Stochastic differential equations with sticky reflection and boundary diffusion

We construct diffusion processes in bounded domains Ω with sticky reflection at the boundary Γ in use of Dirichlet forms. In particular, the occupation time on the boundary is positive. The construction covers a static boundary behavior and an optional diffusion along Γ. The process is a solution to a given SDE for q.e. starting point. Using regularity results for elliptic PDE with Wentzell boundary conditions we show strong Feller properties and characterize the constructed process even for every starting point in Ω\Ξ, where Ξ is given explicitly by the involved densities. By a time change we obtain pointwise solutions to SDEs with immediate reflection under weak assumptions on Γ and the drift. A non-trivial extension of the construction yields N-particle systems with the stated boundary behavior and singular drifts. Finally, the setting is applied to a model for particles diffusing in a chromatography tube with repulsive interactions.


Introduction
In the first part of the present paper, we construct via Dirichlet form techniques diffusions on Ω for bounded domains Ω of R d , d ≥ 1, with boundary Γ, and identify them as weak solutions of SDEs given by for q.e. x ∈ Ω under weak assumptions on the drifts given by α and β, where n(y) is the outward normal at y ∈ Γ, P is the projection on the tangent space and δ ∈ {0, 1}. In the case δ = 1 we additionally assume that d ≥ 2. A solution to (1.1) can be characterized as Brownian motion with drift inside Ω and if the process reaches Γ, Brownian motion with drift along Γ may take place, while a further drift term in normal direction is directed back into the interior of Ω. In addition, the Brownian motion B Γ = (B Γ t ) t≥0 on Γ is the projection of the d-dimensional Brownian motion B = (B t ) t≥0 onto the manifold Γ (in the sense of a Stratonovich SDE).
In this situation, the boundary behavior is called sticky and is connected to so-called Wentzell boundary conditions. In contrast to reflecting (Neumann) boundary conditions which provide an immediate reflection, Wentzell boundary conditions yield sojourn on Γ. The infinitesimal generator and semigroup associated to such kind of diffusions were first investigated in [Fel52] and in [Wen59] on [0, ∞). This kind of diffusion is also considered in [IW89, Chap. IV, Sect. 7] on the half-space R d with Lipschitz continuous, bounded drifts. In [Car09] the author uses a Dirichlet form approach in order to construct Brownian motion with boundary diffusion in a similar setting to ours with the essential difference that the boundary behavior is not sticky and also a drift does not occur, i.e., only constant densities are admissible. More precisely, the considered approach corresponds to ordinary reflecting boundary conditions (with the d-dimensional Lebesgue measure as reference measure) instead of a sticky boundary behavior. Moreover, in [VV03] and [MR06] diffusion operators on Ω with sticky boundary behavior are considered, but without introducing a boundary diffusion operator on Γ. This is in accordance with our setting for δ = 0, but the authors assume stronger conditions and in particular, a drift is not included. Furthermore, we also construct and analyze the underlying dynamics. Additionally, we deduce regularity properties of the associated semigroup and resolvent in use of the regularity results given in [Nit11], [War12] and [War13]. In application, the required additional conditions on the density can be verified in use of a criterion by [Stu95]. Neither in [Car09] nor in [VV03] and [MR06] Feller properties of the associated semigroup are investigated. Moreover, for δ = 0 it is possible to use a random time change in order to obtain solutions to SDEs with immediate reflection from sticky reflection for a Lipschitz boundary Γ. In this generality, the existence result seems also new.
In the second part of the present paper, we use the previous results as well as tensor products and Girsanov transformations of Dirichlet forms in order to construct a solution of the system of SDEs given by reasonable way, since naturally two particles are not allowed to be located at the same position at the same time. For example, a Gibbs measure defined by the Lennard-Jones potential is an admissible choice and models the repulsion of the particles. Thus, our results prove the existence of solutions to systems of singular SDEs with sticky boundary.
In the case of a bounded drift, it is not possible to model repulsive behavior.
