\mu _1^{\Phi })}\\ {}&=\Vert g\Vert _{L^2(\mu _1^{\Phi })} \Vert R(\alpha ,N)g\Vert _{L^2(\mu _1^{\Phi })}\\&\le \frac{1}{\alpha } \Vert g\Vert _{L^2(\mu _1^{\Phi })}^2. \end{aligned}$$

Now additionally suppose that $\Phi$ is a convex function. Hence we can estimate using the third equation in Theorem 3.7 and the convexity of $\Phi$

$$\begin{aligned}&\int _U \mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2 \mathrm {d}\mu _1^{\Phi }\le \int _U (Nf)^2\mathrm {d}\mu _1^{\Phi }= \int _U(\alpha f-g)^2\mathrm {d}\mu _1^{\Phi }\nonumber \\&\quad \le 2\int _U(\alpha f)^2+g^2\mathrm {d}\mu _1^{\Phi }\le 4\int _U g^2\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

(3.7)

Hypothesis 3.9

The potential $\Phi$ is in $W^{1,2}(U,\mu _1,{\mathbb {R}})$, convex, bounded from below and lower semicontinuous.

Remark 3.10

For a potential $\Phi$ fulfilling Hypothesis 3.9 one can introduce the so called Yoshida approximation $\Phi _{t}$, $t>0$, defined by

$$\begin{aligned} \Phi _{t}(u)=\inf _{x\in U}\big \lbrace \Phi (x)+\frac{\Vert u-x\Vert _U^2}{2t}\big \rbrace . \end{aligned}$$

One can show that for all ${t>0}$ is the Yoshida approximation $\Phi _{t}:U\rightarrow (-\infty ,\infty ]$ is convex and Fréchet differentiable with

• 1. $-\infty <\inf _{x\in U}\Phi (x)\le \Phi _{t}(u)\le \Phi (u)$ for all $u\in U$,

• 2. $\lim _{t\rightarrow 0}\Phi _{t}(u)=\Phi (u)$ for all $u\in U$,

• 3. $\Vert D\Phi _{t}(u)\Vert _U\le \Vert D\Phi (u)\Vert _U$ for $\mu _1$-a.e. $u\in U$ and

• 4. $\lim _{t\rightarrow 0}D\Phi _{t}(u)=D\Phi (u)$ for $\mu _1$-a.e. $u\in U$.

Furthermore $D\Phi _{t}$ is Lipschitz continuous for all $t >0$. A proof of these statements can be found in [12].

Theorem 3.11

Suppose $\Phi$ fulfills Hypothesis 3.9. For $f\in \mathcal {F}C_b^{\infty }(B_U)$ and $g=\alpha f-Nf$, $\alpha \in (0,\infty )$, as in Equation (3.6) we have

$$\begin{aligned} \int _U \alpha f^2+(CDf,Df)_U\mathrm {d}\mu _1^{\Phi }&\le \frac{1}{\alpha } \int _U g^2\mathrm {d}\mu _1^{\Phi },\\ \int _U \mathrm {tr}[(CD^2f)^2]+ \Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2\mathrm {d}\mu _1^{\Phi }&\le 4\int _U g^2\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

Proof

Let $(\Phi _{t})_{t>0}$ be the Yoshida approximation of $\Phi$. For all $t>0$, define $g_{t}$ by

$$\begin{aligned} \alpha f-\mathrm {tr}[CD^2f]+(\cdot ,Q_1^{-1}CD f)_U-(D\Phi _{t},CDf)_U=g_{t}. \end{aligned}$$

By Remark 3.8 we obtain

$$\begin{aligned} \int _U (\alpha f^2+(CDf,Df)_U)\rho _{\Phi _{t}}\mathrm {d}\mu _1&\le \frac{1}{\alpha }\int _U g_{t}^2\rho _{\Phi _{t}}\mathrm {d}\mu _1,\\ \int _U (\mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2) \rho _{\Phi _{t}}\mathrm {d}\mu _1&\le 4\int _U g_{t}^2 \rho _{\Phi _{t}}\mathrm {d}\mu _1. \end{aligned}$$

By Remark 3.10, $\rho _{\Phi _{t}}=\frac{1}{c_{\Phi _{t}}}e^{-\Phi _{t}}$ is bounded by a constant $\theta \in (0,\infty )$ independent of t. In particular $(\mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2)\rho _{\Phi _{t}}$ and $(\alpha f^2+(CDf,Df)_U)\rho _{\Phi _{t}}$ are bounded by $(\mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2)\theta$ and $(\alpha f^2+(CDf,Df)_U)\theta$, respectively. Since the expressions converge pointwisely to $(\mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2) \rho _{\Phi }$ and $(\alpha f^2+(CDf,Df)_U)\rho _{\Phi }$ we know that the left hand sides of the inequalities above converge to $\int _U (\mathrm {tr}[(CD^2f)^2]+\Vert Q_1^{-\frac{1}{2}}CDf\Vert _U^2)\mathrm {d}\mu _1^{\Phi }$ and $\int _U \alpha f^2+(CDf,Df)_U\mathrm {d}\mu _1^{\Phi }$, respectively. It also holds

$$\begin{aligned} |\int _U g_{t}^2 \rho _{\Phi _{t}}-g^2\rho _{\Phi }\mathrm {d}\mu _1|&\le |\int _U g_{t}^2 (\rho _{\Phi _{t}}-\rho _{\Phi })\mathrm {d}\mu _1|+ |\int _U (g_{t}^2-g^2)\rho _{\Phi }\mathrm {d}\mu _1|\\&=|\int _U g_{t}^2 (\rho _{\Phi _{t}}-\rho _{\Phi })\mathrm {d}\mu _1|+|\Vert g_{t}\Vert ^2_{L^2(\mu _1^{\Phi })} -\Vert g\Vert ^2_{L^2(\mu _1^{\Phi })}|. \end{aligned}$$

Note that $g_{t}^2$ can be bounded independent of t by an $\mu _1$-integrable function, hence the first term in the above inequality goes to zero as t goes to zero by another application of the dominated convergence theorem. The second term also tends to zero. Indeed the Cauchy-Schwarz inequality and the definitions of g and $g_{t}$ yields

$$\begin{aligned} \int _U (g-g_{t})^2\mathrm {d}\mu _1^{\Phi }&=\int _U(D\Phi -D\Phi _{t},CDf)_U^2 \mathrm {d}\mu _1^{\Phi }\\&\le \int _U \Vert D\Phi -D\Phi _{t}\Vert _U^2\Vert CDf\Vert _U^2\mathrm {d}\mu _1^{\Phi }. \end{aligned}$$

Invoking the third and the fourth item of Remark 3.10 and another application of the dominated convergence theorem yields that $g_{t}$ converges to g in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$ as t goes to zero. In particular the corresponding norms in $L^2(U,\mu _1^{\Phi },{\mathbb {R}})$ converge. All together this finishes the proof. $\square$

Note that the regularity estimate we derived in Theorem 3.11 relies on Hypothesis 3.9, i.e. not necessarily demanding the restriction $\Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}< \frac{1}{4\lambda _1}$, we needed to show Theorem 3.6.

