Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and R\"ockner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.


Introduction
We study the exponential decay to equilibrium of Langevin dynamics with multiplicative noise. The corresponding evolution equation is given by the following stochastic differential equation on R 2d , d ∈ N, as where Φ : R d → R is a suitable potential whose properties are specified later, B = (B t ) t≥0 is a standard d-dimensional Brownian motion, σ : R d → R d×d a variable diffusion matrix with at least weakly differentiable coefficients, and b : R d → R d given by where a ij = Σ ij with Σ = σσ T .
This equation describes the evolution of a particle described via its position (X t ) t≥0 and velocity (V t ) t≥0 coordinates, which is subject to friction, stochastic perturbation depending on its velocity, and some outer force ∇Φ. To simplify notation, we split R 2d into the two components x, v ∈ R d corresponding to position and velocity respectively. This extends to differential operators ∇ x , ∇ v , and the Hessian matrix H v .
Using Itô's formula, we obtain the associated Kolmogorov operator L as Here a · b or alternatively (a, b) euc denotes the standard inner product of a, b ∈ R d . We introduce the measure µ = µ Σ,Φ on (R 2d , B(R 2d )) as i.e. ν is the normalized standard Gaussian measure on R d . We consider the operator L on the Hilbert space H . . = L 2 (R 2d , µ).
We note that the results below on exponential convergence to equilibrium can also be translated to a corresponding Fokker-Planck setting, with the differential operator L FP given as the adjoint, restricted to sufficiently smooth functions, of L in L 2 (R 2d , d(x, v)). The considered Hilbert space there isH . . = L 2 (R 2d ,μ), wherẽ Indeed, this is the space in which hypocoercivity of the kinetic Fokker-Planck equation associated with the classical Langevin dynamics was proven in [1]. The rigorous connection to the Kolmogorov backwards setting considered throughout this paper and convergence behaviour of solutions to the abstract Cauchy problem ∂ t f (t) = L FP f (t) are discussed in Section 5.3.
The concept of hypocoercivity was first introduced in the memoirs of Cédric Villani ([2]), which is recommended as further literature to the interested reader. The approach we use here was introduced algebraically by Dolbeault, Mouhot and Schmeiser (see [3] and [1]), and then made rigorous including domain issues in [4] by Grothaus and Stilgenbauer, where it was applied to show exponential convergence to equilibrium of a Fiber laydown process on the unit sphere. This setting was further generalized by Wang and Grothaus in [5], where the coercivity assumptions involving in part the classical Poincaré inequality for Gaussian measures were replaced by weak Poincaré inequalities, allowing for more general measures for both the spatial and the velocity component. In this case, the authors still obtained explicit, but subexponential rates of convergence.
On the other hand, the stronger notion of hypercontractivity was explored in [6] on general separable Hilbert spaces without the necessity to explicitly state the invariant measure. The specific case of hypocoercivity for Langevin dynamics on the position space R d has been further explored in [7] and serves as the basis for our hypocoercivity result. However, all of these prior results assume the diffusion matrix to be constant, while we allow for velocity-dependent coefficients.