In [Gra88] a sticky diffusion is constructed by probabilistic methods with regard to a propagation of chaos result in the follow-up paper [GM89]. The constructed process coincides with our setting in special cases, but the considered domain is determined by the zero set of a C 2 (R d )-function and the drift is assumed to be Lipschitz continuous and bounded which is quite restrictive, especially with regard to singular interactions. The author uses the constructed diffusion to model a system of particles interacting at the boundary. This interacting particle system in turn is used to model the behavior of molecules in a chromatography tube. We apply our results to this kind of application.
In [EP14] the authors analyze Brownian motion on [0, ∞) which is sticky in 0. They show that strong solutions do not exist and that the sticky Brownian motion is the limit of time scaled reflected Brownian motions. This suggests that a strong solution in our framework also does not exist and hence, the solutions we construct in this paper are optimal in this sense.
The main novelties summarize as follows: • Existence of a weak solution to the SDE (1.1) with Lipschitz boundary Γ for δ = 0, C 2 boundary for δ = 1 and possibly unbounded, non-Lipschitz drifts. (Theorem 3.17) • Ergodicity of the solution to (1.1). (Theorem 3.23) • Regularity properties of the associated semigroup and resolvent as well as existence of a solution of (1.1) for an explicitly known set of starting points and singular drifts. (Section 3.3, Theorem 3.38) • Existence of weak solutions to SDEs with immediate reflection at Lipschitz boundaries Γ and with singular drifts for an explicitly known set of starting points.
(Corollary 3.39) • Existence of a weak solution to the N -particle SDE (1.2). (Theorem 4.22) • Application of the concepts to N -particle dynamics in a chromatography tube with singular interactions. (Section 4.3)

Preliminaries
Throughout this paper, Ω ⊂ R d , d ≥ 1, denotes a non-empty bounded Lipschitz domain, λ the Lebesgue measure on Ω and σ the surface measure on Γ := ∂Ω. In the case δ = 1 we assume that d ≥ 2. The standard scalar product on R d is given by (·, ·) and norms on R d by | · | (in particular, for the modulus in R; eventually labeled by a lower index in order to distinguish norms). Similarly, · denotes norms on function spaces. The metric on R d induced by the euclidean metric is denoted by d euc .
For a vector x ∈ Ω N , N ∈ N, we use the representation x = (x 1 , . . . , x N ), where x i ∈ Ω, i = 1, . . . , N , is represented in the form x i = (x i 1 , . . . , x i d ). We denote by ∇ the gradient of a smooth function and by ∂ x i k , i = 1, . . . , N , k = 1, . . . , d, its partial derivatives. In the case N = 1 we simply write ∂ k for k = 1, . . . , d. By ∇ i , i = 1, . . . , N , we denote the d-dimensional vector given by the partial derivatives with respect to the coordinates x i k , k = 1, . . . , d. Moreover, ∇ 2 denotes the Hessian for functions mapping from subsets of EJP 22 (2017), paper 7. R d to R and ∆ = Tr(∇ 2 ) the Laplacian. ∇ 2 i and ∆ i are defined analogously. In the case of Sobolev functions we use the same notations in the weak sense. In order to fix the notation we make the following definitions: Definition 2.1. Let Ω ⊂ R d be a bounded domain. The boundary Γ of Ω is said to be Lipschitz continuous (respectively C k -smooth) if for every x ∈ Γ there exists a neighborhood V of x in R d such that Γ ∩ V is the graph of a Lipschitz continuous (respectively C k -smooth) function and Ω ∩ V is located at one side of the graph, i.e., it exist new orthogonal coordinates (y 1 , . . . , y d ) (given by an orthogonal map T ), a reference point z ∈ R d−1 , real numbers r, h > 0 and a Lipschitz continuous (respectively C k -smooth) function ϕ : R d−1 → R such that in the new coordinates it holds So Γ is Lipschitz continuous (respectively C k -smooth) if Ω is locally below the graph of a Lipschitz continuous (respectively C k -) function and the graph coincides with Γ. In this case, we also simply say that Γ is Lipschitz (respectively C k ) or that Ω has Lipschitz boundary (respectively C k -boundary). Note that in the case of a Lipschitz continuous boundary Γ, each ϕ is almost everywhere differentiable by Rademacher's theorem. Moreover, it is possible to find a finite open cover (V i ) i=1,...,l of Γ such that the assumptions in Definition 2.1 are fulfilled for each V i , i = 1, . . . , l, since Γ is compact.