4 The Infinite-dimensional Langevin Operator

The essential m-dissipativity of finite-dimensional Langevin operators have been extensively studied in [4] and [5] for singular potentials and even in a manifold setting in [20]. In this section we want to extend these result to an infinite-dimensional setting, where as in the above references the non-sectorality of $L_{\Phi }$ causes difficulties.

As described in the introduction, we fix two real separable Hilbert spaces $(U,(\cdot ,\cdot )_U)$ and $(V,(\cdot ,\cdot )_V)$ and consider the real separable Hilbert space $(W,(\cdot ,\cdot )_W)$ defined by $W=U\times V$ and

$$\begin{aligned} ((u_1,v_1),(u_2,v_2))_W=(u_1,u_2)_U+(v_1,v_2)_V,\quad (u_1,v_1),(u_2,v_2)\in W. \end{aligned}$$

By $\mu _1$ and $\mu _2$ we denote two centered non-degenerate Gaussian measures on $(U,\mathcal {B}(U))$ and $(V,\mathcal {B}(V))$, respectively. The corresponding covariance operators are denoted by $Q_1\in \mathcal {L}_1^+(U)$ and $Q_2\in \mathcal {L}_1^+(V)$. We also fix two orthonormal basis $B_U=(d_i)_{i\in {\mathbb {N}}}$ and $B_V=(e_i)_{i\in {\mathbb {N}}}$ of eigenvectors with corresponding eigenvalues $(\lambda _i)_{i\in {\mathbb {N}}}$ and $(\nu _i)_{i\in {\mathbb {N}}}$ of $Q_1$ and $Q_2$, respectively. W.l.o.g. we assume that $(\lambda _i)_{i\in {\mathbb {N}}}$ and $(\nu _i)_{i\in {\mathbb {N}}}$ are decreasing to zero. Furthermore we set $B_W=(B_U,B_V)$.

As in Definition 2.3 one can consider the orthogonal projections to $B_U^n$ and $B_V^n$, $n\in {\mathbb {N}}$. To avoid an overload of notation we omit to indicate if we project to $B_U^n$ and $B_V^n$ as it is clear from the context.

On $(W,\mathcal {B}(W))$ we consider the product measure $\mu =\mu _1\otimes \mu _2$. Using the separability of U and V, [19, Lemma 1.2] it holds $\mathcal {B}(W)=\mathcal {B}(U)\otimes \mathcal {B}(V)$. Applying [10, Theorem 1.12] one can check that $\mu$ is a non-degenerate centered Gaussian measure with centered non-degenerate covariance operator $Q\in \mathcal {L}_1^+(W)$ defined by

$$\begin{aligned} W\ni (u,v)\mapsto Q(u,v)=(Q_1u,Q_2v)\in W. \end{aligned}$$

Definition 4.1

In $L^2(\mu )$ we denote by $\mathcal {F}C_b^{\infty }(B_W)$ the space of finitely based smooth and bounded functions on W defined by

$$\begin{aligned}&\mathcal {F}C_b^{\infty }(B_W)\\&=\lbrace W\ni (u,v)\mapsto \varphi (P_m(u),P_m(v))\in {\mathbb {R}}\mid m\in {\mathbb {N}}, \; \varphi \in C_b^{\infty }({\mathbb {R}}^{m}\times {\mathbb {R}}^m)\rbrace \end{aligned}$$

and correspondingly the space of finitely based smooth and bounded functions on W only dependent on the first n directions by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_W,n)=\lbrace W\ni (u,v)\mapsto \varphi (P_n(u),P_n(v))\in {\mathbb {R}}\mid \; \varphi \in C_b^{\infty }({\mathbb {R}}^{n}\times {\mathbb {R}}^n)\rbrace . \end{aligned}$$

Concerning derivatives of sufficient smooth functions $f:W\rightarrow {\mathbb {R}}$ recall the explanation in Remark 2.5. We set $D_1f=\sum _{i=1}^{\infty }(Df,(d_i,0))_Wd_i\in U$ and $D_2f=\sum _{i=1}^{\infty }(Df,(0,e_i))_We_i\in V$ as well as $\partial _{i,1}f=(D_1f,d_i)_U$ and $\partial _{i,2}f=(D_2f,e_i)_V$. In particular we have

$$\begin{aligned} Df=\sum _{i=1}^{\infty }(Df,(d_i,0))_W(d_i,0)+\sum _{i=1}^{\infty } (Df,(0,e_i))_W(0,e_i)=(D_1f,D_2f). \end{aligned}$$

Analogously we define $D_1^2f$, $D_2^2f$ as well as $\partial _{ij,1}f$ and $\partial _{ij,2}f$.

For given $n\in {\mathbb {N}}$, recall the image measures $\mu _1^n$ and $\mu _2^n$ from Lemma 2.1 w.r.t. $B_U$ and $B_V$, respectively and set $\mu ^n=\mu _1^n\otimes \mu _2^n$ on $({\mathbb {R}}^{n}\times {\mathbb {R}}^n,\mathcal {B}({\mathbb {R}}^n)\otimes \mathcal {B}({\mathbb {R}}^n))$.

Remark 4.2

Arguing as in Lemma 2.4 one can show that $\mathcal {F}C_b^{\infty }(B_W)$ is dense in $L^2(\mu )$.

Moreover we fix operators $K_{12}\in \mathcal {L}(U;V)$, $K_{21}\in \mathcal {L}(V;U)$ and $K_{22}\in \mathcal {L}^+(U)$. Last but not least we consider a measurable potential $\Phi :U\rightarrow (-\infty ,\infty ]$ which is bounded from below and recall the measures $\mu _1^{\Phi }$ and $\mu ^{\Phi }$. During the whole section we will assume the following hypothesis.