In contrast to [7], we do not know if our operator (L, C ∞ c (R 2d )) is essentially m-dissipative, and are therefore left to prove that first. This property of the Langevin operator has been shown by Helffer and Nier in [8] for smooth potentials and generalized to locally Lipschitz-continuous potentials by Conrad and Grothaus in [9, Corollary 2.3]. However, a corresponding result for a non-constant second order coefficient matrix Σ is not known to the authors.
Moreover, the symmetric part S of our operator L does not commute with the linear operator B as in [7], hence the boundedness of the auxiliary operator BS needs to be shown in a different way, which we do in Proposition 3.10.
In Theorem 3.4, we show under fairly light assumptions on the coefficients and the potential that the operator (L, C ∞ c (R 2d )) is essentially m-dissipative and therefore generates a strongly continuous contraction semigroup on H. The proof is given in Section 4 and follows the main ideas as in the proof of [9, Theorem 2.1], where a corresponding result for Σ = I was obtained.
For that proof we rely on perturbation theory of m-dissipative operators, starting with essential m-dissipativity of the symmetric part of L. To that end, we state an essential self-adjointness result for a set of non-degenerate elliptic Dirichlet differential operators (S, C ∞ c (R d )) on L 2 -spaces where the measure is absolutely continuous wrt. the Lebesgue measure. This result is stated in Theorem 4.5 and combines regularity results from [10] and [11] with the approach to show essential self-adjointness from [12].
Finally, our main hypocoercivity result reads as follows: which satisfies a Poincaré inequality of the form for some Λ ∈ (0, ∞) and all f ∈ C c (R d ). Furthermore assume the existence of a constant c < ∞ such that where H denotes the Hessian matrix and |HΦ| the Euclidian matrix norm. If β > −1, then also assume that there are constants N < ∞, γ < 2 1+β such that Then the Langevin operator (L, C ∞ c (R 2d )) as defined in (1.2) is closable on H and its closure (L, D(L)) generates a strongly continuous contraction semigroup (T t ) t≥0 on H. Further, it holds that for each θ 1 ∈ (1, ∞), there is some θ 2 ∈ (0, ∞) such that for all g ∈ H and all t ≥ 0. In particular, θ 2 can be specified as , and the coefficients n i ∈ (0, ∞) only depend on the choice of Φ.
Finally, our main results may be summarized by the following list: • Essential m-dissipativity (equivalently essential self-adjointness) of non-degenerate elliptic Dirichlet differential operators with domain C ∞ c (R d ) on Hilbert spaces with measure absolutely continuous wrt. the d-dimensional Lebesgue measure is proved, see Theorem 4.5.
• Essential m-dissipativity of the backwards Kolmogorov operator (L, C ∞ c (R d )) associated with the Langevin equation with multiplicative noise (1.1) on the Hilbert space H under weak assumptions on the coefficient matrix Σ and the potential Φ, in particular not requiring smoothness, is shown, see Theorem 3.4.
• Exponential convergence to a stationary state of the corresponding solutions to the abstract Cauchy problem ∂ t u(t) = Lu(t), see (5.1) on the Hilbert space H with explicitly computable rate of convergence, as stated in Theorem 1.1, is proved.
• Adaptation of this convergence result to the equivalent formulation as a Fokker-Planck PDE on the appropriate Hilbert spaceH . . = L 2 (R 2d ,μ) is provided. In particular, this yields exponential convergence of the solutions to the abstract Fokker-Planck Cauchy problem ∂ t u(t) = L FP u(t), with L FP given by (5.3), to a stationary state, see Section 5.3.
• A stochastic interpretation of the semigroup as a transition kernel for a diffusion process is worked out. Moreover, we prove this diffusion process to be a weak solution to the Langevin SDE (1.1) and derive for it strong mixing properties with explicit rates of convergence, see Section 5.2.