Hypothesis 4.3

• 1. $K_{22}$ is symmetric and positive.

• 2. $K_{12}^*=K_{21}$ and $K_{21}$ is injective.

• 3. There is a strictly increasing sequence $(m_k)_{k\in {\mathbb {N}}}\subset {\mathbb {N}}$ such that for each $n\in {\mathbb {N}}$ with $n\le m_k$, it holds

  $$\begin{aligned} K_{22}(B_V^n)\subset B_V^{m_k},\quad K_{12}(B_U^n)\subset B_V^{m_k}\quad \text {and}\quad K_{21}(B_V^n)\subset B_U^{m_k}. \end{aligned}$$

• 4. $\Phi \in W^{1,2}(U,\mu _1,{\mathbb {R}})$.

For $n\in {\mathbb {N}}$, set $n^*=\min _{k\in {\mathbb {N}}}\lbrace m_k\vert n\le m_k\rbrace$.

We will realize in Remark 4.6, that the invariance properties of $K_{12}$, $K_{21}$ and $K_{22}$ included in the hypothesis above, ensures that the infinite dimensional Langevin operator $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$, defined below, has a useful decomposability property.

Definition 4.4

We define $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ in $L^2(\mu ^{\Phi })$ by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B_W) \ni f \mapsto L_{\Phi }f=S_{\Phi }f-A_{\Phi }f\in L^2(\mu ^{\Phi }), \end{aligned}$$

where for $f\in \mathcal {F}C_b^{\infty }(B_W)$, $S_{\Phi }f$ and $A_{\Phi }f$ are given by

$$\begin{aligned} S_{\Phi }f=&\mathrm {tr}[K_{22}D^2_2f]-(v,Q_2^{-1}K_{22}D_2f)_V ,\\ A_{\Phi }f=&(u,Q_1^{-1}K_{21}D_2f)_U+(D \Phi (u),K_{21}D_2f)_U-(v,Q_2^{-1}K_{12}D_1f)_V. \end{aligned}$$

The designation of $S_{\Phi }$ and $A_{\Phi }$ is not accidental, as we see show in the next lemma, that $(S_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ is symmetric and $(A_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ antisymmetric.

Remember that $\mathcal {F}C_b^{\infty }(B_W)$ is dense in $L^2(\mu ^{\Phi })$ by Remark 4.2 and the fact that $\Phi$ is bounded from below. For $f\in \mathcal {F}C_b^{\infty }(B_W)$, expressions like $Q_2^{-1}K_{22}D_2f$, $Q_2^{-1}K_{22}D_2f$ and $Q_1^{-1}K_{21}D_2f$ are reasonable due to Hypothesis 4.3, compare also Remark 2.8.

Using the integration by parts formula from Corollary  2.11 together with the invariance properties of $K_{22}$, $K_{21}$ and $K_{12}$ one can derive the following lemma.

Lemma 4.5

  It holds

• (1) $(S_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ is symmetric and dissipative in $L^2(\mu ^{\Phi })$.

• (2) $(A_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ is antisymmetric in $L^2(\mu ^{\Phi })$.

• (3) $1\in \mathcal {F}C_b^{\infty }(B_W)$ with $L_{\Phi }1=0$ and in particular $\mu ^{\Phi }$ is invariant for $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ in the sense that

  $$\begin{aligned} \int _W L_{\Phi }f\mathrm {d}\mu ^{\Phi }=0\quad \text {for all}\quad f\in \mathcal {F}C_b^{\infty }(B_W). \end{aligned}$$

• (4) $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ is dissipative in $L^2(\mu ^{\Phi })$ and for all $f,g\in \mathcal {F}C_b^{\infty }(B_W)$ it holds

  $$\begin{aligned} -&\int _W L_{\Phi }f g\mathrm {d}\mu ^{\Phi }\\ =&\int _W (D_2f,K_{22}D_2g)_V-(D_1f,K_{21}D_2g)_U+(D_2f,K_{12}D_1g)_V \mathrm {d}\mu ^{\Phi }. \end{aligned}$$

By [14, Proposition 3.14] densely defined dissipative operators are closable. Since $(L,\mathcal {F}C_b^{\infty }(B_W))$, $(S_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ and $(A,\mathcal {F}C_b^{\infty }(B_W))$ are densely defined dissipative operators in $L^2(\mu ^{\Phi })$, it is reasonable to denote their closures by $(L_{\Phi },D(L_{\Phi }))$, $(S_{\Phi },D(S_{\Phi }))$ and $(A_{\Phi },D(A_{\Phi }))$. The overall goal is to show essential m-dissipativity of $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$, i.e. m-dissipativity of $(L_{\Phi },D(L_{\Phi }))$ in $L^2(\mu ^{\Phi })$.

The idea is to start with $\Phi =0$, and use a similar argumentation as in the previous section to conclude essential m-dissipativity for $\Phi$ as in Theorem 4.11. If $\Phi =0$, the infinite-dimensional measure $\mu ^{\Phi }$ reduces to the infinite-dimensional centered non-degenerate Gaussian measure $\mu$ with covariance operator Q.

Remark 4.6

Given $n\in {\mathbb {N}}$ with corresponding ${n^*}$ provided by Hypothesis 4.3. Set

$$\begin{aligned}&K_{22,{n^*}}=((K_{22}e_i,e_j)_V)_{ij=1}^{n^*},\quad K_{12,{n^*}}=((K_{12}d_i,e_j)_V)_{ij=1}^{n^*},\\&K_{21,{n^*}}=K_{12,{n^*}}^*,\quad Q_{1,{n^*}}=((Q_{1}d_i,d_j)_U)_{ij=1}^{n^*},\quad Q_{2,{n^*}}=((Q_{2}e_i,e_j)_V)_{ij=1}^{n^*}. \end{aligned}$$

and consider the matrices

$$\begin{aligned}&\tilde{K}_{22,{n^*}}=\begin{pmatrix} 0 &{} 0 \\ 0 &{} K_{22,{n^*}} \\ \end{pmatrix}\quad Q_{{n^*}}=\begin{pmatrix} Q_{1,{n^*}} &{} 0 \\ 0 &{} Q_{2,{n^*}} \\ \end{pmatrix}\\&K_{n^*}=\begin{pmatrix} 0 &{} K_{21,n} \\ -K_{12,{n^*}} &{} -K_{22,{n^*}} \\ \end{pmatrix}. \end{aligned}$$