The abstract hypocoercivity setting
We start by recalling some basic facts about closed unbounded operators on Hilbert spaces: (iv) Let (T, D(T )) be closed. Then the operator T L with domain is also closed.
(v) LT with domain D(T ) need not be closed, however Let us now briefly state the abstract setting for the hypocoercivity method as in [4].

Data conditions (D).
We require the following conditions which are henceforth assumed without further mention. As in the given source, we also require the following assumptions: Assumption (H1). Algebraic relation: It holds that P AP | D = 0.
Assumption (H2). Microscopic coercivity: There exists some Λ m > 0 such that Assumption (H3). Macroscopic coercivity: Define (G, D) via G = P A 2 P on D. Assume that (G, D) is essentially self-adjoint on H. Moreover, assume that there is some More specifically, if there exist δ > 0, ε ∈ (0, 1) and 0 < κ < ∞ such that for all g ∈ D(L), t ≥ 0, it holds where f t . . = T t g − (g, 1) H , then the constants κ 1 and κ 2 are given by In order to prove (H4), we will make use of the following result: If there exists some C < ∞ such that By essential self-adjointness and hence essential m-dissipativity of G,

Hypocoercivity for Langevin dynamics with multiplicative noise
As stated in the introduction, the aim of this section is to prove exponential convergence to equilibrium of the semigroup solving the abstract Kolmogorov equation corresponding to the Langevin equation with multiplicative noise (1.1).
We remark that most of the conditions are verified analogously to [7], the main difference being the proof of essential m-dissipativity for the operator (L, C ∞ c (R 2d )) as well as the first inequality in (H4). Nevertheless, some care has to be taken whenever S is involved, as it doesn't preserve regularity to the same extent as in the given reference.

The data conditions
We start by introducing the setting and verifying the data conditions (D). The notations introduced in this part will be used for the remainder of the section without further mention.
Let d ∈ N and set the state space as E = R 2d , F = B(R 2d ). In the following, the first d components of E will be written as x, the latter d components as v. Let ν be the normalised Gaussian measure on R d with mean zero and covariance matrix I, i.e.
Assumption (P). The potential Φ : R d → R is assumed to depend only on the position variable x and to be locally Lipschitz-continuous. We further assume e −Φ(x) dx to be a probability measure on (R d , B(R d )).
Note that the first part implies Φ ∈ H 1,∞ loc (R d ). Moreover, Φ is differentiable dx-a.e. on R d , such that the weak gradient and the derivative of Φ coincide dx-a.e. on R d . In the following, we fix a version of ∇Φ.
The probability measure µ on (E, F) is then given by µ = e −Φ(x) dx ⊗ ν, and we set H . . = L 2 (E, µ), which satisfies condition (D1). Next we assume the following about Σ = (a ij ) 1≤i,j≤d with a ij : R d → R: Assumption (Σ1). Σ is symmetric and uniformly strictly elliptic, i.e. there is some Additionally, we will consider one of the following conditions on the growth of the partial derivatives: We note that any of these imply ∂ j a ij ∈ L 2 (R d , ν) for all 1 ≤ i, j ≤ d.

Definition 3.1. Let Σ satisfy (Σ2). Then we set
If Σ additionally satisfies (Σ3), then we define If instead (Σ3 ′ ) is fulfilled, then we consider instead be the space of compactly supported smooth functions on E. We define the linear operators S, A and L on D via Integration by parts shows that ( For f ∈ D and g ∈ H 1,2 (E, µ), integration by parts yields In particular, (D6) is obviously fulfilled. Next we provide an estimate which will be needed later: Proof: Due to integration by parts, it holds that Hence we obtain in the case (Σ3 ′ ) We now state the essential m-dissipativity result, which will be proven in the next section.
Let us now introduce the orthogonal projections P S and P : where integration is understood w.r.t the velocity variable v. By Fubini's theorem and the fact that ν is a probability measure on (E, F), it follows that P S is a well-defined orthogonal projection on H with Additionally, for each f ∈ D, P S f admits a unique representation in In order to show the last remaining conditions (D5) and (D7), we will make use of a standard sequence of cutoff functions as specified below: Moreover 0 ≤ ϕ n ≤ 1 for all n ∈ N and ϕ n → 1 pointwisely on R d as n → ∞.
Then the operator L satisfies the following: In particular, (D5) and (D7) are fulfilled.

Proof:
We only show (i), as the other parts can be shown exactly as in [7]. First, let f ∈ C ∞ c (R d ) and define f n ∈ D via f n (x, y) . . = f (x)ϕ n (v). Then by Lebesgue's dominated convergence theorem and the inequalities in the previous definition, Since f n → f in H and by closedness of (S, D(S)), this implies f ∈ D(S) with Sf = 0, where f is interpreted as an element of H. Now let g ∈ P (H) and identify g as an element of Identifying all g n and g with elements in H then yields g n → g in H as n → ∞ and g n ∈ D(S), Sg n = 0 for all n ∈ N. Therefore, again by closedness of (S, D(S)), g ∈ D(S) and Sg = 0.