We have for $f=\varphi (P_n(\cdot ),P_n(\cdot ))\in \mathcal {F}C_b^{\infty }(B_W)$ and $(u,v)\in W$

$$\begin{aligned} Lf(u,v)=\mathrm {tr}[\tilde{K}_{22,{n^*}}D^2\tilde{\varphi }(z_{n^*})]+\langle K_{n^*}Q_{n^*}^{-1}z_{n^*},D\tilde{\varphi }(z_{n^*})\rangle , \end{aligned}$$

where $z_{n^*}=(P_{n^*}(u),P_{n^*}(v))$ and $\tilde{\varphi }\in C_b^{\infty }({\mathbb {R}}^{{n^*}}\times {\mathbb {R}}^{n^*})$ is the canonical extension of $\varphi \in C_b^{\infty }({\mathbb {R}}^{{n}}\times {\mathbb {R}}^{n})$. Hence, it is reasonable to consider the operator $(L_{n^*},C_b^{\infty }({\mathbb {R}}^{{n^*}}\times {\mathbb {R}}^{n^*}))$ defined for $\tilde{\varphi }\in C_b^{\infty }({\mathbb {R}}^{{n^*}}\times {\mathbb {R}}^{n^*})$ and $z\in {\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}$ by

$$\begin{aligned} L_{n^*}\varphi (z)=\mathrm {tr}[\tilde{K}_{22,{n^*}}D^2\tilde{\varphi }(z)]+\langle K_{n^*}Q_{n^*}^{-1}z,D\tilde{\varphi }(z)\rangle . \end{aligned}$$

Invoking the assumptions from Hypothesis 4.3 the following proposition shows that the operator $(L_{n^*},C_b^{\infty }({\mathbb {R}}^{{n^*}}\times {\mathbb {R}}^{n^*}))$ is essentially m-dissipative in $L^2(\mu ^{n^*})$. The idea of the proof is inspired by the consideration in [21] and [6]. By $C_b({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*})$ we denote the space of bounded continuous functions from ${\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}$ to ${\mathbb {R}}$.

Proposition 4.7

For all $n\in {\mathbb {N}}$, the operator $(L_{n^*},C_b^{\infty }({\mathbb {R}}^{{n^*}}\times {\mathbb {R}}^{n^*}))$ is essentially m-dissipative in $L^2(\mu ^{n^*})$.

Proof

Note that the Markov semigroup $(S_t)_{t\ge 0}$ associated to $L_{n^*}$ can be represented for all $t\in (0,\infty )$, applied to $\varphi \in C_b({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*})$, evaluated at $z\in {\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}$ as

$$\begin{aligned} S_t\varphi (z)=\frac{1}{c_t}\int _{{\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}}e^{-\langle Q(t)^{-1}w,w\rangle /4}\varphi (e^{tK_{n^*}Q_{n^*}^{-1}}z-w)\mathrm {d} (\lambda ^{n^*}\otimes \lambda ^{n^*})(w), \end{aligned}$$

where for $t\in (0,\infty )$

$$\begin{aligned} c_t=(4\pi )^{n^*}(\det Q(t))^{1/2}\;\; \text {and}\;\;Q(t)=\int _{[0,t)} e^{sK_{n^*}Q_{n^*}^{-1}}\tilde{K}_{22,{n^*}}e^{sQ_{n^*}^{-1} K_{n^*}^*}\mathrm {d}\lambda (s), \end{aligned}$$

is a symmetric and nonnegative matrix. Using the first and the second item of Hypothesis 4.3, in particular the positivity of $K_{22}$ and the injectivity of $K_{21}$ it is easy to see that the matrix

$$\begin{aligned} \begin{pmatrix} \tilde{K}_{22,{n^*}}^{\frac{1}{2}}&K_{n^*}Q_{n^*}^{-1}\tilde{K}_{22,{n^*}}^{\frac{1}{2}}\end{pmatrix}= \begin{pmatrix} 0 &{} 0 &{} 0 &{} K_{21,{n^*}}Q_{2,{n^*}}^{-1}K_{22,{n^*}}^{\frac{1}{2}} \\ 0 &{} K^{\frac{1}{2}}_{22,{n^*}} &{}0 &{} -K_{22,{n^*}}Q_{2,{n^*}}^{-1}K_{22,{n^*}}^{\frac{1}{2}} \\ \end{pmatrix} \end{aligned}$$

has full rank. Hence the so called Kalman rank condition is satisfied and therefore by [26] $\det Q(t)>0$. In particular the representation of $(S_t)_{t\ge 0}$ is reasonable. As $\det Q(t)>0$, we know by the arguments from [9, Section 11.3.1] that the existence of an invariant measure $\nu$ for $(S_t)_{t\ge 0}$ i.e.

$$\begin{aligned} \int _{{\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}}S_t\varphi \mathrm {d}\nu =\int _{{\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}}\varphi \mathrm {d}\nu , \end{aligned}$$

for every $t\in [0,\infty )$ and $\varphi \in C_b({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*})$ is equivalent to the existence of a nonnegative symmetric matrix P s.t.

$$\begin{aligned} \langle P Q_{n^*}^{-1}K_{n^*}^*z,z\rangle +\langle \tilde{K}_{22,{n^*}}z,z\rangle =0\quad \text {for all }z\in {\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}. \end{aligned}$$

As the equality above is satisfied for $P=Q_{n^*}$ the existence of the invariant measure is justified. Moreover by [9, Section 11.3.3] the invariant measure is unique and given by $\mu ^{n^*}$.

The invariance of the measure $\mu ^{n^*}$ allows us to extend $(S_t)_{t\ge 0}$ to a strongly continuous contraction semigroup in $L^2(\mu ^{n^*})$. Using [21, Lemma 2.1], we can show that the generator of the strongly continuous contraction semigroup $(S_t)_{t\ge 0}$ is given by $L_{n^*}\varphi$ for all $\varphi \in C_b^{\infty }({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*})$ and that $C_b^{\infty }({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*})$ is a core for the generator of $(S_t)_{t\ge 0}$ in $L^2(\mu ^{n^*})$, i.e. the assertion is shown. $\square$

Theorem 4.8

The operator $(L,\mathcal {F}C_b^{\infty }(B_W))$ defined by

$$\begin{aligned} \mathcal {F}C_b^{\infty }(B)\ni f\mapsto Lf=&\mathrm {tr}[K_{22}D^2_2f]-(v,Q_2^{-1}K_{22}D_2f)_V\\&-(u,Q_1^{-1}K_{21}D_2f)_U+(v,Q_2^{-1}K_{12}D_1f)_V\in L^2(\mu ), \end{aligned}$$

is essentially m-dissipative in $L^2(\mu )$ with m-dissipative closure (L, D(L)). The resolvent in $\alpha \in (0,\infty )$, of (L, D(L)) is denoted by $R(\alpha ,L)$.

Proof

Since by Lemma 4.5 we already know that $(L,\mathcal {F}C_b^{\infty }(B_W))$ is dissipative it is left to show that $(Id-L)(\mathcal {F}C_b^{\infty }(B_W))$ is dense in $L^2(\mu )$. Let $\varepsilon >0$ and $h\in L^2(\mu )$ be given. Remark 4.2 provides $g=\psi (P_n(\cdot ),P_n(\cdot ))\in \mathcal {F}C_b^{\infty }(B_W,n)\subset L^2(\mu )$ such that

$$\begin{aligned} \Vert h-g\Vert _{L^2(\mu )}< \frac{\varepsilon }{2}. \end{aligned}$$

Let $\tilde{\psi }\in C_b^{\infty }({\mathbb {R}}^{{n^*}}\times {\mathbb {R}}^{n^*})$ be the canonical extension of $\psi \in C_b^{\infty }({\mathbb {R}}^{{n}}\times {\mathbb {R}}^{n})$. Since $(Id-L_{n^*})(C_b^{\infty }({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*}))$ is dense in $L^2(\mu ^{n^*})$, by Proposition 4.7, we find a $\varphi \in C_b^{\infty }({\mathbb {R}}^{n^*}\times {\mathbb {R}}^{n^*})$ such that

$$\begin{aligned} \Vert (Id-L_{n^*})\varphi -\tilde{\psi }\Vert _{L^2(\mu ^{n^*})}< \frac{\varepsilon }{2}. \end{aligned}$$

Set $f=\varphi (P_{n^*}(\cdot ),P_{n^*}(\cdot ))$ and use the triangle inequality together with Lemma 2.1 to observe

$$\begin{aligned}&\Vert (Id-L)f-h\Vert _{L^2(\mu )}\le \Vert (Id-L)f-g\Vert _{L^2(\mu )}+\Vert h-g\Vert _{L^2(\mu )}\\&\quad =\Vert (Id-L_{n^*})\varphi -\tilde{\psi }\Vert _{L^2(\mu ^{n^*})}+\Vert h-g\Vert _{L^2(\mu )}< \frac{\varepsilon }{2}+ \frac{\varepsilon }{2}=\varepsilon . \end{aligned}$$

$\square$

Before we proof the main result of this section, we discuss regularity estimates similar to the ones from Remark 3.8. In contrast to the estimates in Sect. 3 we don’t have to deal with unbounded diffusion operators (C, D(C)) as coefficients, but the degenerate structure of $(L,\mathcal {F}C_b^{\infty }(B_W))$ is more challenging. Indeed, we are only able to derive first order regularity results.

Proposition 4.9

For $f\in D(L)$ and $\alpha \in (0,\infty )$, set $g=\alpha f-Lf$. Then the following equation hold

$$\begin{aligned}&\int _W\alpha f^2+\Vert K_{22}^{\frac{1}{2}}D_2f\Vert ^2_V\mathrm {d}\mu =\int _W fg\mathrm {d}\mu . \end{aligned}$$

In particular

$$\begin{aligned} \int _W\Vert K_{22}^{\frac{1}{2}}D_2f\Vert ^2_V\mathrm {d}\mu \le \frac{1}{2} \int _Wf^2+(Lf)^2\mathrm {d}\mu \quad \text {and} \end{aligned}$$

(4.1)

$$\begin{aligned} \int _W\Vert K_{22}^{\frac{1}{2}}D_2f\Vert ^2_V\mathrm {d}\mu \le \frac{1}{4\alpha } \int _Wg^2\mathrm {d}\mu . \end{aligned}$$

(4.2)

Proof

Assume $f\in \mathcal {F}C_b^{\infty }(B_W)$ and $g=\alpha f-Lf$. Now Multiply $g=\alpha f-Lf$ with f, integrate over W w.r.t. $\mu$ and use Lemma 4.5 item (iv) to obtain the first identity. Rearranging the terms we obtain

$$\begin{aligned} \int _W\Vert K_{22}^{\frac{1}{2}}D_2f\Vert ^2_V\mathrm {d}\mu&=\int _Wfg -\alpha f^2\mathrm {d}\mu =\int _W f(g-\alpha f)\mathrm {d}\mu =-\int _W fLf\mathrm {d}\mu .\\&\le \frac{1}{2} \int _Wf^2+(Lf)^2\mathrm {d}\mu . \end{aligned}$$

Moreover by completing the square we have

$$\begin{aligned} \int _W\Vert K_{22}^{\frac{1}{2}}D_2f\Vert ^2_V\mathrm {d}\mu =-\int _W \alpha f^2-fg\mathrm {d}\mu \le \frac{1}{4\alpha } \int _Wg^2\mathrm {d}\mu . \end{aligned}$$

Since $\mathcal {F}C_b^{\infty }(B_W)$ is dense in the (L, D(L)) graph norm the (in)equalities above are also valid for $f\in D(L)$. Note that for $f\in D(L)$ the expression $K_{22}D_2f$ is understood in the sense of Theorem 2.9, compare also Corollary 3.5. $\square$

Having the regularity estimates at hand we need one last hypothesis to derive our final essential m-dissipativity result. The hypothesis includes condition on the potential $\Phi$ and a coercivity assumption on $K_{22}$ in terms of the operator $K_{12}K_{21}$.

Hypothesis 4.10

• 1. There is a constant $c_K\in (0,\infty )$ such that

  $$\begin{aligned} (K_{12}K_{21}v,v)_V\le c_K (K_{22}v,v)_V\quad \text {for all } v\in V. \end{aligned}$$

• 2. The potential $\Phi$ is bounded from below and $D\Phi$ is bounded.

Note that the second item implies that $\rho _{\Phi }=\frac{1}{c_{\Phi }}e^{-\Phi }$ is bounded.

Theorem 4.11

Suppose Hypothesis 4.10 is valid, then $D(L)\subset D(L_{\Phi })$ with

$$\begin{aligned} L_{\Phi }f=Lf-(D\Phi ,K_{21}D_2f)_U, \quad f\in D(L), \end{aligned}$$

and the infinite-dimensional Langevin operator $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ is essentially m-dissipative in $L^2(\mu ^{\Phi })$.