The hypocoercivity conditions
Now we verify the hypocoercivity conditions (H1)-(H4) for the operator L. From here on, we will assume Σ to satisfy (Σ1), (Σ2) and either (Σ3) or (Σ3 ′ ), with N Σ referring to the appropriate constant as in Definition 3.1. Analogously to [7] we introduce the following conditions: Hypocoercivity assumptions (C1)-(C3). We require the following assumptions on Φ : R d → R: (C1) The potential Φ is bounded from below, is an element of C 2 (R d ) and e −Φ(x) dx is a probability measure on (R d , B(R d )).
Since we only change the operator (S, D(S)) in comparison to the framework of [7], the results stated there involving only (A, D(A)) and the projections also hold here and are collected as follows:
Then the first inequality of (H4) is also satisfied with Proof: We want to apply Lemma 2.3 to the operator (S, D(S)). Let f ∈ D, h ∈ D(S) and h n ∈ D such that h n → h and Sh n → Sh in H as n → ∞. Then, by integration by parts, This A final integration by parts then yields where the last inequality is due to dissipativity of (G, D).

Remark 3.11.
We remark here that all previous considerations up to the explicit rate of convergence can also be applied to the formal adjoint operator (L * , D) with L * = S + A, the closure of which generates the adjoint semigroup (T * t ) t≥0 on H. For example, the perturbation procedure to prove essential m-dissipativity is exactly the same as for L, since the sign of A does not matter due to antisymmetry. We can use this to construct solutions to the corresponding Fokker-Planck PDE associated with our Langevin dynamics, see Section 5.3.

Essential m-dissipativity of the Langevin operator
The goal of this section is to prove Theorem 3.4. We start by giving some basics on perturbation of semigroup generators.

Definition 4.1. Let (A, D(A)) and (B, D(B)) be linear operators on H. Then B is said to be A-bounded if D(A) ⊂ D(B) and there exist constants a, b < ∞ such that
holds for all f ∈ D(A). The number inf{a ∈ R | (4.1) holds for some b < ∞} is called the A-bound of B.

Theorem 4.2. Let D ⊂ H be a dense linear subspace, (A, D) be an essentially mdissipative linear operator on H and let (B, D) be dissipative and A-bounded with Abound strictly less than 1. Then (A + B, D) is essentially m-dissipative and its closure is given by (A + B, D(A)).
A useful criterion for verifying A-boundedness is given by:

Lemma 4.3. Let D ⊂ H be a dense linear subspace, (A, D) be essentially m-dissipative and (B, D) be dissipative. Assume that there exist constants c, d < ∞ such that
We also require the following generalization of the perturbation method:

The symmetric part
We first prove essential self-adjointness, equivalently essential m-dissipativity, for a certain class of symmetric differential operators on specific Hilbert spaces. This is essentially a combination of two results by Bogachev, Krylov, and Röckner, namely [10, Corollary 2.10] and [12,Theorem 7], however, the combined statement does not seem to be well known and might hold interest as the basis for similar m-dissipativity proofs. We use the slightly more general statement from [11, Theorem 5.1] in order to relax the assumptions.  ) and

Theorem 4.5. Let d ≥ 2 and consider
Define further the linear operator (S, D) via Then (S, D) is essentially self-adjoint on H.