Proof

Using Proposition 4.9 and the first item from Hypothesis 4.10 it holds for all $f\in \mathcal {F}C_b^{\infty }(B_W)$

$$\begin{aligned} \int _W(K_{21}D_2f,K_{21}D_2f)_U\mathrm {d}\mu&\le \int _W c_K (K_{22}D_2f,D_2f)_V\mathrm {d}\mu \\&\le \frac{c_K}{2} \int _Wf^2+(Lf)^2\mathrm {d}\mu . \end{aligned}$$

Since $\mathcal {F}C_b^{\infty }(B_W)$ is dense D(L) w.r.t. the (L, D(L)) graph norm the estimate above also holds for $f\in D(L)$. Again, note that for $f\in D(L)$ the expressions $K_{22}D_2f$ and $K_{21}D_2f$ are understood in the sense of Theorem 2.9, compare also Corollary 3.5.

Let $(f_n)_{n\in {\mathbb {N}}}\subset \mathcal {F}C_b^{\infty }(B_W)$ be a sequence converging to $f\in D(L)$ wit respect to the (L, D(L)) graph norm. Since $\Phi$ is bounded from below it is easy to check that $(f_n)_{n\in {\mathbb {N}}}$ converges to f in $L^2(\mu ^{\Phi })$. Moreover we can estimate

$$\begin{aligned}&\int _W(L_{\Phi }f_n-Lf+(D\Phi ,K_{21}D_2f)_U)^2\mathrm {d}\mu ^{{\Phi }}\\&\quad \le 2\int _W (Lf_n-Lf)^2\mathrm {d}\mu ^{\Phi } +2 \int _W (D\Phi ,K_{21}D_2(f_n-f))^2_U\mathrm {d}\mu ^{\Phi }\\&\quad \le 2\Vert \rho _{\Phi }\Vert _{L^{\infty }(\mu _1)}\big (\int _W (Lf_n-Lf)^2\mathrm {d}\mu \\&\quad +\frac{c_K}{2}\Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\int _W(f_n-f)^2+(Lf_n-Lf)^2\mathrm {d}\mu \big ). \end{aligned}$$

Hence the sequence $(Lf_n)_{n\in {\mathbb {N}}}$ converges to $Lf-(D\Phi ,K_{21}D_2f)_U$ in $L^2(\mu ^{\Phi })$. As $(L_{\Phi },D(L_{\Phi }))$ is closed we get $D(L)\subset D(L_{\Phi })$ and for all $f\in D(L)$

$$\begin{aligned} L_{\Phi }f=Lf-(D\Phi ,K_{21}D_2f)_U. \end{aligned}$$

By Lemma 4.5 item (iv) we already know that $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$ is dissipative. In view of the Lumer-Phillips theorem we are left to show the dense range condition. For $f\in L^2(\mu )$ and $\alpha >0$ set

$$\begin{aligned} T_{\alpha }f=-(D\Phi ,K_{21}D_2R(\alpha ,L)f)_U. \end{aligned}$$

We calculate using the Cauchy-Schwarz inequality, the assumption on $\Phi$, Hypothesis 4.10 and Inequality (4.2)

$$\begin{aligned} \int _W (T_{\alpha }f)^2\mathrm {d}\mu&=\int _W (D\Phi ,K_{21}D_2R(\alpha ,L)f)_U^2\mathrm {d}\mu \\&\le \Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\int _W (K_{21}D_2R(\alpha ,L)f,K_{21} D_2R(\alpha ,L)f)_U\mathrm {d}\mu \\&\le \Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\int _W c_K(K_{22}D_2R(\alpha ,L)f,D_2 R(\alpha ,L)f)_U\mathrm {d}\mu \\&\le \Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\frac{c_K}{4\alpha }\int _W f^2\mathrm {d}\mu . \end{aligned}$$

Hence the operator $T_{\alpha }:L^2(\mu )\rightarrow L^2(\mu )$ is well-defined. Moreover, if

$$\begin{aligned} \Vert D\Phi \Vert ^2_{L^{\infty }(\mu _1)}\frac{c_K}{4\alpha }<1, \end{aligned}$$

we can apply Neumann-Series theorem to get $(Id-T_{\alpha })^{-1}\in \mathcal {L}(L^2(\mu ))$. In particular, for such $\alpha$, for all $g\in L^2(\mu )$ we find $f\in L^2(\mu )$ with $f-T_{\alpha }f=g$ in $L^2(\mu )$. Furthermore there is $h\in D(L)$ with $(\alpha -L)h=f$. Therefore,

$$\begin{aligned} (\alpha -L_{\Phi })h&=(\alpha -L)h+(D\Phi ,K_{21}D_2h)_U=f+(D\Phi , K_{21}D_2R(\alpha ,L)f)_U\\&=f-T_{\alpha }f=g. \end{aligned}$$

This yields $L^2(\mu )\subset (\alpha -L_{\Phi })(D(L))$. Since $L^2(\mu )$ is dense in $L^2(\mu ^{\Phi })$ and $D(L)\subset D(L_{\Phi })$ the dense range condition is shown and the proof is finished.$\square$

5 Examples and Outlook

In this section we have a look at certain examples, where the results we derived above are applicable. We consider the following situation, which is inspired by the one in [7, Section 5].

Let $U=V=L^2((0,1),\lambda ,{\mathbb {R}})$, $W=U\times V$ and $K_{12},K_{21},K_{22}$ such that they fulfill the properties in Hypothesis 4.3 and 4.10(e.g. $K_{12}=K_{21}=K_{22}=Id$). Moreover, let $(-\Delta , D(\Delta ))$ be the negative Dirichlet Laplacian, i.e.

$$\begin{aligned} D(\Delta )&=W^{1,2}_0((0,1),\lambda ,{\mathbb {R}})\cap W^{2,2}((0,1),\lambda ,{\mathbb {R}}) \subset L^2((0,1),\lambda ,{\mathbb {R}}),\\ -\Delta x&=-x''. \end{aligned}$$

On $(U,\mathcal {B}(U))$ and $(V,\mathcal {B}(V))$ we consider two centered non-degenerate infinite-dimensional Gaussian measures $\mu _1$ and $\mu _2$ with covariance operators

$$\begin{aligned} Q_1=Q_2=-\Delta ^{-1}:L^2((0,1),\lambda ,{\mathbb {R}})\rightarrow D(\Delta ), \end{aligned}$$

respectively. Recall the definition of the measures $\mu _1^{\Phi }$ and $\mu ^{\Phi }$ and denote by $B_U=B_V=(d_k)_{k\in {\mathbb {N}}}=(e_k)_{k\in {\mathbb {N}}}=(\sqrt{2}\sin (k\pi \cdot ))_{k\in {\mathbb {N}}}$ the orthonormal basis of $L^2((0,1),\lambda ,{\mathbb {R}})$ diagonalizing $Q_1$ and $Q_2$ with corresponding eigenvalues $(\lambda _k)_{k\in {\mathbb {N}}}=(\nu _k)_{k\in {\mathbb {N}}}=(\frac{1}{k^2\pi ^2})_{k\in {\mathbb {N}}}$. Additionally we fix a continuous differentiable (convex) function $\phi :{\mathbb {R}}\rightarrow {\mathbb {R}}$, which is bounded from below. Assume that there are constants $C_{1},C_2\in [0,\infty )$, $p_1\in [2,\infty )$ and $p_2\in [1,\infty )$ such that

$$\begin{aligned} |\phi (t)|&\le C_{1}(1+|t|^{p_1}),\quad t\in {\mathbb {R}},\\ |\phi '(t)|&\le C_{2}(1+|t|^{p_2}),\quad t\in {\mathbb {R}}. \end{aligned}$$

I.e. $\phi$ and its derivative have at most polynomial growth. For such $\phi$ we consider potentials $\Phi :L^2((0,1),\lambda ,{\mathbb {R}})\rightarrow (-\infty ,\infty ]$ defined by

$$\begin{aligned} \Phi (u)={\left\{ \begin{array}{ll}\int _{(0,1)}\phi \circ u\mathrm {d}\lambda &{} u\in L^{p_1}((0,1),\lambda ,{\mathbb {R}}) \\ \infty &{} u\notin L^{p_1}((0,1),\lambda ,{\mathbb {R}}) \end{array}\right. }. \end{aligned}$$

Remark 5.1

Note that potentials as defined above are lower semicontinuous by Fatou’s lemma, bounded from below and in $L^p(U,\mu _1,{\mathbb {R}})$ for all $p\in [1,\infty )$. If $\phi$ is convex the same holds true for $\Phi$. Using [7, Proposition 5.2] we know that $\Phi$ is bounded from below, lower semicontinuous and in $W^{1,2}(U,\mu _1,{\mathbb {R}})$ with $D\Phi (u)=\phi '\circ u$ for a.e. $u\in L^2((0,1),\lambda ,{\mathbb {R}})$ (namely, for all $u\in L^{2p_2}((0,1),\lambda ,{\mathbb {R}})$).

Choose a continuous differentiable (convex) $\phi$, which is bounded from below with at most polynomial growth. Further assume that the derivative of $\phi$ is bounded. Now define the potential $\Phi$ in terms of $\phi$. In Hypothesis 4.3 and 4.10 all items are valid, were by construction and Remark 5.1 we have

$$\begin{aligned} \Vert D\Phi \Vert _{L^{\infty }(\mu _1)}=\sup _{t\in {\mathbb {R}}}|\phi '(t)|<{\infty }. \end{aligned}$$

Therefore Theorem 4.11 is applicable and we obtain essential m-dissipativity in $L^2(L^{2}((0,1),\lambda ,{\mathbb {R}})\times L^{2}((0,1),\lambda ,{\mathbb {R}}),\mu ^{\Phi },{\mathbb {R}})$ of $(L_{\Phi },\mathcal {F}C_b^{\infty }(B_W))$, where for $f\in \mathcal {F}C_b^{\infty }(B_W)$ we have

$$\begin{aligned} L_{\Phi }f=&\mathrm {tr}[K_{22}D^2_2f]+(v,K_{22}\Delta D_2f)_{L^2(\lambda )}\\&+(u,K_{21}\Delta D_2f)_{L^2(\lambda )}-(\phi '\circ u, K_{21}D_2f)_{L^2(\lambda )}-(v,K_{12}\Delta D_1f)_{L^2(\lambda )}. \end{aligned}$$

This is the starting point to construct a martingale and even a weak solution to the non-linear infinite-dimensional stochastic differential equation given by

$$\begin{aligned} \begin{aligned}&\mathrm {d}U_t=-K_{21}\Delta V_t\mathrm {d}t\\&\mathrm {d}V_t=(K_{22}\Delta V_t+K_{12}\Delta U_t-K_{12}\phi '(U_t))\mathrm {d}t+\sqrt{2K_{22}}\mathrm {d}W_t. \end{aligned} \end{aligned}$$

(5.1)

In the equation above $(W_t)_{t\ge 0}$ is a cylindrical Brownian motion on $(V,\mathcal {B}(V))$. A heuristically application of the Itô-formula suggest that this stochastic differential equation corresponds to $(L_{\Phi },D(L_{\Phi }))$. In order to make this correspondence rigorous and to construct weak solutions we want to apply general resolvent methods described in [1].

I.e. we plan to construct a $\mu ^{\Phi }$-standard right process (see [1, Appendix B.])

$$\begin{aligned} X=(\Omega , \mathcal {F}, (\mathcal {F}_t)_{t\ge 0}, (U_t,V_t)_{t\ge 0}, (\theta _t)_{t\ge 0}, (\mathbbm {P}^{w})_{w\in W}), \end{aligned}$$

providing a martingale and even a weak solution, with infinite lifetime $\mathbbm {P}_{\mu ^{\Phi }}$-a.e. and $\mathcal {T}$-continuous paths, $\mathbbm {P}_{\mu ^{\Phi }}$-a.e.. Here $\mathcal {T}$ denotes the weak topology on the Hilbert space W and $\mathbbm {P}_{\mu ^{\Phi }}$ the probability measure on $(\Omega ,\mathcal {F})$ defined for $A\in \mathcal {F}$ by

$$\begin{aligned} \mathbbm {P}_{\mu ^{\Phi }}(A)=\int _W\mathbbm {P}^{w}(A)\mathrm {d}\mu ^{\Phi }(w)\quad A\in \mathcal {F}. \end{aligned}$$

Moreover, the transition semigroup corresponding to the process X and the strongly continuous semigroup $(T_t)_{t\ge 0}$ generated by $(L_{\Phi },D(L_{\Phi }))$ coincide in $L^2(\mu ^{\Phi })$. Using this correspondence we like to study the long-time behavior of the Process X via the long-time behavior of the semigroup $(T_t)_{t\ge 0}$. I.e. we plan to apply the abstract Hilbert space hypocoercivity method from [15], to show exponential convergence to equilibrium of the semigroup.