Proof:
Analogously to the proof of [12,Theorem 7], it can be shown that ρ is continuous, hence locally bounded. Assume that there is some g ∈ H such that (4.2) Define the locally finite signed Borel measure ν via ν = gρ dx, which is then absolutely continuous with respect to the Lebesgue measure. By definition it holds that so by [11,Theorem 5.1], the density g·ρ of ν is in H 1,p loc (R d ) and locally Hölder continuous, hence locally bounded. This implies , is locally bounded, and g · b i ∈ L p loc (R d ) for all 1 ≤ i ≤ d. Therefore, we can apply integration by parts to (4.2) and get for every f ∈ D: Note that this equation can then be extended to all f ∈ H 1,2 (R d ) with compact support, since p > 2 by definition. Now let ψ ∈ C ∞ c (R d ) and set η = ψg ∈ H 1,2 (R d ), which has compact support. The same then holds for f . . = ψη ∈ H 1,2 (R d ). Elementary application of the product rule yields (∇η, A∇(ψg)) euc = (∇f, A∇g) euc − η(∇ψ, A∇g) euc + g(∇η, A∇ψ) euc . (4.4) From now on, for a, b : R d → R d , let (a, b) always denote the evaluation of the Euclidean inner product (a, b) euc . By using (4.4) and applying (4. 3) to f , we get where the last step follows from the product rule and symmetry of A. Since A is locally strictly elliptic, there is some c > 0 such that and therefore it follows that Let (ψ n ) n∈N be as in Definition 3.6. Then (4.5) holds for all ψ = ψ n . By dominated convergence, the left part converges to g 2 H as n → ∞. The integrand of the right hand side term is dominated by d 2 C 2 M · g 2 ∈ L 1 (µ), where C is from Def. 3.6 and M . . = max 1≤i,j≤d a ij ∞ . By definition of the ψ n , that integrand converges pointwisely to zero as n → ∞, so again by dominated convergence it follows that g = 0 in H. H and therefore that (S, D) is essentially selfadjoint.

This implies that (S − I)(D) is dense in
Remark 4.6. The above theorem also holds for d = 1, as long as p ≥ 2. Indeed, continuity of ρ follows from similar regularity estimates, see [12,Remark 2]. The proof of [11,Theorem 5.1] mirrors the proof of [10, Theorem 2.8], where d ≥ 2 is used to apply [10,Theorem 2.7]. However, in the cases where it is applied, this distinction is not necessary (since p ′ < q always holds). Finally, the extension of (4.3) requires p ≥ 2.
We use this result to prove essential m-dissipativity of the symmetric part (S, D) of our operator L:

Proof:
The density ρ of ν wrt. the Lebesgue measure is given by ρ(v) = e −v 2 /2 = (e −v 2 /4 ) 2 . Due to the conditions (Σ1), (Σ2) and either (Σ3) or (Σ3 ′ ), all assumptions from Theorem 4.5 are fulfilled and therefore, (S, Then which converges to zero as n → ∞. By taking linear combinations, this shows that is dense in H, (S, D) is essentially m-dissipative and its closure (S, D(S)) generates a strongly continuous contraction semigroup.
It can easily be shown that (Sf, f + ) H ≤ 0 for all f ∈ D. Parallelly to the proof of (D7), it holds that 1 ∈ D(S) and S1 = 0. This together implies that (S, D(S)) is a Dirichlet operator and the generated semigroup is sub-Markovian.

Perturbation of the symmetric part for nice coefficients
Now we extend the essential m-dissipativity stepwise to the non-symmetric operator L by perturbation. This follows and is mostly based on the method seen in the proof of [15, Theorem 6.3.1], which proved that result for Σ = I.
, is again essentially m-dissipative. Note that S 1 is explicitly given by We show the following perturbation result: d(x, v)).

Proof:
Define the orthogonal projections P n via P n f (x, v) . . = ξ n (x)f (x, v), where ξ n is given by ξ n = 1 [n−1,n) (|x|), which leave D 1 invariant. Then the conditions for Lemma 4.4 are fulfilled, and we are left to show the A n -bounds. Note that due to the restriction on β, For each fixed n ∈ N it holds for all f ∈ P n D 1 : Hence by Lemma 4.3, (ivxIP n , P n D 1 ) is S 1 P n -bounded with Kato-bound zero. Application of Lemma 4.4 yields the statement.
wrt. the graph norm of S 1 + ivxI, we obtain essential m-dissipativity of ( ) and therefore also of its dissipative extension (S 1 + ivxI,   d(x, v)).