As announced in the introduction the results derived in Sect. 3 are naturally needed while applying the abstract Hilbert space hypocoercivity method. A rigorous application in our infinite-dimensional setting goes beyond the aim of this article. At this point we illustrate the rough idea and explain how to achieve such an exponential convergence result and how Ornstein-Uhlenbeck operators perturbed by the gradient of a potential with unbounded diffusions (C, D(C)) as coefficients appear during the application process. As in the article [13], of Dolbeault, Mouhot and Schmeiser we define the modified entropy functional $H_{\varepsilon }:{L^2(\mu ^{\Phi })}\rightarrow {\mathbb {R}}$, $\varepsilon \in [0,1)$ by

$$\begin{aligned} H_{\varepsilon }[f]=\frac{1}{2}\Vert f\Vert _{L^2(\mu ^{\Phi })}^2+\varepsilon (Bf,f)_{L^2(\mu ^{\Phi })},\quad f\in {L^2(\mu ^{\Phi })}. \end{aligned}$$

The bounded linear operator B, specified below is chosen such that $H_{\varepsilon }$ defines a norm which is equivalent to the norm $\Vert \cdot \Vert _{L^2(\mu ^{\Phi })}$, compare [15]. To construct B, define $P_S:L^2(\mu ^{\Phi })\rightarrow L^2(\mu ^{\Phi })$ by

$$\begin{aligned} P_Sf= \int _V f\mathrm {d}\mu _2, \end{aligned}$$

where the integration is understood w.r.t. the second variable. Using that $\mu ^{\Phi }$ is a probability measure one can check that the map $P:L^2(\mu ^{\Phi })\rightarrow L^2(\mu ^{\Phi })$ given as

$$\begin{aligned} Pf=P_Sf-(f,1)_{L^2(\mu ^{\Phi })},\quad f\in L^2(\mu ^{\Phi }), \end{aligned}$$

is an orthogonal projection with

$$\begin{aligned} Pf\in L^2(\mu _1^{\Phi })\quad \text {and}\quad \Vert Pf\Vert _{L^2(\mu _1^{\Phi })}= \Vert Pf\Vert _{L^2(\mu ^{\Phi })},\quad f\in L^2(\mu ^{\Phi }), \end{aligned}$$

where we canonically embed $L^2(\mu _1^{\Phi })$ into $L^2(\mu ^{\Phi })$. Now the bounded linear operator B on $L^2(\mu ^{\Phi })$ is defined as the unique extension of $(B,D((A_{\Phi }P)^*))$ to a continuous linear operator on $L^2(\mu ^{\Phi })$ where

$$\begin{aligned} B= (Id+(A_{\Phi }P)^*A_{\Phi }P)^{-1}(A_{\Phi }P)^*\quad \text {on}\quad D((A_{\Phi }P)^*). \end{aligned}$$

Here $(A_{\Phi }P,D(A_{\Phi }P))$ is the linear operator $A_{\Phi }P$ with domain

$$\begin{aligned} D(A_{\Phi }P)=\lbrace f\in {L^2(\mu ^{\Phi })}\mid Pf\in D(A_{\Phi })\rbrace . \end{aligned}$$

and $((A_{\Phi }P)^*,D((A_{\Phi }P)^*))$ denotes its adjoint on ${L^2(\mu ^{\Phi })}$. Note that by the Neumann-Series theorem, the operator

$$\begin{aligned} Id+(A_{\Phi }P)^*A_{\Phi }P:D((A_{\Phi }P)^*A_{\Phi }P)\rightarrow {L^2(\mu ^{\Phi })} \end{aligned}$$

with domain $D((A_{\Phi }P)^*A_{\Phi }P)=\lbrace f\in D(A_{\Phi }P)\mid A_{\Phi }Pf\in D((A_{\Phi }P)^*)\rbrace$ is bijective and admits a bounded inverse. Hence B is indeed well-defined on $D((A_{\Phi }P)^*)$. For the fact that B extends to a bounded linear operator on ${L^2(\mu ^{\Phi })}$, see [22, Theo. 5.1.9]. Now one can calculate

$$\begin{aligned}&\frac{d}{dt}H_{\varepsilon }[T_tf]\\&=(L_{\Phi }T_tf,T_tf)_{L^2(\mu ^{\Phi })}+\varepsilon (BL_{\Phi }T_tf,T_tf)_{L^2(\mu ^{\Phi })} +\varepsilon (BT_tf,L_{\Phi }T_tf)_{L^2(\mu ^{\Phi })}. \end{aligned}$$

Define the possible unbounded operator (C, D(C)) in U by

$$\begin{aligned} C=K_{21}Q_2^{-1}K_{12}\quad \text {with}\quad D(C)=\lbrace u\in U\vert K_{12}u\in D(Q_2^{-1})\rbrace . \end{aligned}$$

One can show that for all $f\in \mathcal {F}C_b^{\infty }(B_W)$, the operator $(PA_{\Phi }^2P,\mathcal {F}C_b^{\infty }(B_W))$ is given by the formula

$$\begin{aligned} PA_{\Phi }^2Pf=&\mathrm {tr}[CD^2P_Sf]-(u,Q_1^{-1}CD P_Sf)_U-(D\Phi (u),CD P_Sf)_U. \end{aligned}$$

The essential m-dissipativity of $(PA_{\Phi }^2P,\mathcal {F}C_b^{\infty }(B_W))$ in $L^2(\mu ^{\Phi })$ and corresponding regularity estimates, which are applicable in view of Theorem 3.6 and Theorem 3.11 derived in Sect. 3, are fundamental to show [15, Corollary 2.13] and [15, Proposition 2.15], i.e. to derive

$$\begin{aligned} \frac{d}{dt}H_{\varepsilon }[T_tf]\le -\kappa H_{\varepsilon }[T_tf], \end{aligned}$$

for an appropriate chosen $\varepsilon \in [0,1)$ and a positive constant $\kappa \in (0,\infty )$, compare [15, Theorem 2.18]. Applying Grönwall’s lemma yields the desired exponential convergence to equilibrium.