Proof:
It holds due to antisymmetry of v∇ x that analogously to the proof of Proposition 4.8, which again implies that the antisymmetric, hence dissipative operator (∇Φ∇ v , D 2 ) is L 2 -bounded with bound zero. This shows the claim.
Due to boundedness of e −Φ and v j e −v 2 /2 for all 1 ≤ j ≤ d, it follows immediately that g n ⊗ h → f and L(g n ⊗ h) → Lf in H as n → ∞. This extends to arbitrary f ∈ D ′ via linear combinations and therefore shows that and hence also D, is a core for (L, D(L)).

Proof of Theorem 3.4
It is now left to relax the assumptions on Σ and Φ by approximation. Let the assumptions of Theorem 3.4 hold and wlog Φ ≥ 0. For n ∈ N we define Σ n via Then each Σ n also satisfies (Σ1)-(Σ3) with β = −1, since ∂ k a ij,n = ∂ k a ij on B n (0) and |∂ k a ij,n | ≤ Then Theorem 4.10 shows that for each n, m ∈ N, (L n,m , D) is essentially m-dissipative on H m , and it holds that L n,m f = Lf for all f ∈ D on B m (0) × B n (0). Note further that · H ≤ · Hm .
We need the following estimates: Analogously to the proof of Proposition 4.8 and due to antisymmetry of v∇ as well as Further, it clearly holds that this proves the first statement with For the second part, note that Repeating all calculations of the first part yields We prove that for every ε > 0, we can find some f ∈ D such that (I − L)f − g H < ε. This then extends to arbitrary g ∈ C ∞ c (R d )⊗C ∞ c (R d ) via linear combinations and therefore implies essential m-dissipativity of (L, If β ≤ −1, then the proof is easier and follows analogously to the proof of of [15, Theorem 6.3.1]. Therefore we will assume β > −1. Recall that in this case, we have |∇Φ(x)| ≤ N (1 + |x| γ ) for all x ∈ R d , where γ < 2 1+β , see the assumptions of Theorem 3.4.
Denote the support of g by K x × K v , where K x and K v are compact sets in R d . By a standard construction, for each δ for all multi-indices s ∈ N d . Fix α such that 1+β 2 < α < 1 γ . For any δ > 0, we set δ x . . = δ α and δ v . . = δ, and then define For f ∈ D, δ > 0, consider f δ . . = χ δ f , which is an element of D, as χ δ ∈ D. Without loss of generality, we consider δ and hence δ α sufficiently large such that supp(φ δ α ) ⊂ B 2δ α (0), supp(ψ δ ) ⊂ B 2δ (0) and that there are n, m ∈ N that satisfy The following then holds: and φ, ψ as above. Then there is a constant D 2 < ∞ and a function ρ : R → R satisfying ρ(s) → 0 as s → ∞, such that for any δ, n and m satisfying (4.6), holds for all f ∈ D.

Proof:
By the product rule, Due to the choice of n and m, every · H on the right hand side can be replaced with · Hm , a ij by a ij,n , and Φ by Φ m , hence L by L n,m .
We now give estimates for each summand of the right hand side, in their order of appearance: where the last inequality is due to |∂ i Φ(x)| ≤ N (1+|x| γ ) for all x ∈ R d and the support of the cutoff as in (4.6). Application of Lemma 4.11 shows the existence of D 2 independent of n, m, such that Clearly ρ(δ) → 0 as δ → ∞ due to β < 1 and the definition of α.
Now finally we show that for each ε > 0, we can find some f δ ∈ D such that Choose δ > 0 large enough such that ρ(δ) < ε 4D 2 g H (where ρ ans D 2 are provided by Lemma 4.12) and that there exist n, m satisfying (4.6).
Then choose f ∈ D via Theorem 4.10 such that (I −L n,m )f −g Hm < min{ ε 2 , g H } and define f δ as before. Note that due to the choice of the cutoffs, it holds g H = g Hm , therefore As mentioned earlier, this shows essential m-dissipativity of the operator (L, D) on H and therefore concludes the proof of Theorem 3.4.

The associated Cauchy problem
We consider the abstract Cauchy problem associated with the operator L. Given the initial condition u 0 ∈ H, u : [0, ∞) → H should satisfy If u 0 ∈ D(L), then u(t) ∈ D(L) for all t ≥ 0, and ∂ t u(t) = LT t u 0 = Lu(t), so u is even a classical solution to the abstract Cauchy problem associated to L. In particular, this holds for all u 0 ∈ C 2 c (R d×d ), since L is dissipative there and it extends D, which implies C 2 c (R d×d ) ⊂ D(L). In this context, Theorem 1.1 shows exponential convergence of the unique solution u(t) to a constant as t → ∞. More precisely, for each θ 1 > 1 we can calculate θ 2 ∈ (0, ∞) depending on the choice of Σ and Φ such that for all t ≥ 0,

Connection to Langevin dynamics with multiplicative noise
So far, our considerations have been purely analytical, giving results about the core property of D for L and rate of convergence for the generated semigroup (T t ) t≥0 in H. However, this approach is still quite natural in the context of the Langevin SDE (1.1), as the semigroup has a meaningful stochastic representation. The connection is achieved via the powerful theory of generalized Dirichlet forms as developed by Stannat in [16], which gives the following: Assume the context of Theorem 3.4. There exists a Hunt process M = Ω, F, (F t ) t≥0 , (X t , V t ), (P (x,v) ) (x,v)∈R d ×R d with state space E = R d × R d , infinite lifetime and continuous sample paths (P (x,v) -a.s. for all (x, v) ∈ E), which is properly associated in the resolvent sense with (T t ) t≥0 . In particular (see [15,Lemma 2.2.8]), this means that for each bounded measurable f which is also square-integrable with respect to the invariant measure µ and all t > 0, T t f is a µ-version of p t f , where (p t ) t≥0 is the transition semigroup of M with

Corresponding Fokker-Planck equation
In this part we give a reformulation of the convergence rate result detailed in Section 5.1 for readers which are more familiar with the classical Fokker-Planck formulation for probability densities. In the current literature, Fokker-Planck equations are more often expressed as equations on measures, rather than functions. For example, in the nondegenerate case, exponential convergence in total variation to a stationary solution is studied in [18], which includes further references to related works. Our goal here however is simply to make the convergence result immediately applicable to less specialized readers in the form of the estimate (5.4) for solutions to the Cauchy problem associated with the operator defined in (5.3), hence we stick to the expression via probability densities.
Given a Kolmogorov backwards equation of the form −∂ t u(x, t) = L K u(x, t), the corresponding Fokker-Planck equation is given by ∂ t f (x, t) = L FP f (x, t), where L FP = (L K ) ′ is the adjoint operator of L K in L 2 (R d , dx), restricted to smooth functions. In our setting, L K = L produces via integration by parts for f ∈ D: Let (T t ) t≥0 be the semigroup on H generated by (L, D(L)) and denote by (T * t ) t≥0 and L * the adjoint semigroup on H and its generator, respectively. It is evident that for f ∈ D, L * is given as L * f = (S + A)f , where S and A refer to the symmetric and antisymmetric components of L respectively, as defined in Definition 3.2. As mentioned in 3.11, we achieve the exact same results for the equation corresponding to L * as for the one corresponding to L, which we considered in Section 3. In particular, (L * , D) is essentially m-dissipative and its closure (L * , D(L * )) generates (T * t ) t≥0 , which converges exponentially to equilibrium with the same rate as (T t ) t≥0 .
Let T t g . . = T (T * t )T −1 g for t ≥ 0, g ∈ H. Then ( T t ) t≥0 is a strongly continuous contraction semigroup on H with the generator (T L * T −1 , T (D(L * ))). It is easy to see that L FP = T L * T −1 , so for each initial condition u 0 ∈ H, u(t) . . = T t u 0 is a mild solution to the Fokker-Planck Cauchy problem. Note that for Φ ∈ C ∞ (R d ), the transformation T leaves D invariant, which implies D ⊂ T (D(L * )) and essential m-dissipativity of (L FP , D) on H.