Definition 2.2.
Let Ω be open and bounded with Lipschitz continuous boundary Γ. Then we define for y = (y 1 , . . . , y d ) ∈ Ṽ n(y) := (−∇ϕ(y ,d ), 1) |∇ϕ(y ,d )| 2 + 1 supposed that ϕ is differentiable at y ,d := (y 1 , . . . , y d−1 ). Let x ∈ Γ and T ∈ R d×d be the orthogonal coordinate transformation from Definition 2.1. Then define the (outward) normal vector at x by n(x) Analogously, we define higher derivatives of order k ∈ N. Let C k (Γ) be the space of ktimes continuously differentiable functions on Γ. As usual, set C ∞ (Γ) := ∩ k∈N C k (Γ). As before, denote by ∇ Γ the gradient on Γ and by div Γ the divergence operator. Moreover, in the case that n is differentiable at x we define the mean curvature of Γ at x by κ(x) := div Γ n(x).
In the above way, it is possible to obtain from functions in C 1 (Ω) and C 2 (Ω) elements in C 1 (Γ) and C 2 (Γ) respectively by restriction.
We have the following relation for the mean curvature κ: Using that P is the orthogonal projection on span(n) ⊥ , we get that (n, P ∇n i ) = 0 and therfore, the assertion holds true.
Definition 2.7. The Sobolev space H 1,k (Γ), k ≥ 1, is defined by C 1 (Γ) · H 1,k (Γ) ⊂ L k (Γ; σ), i.e., the closure C 1 (Γ) with respect to the norm Remark 2.8. For a vector valued C 1 -function Φ on Γ and g ∈ H 1,k (Γ), we have the divergence theorem We shortly recall some facts about Brownian motion on Γ. For details about stochastic analysis on manifolds, we refer to [HT94], [Hsu02] and [IW89]: By definition, Brownian motion (B Γ t ) t≥0 on a smooth boundary Γ is a Γ-valued stochastic process that is generated by 1 2 ∆ Γ , in analogy to Brownian motion on R d , in the sense that (B Γ t ) t≥0 solves the martingale problem for ( 1 2 ∆ Γ , C ∞ (Γ)). We recall the following: Lemma 2.9. Let Γ be a smooth submanifold of R d as in Definition 2.1. Then a solution of the Stratonovich SDE is a Brownian motion on Γ, where (B t ) t≥0 is a Brownian motion in R d . Remark 2.10. Note that the dimension of the driving Brownian motion (B t ) t≥0 is strictly larger than the dimension of the submanifold Γ and hence, according to [Hsu02] the driving Brownian motion contains some extra information beyond what is usually provided by a Brownian motion on Γ. Furthermore, a solution of the above SDE is naturally Γ-valued, since P (x)z is tangential to Γ at x for every x ∈ Γ and z ∈ R d . In our application, it is natural to construct a Brownian motion on Γ by means of a d-dimensional Brownian motion, since a Brownian motion on R d is involved anyway.
By summarizing the preceding results, we get the following theorem: Theorem 3.8. Assume Condition 3.1 and Condition 3.3. Then the symmetric and positive definite bilinear form (E, D) is densely defined and closable on L 2 (Ω; µ). Its closure (E, D(E)) is a recurrent, strongly local, regular, symmetric Dirichlet form on L 2 (Ω; µ).
By the theory of Dirichlet forms, we obtain immediately the following theorem. For details see e.g. [MR92, Chap. V, Theorem 1.11] or [FOT11, Theorem 7.2.2 and Exercise 4.5.1]. We remark that the definitions of capacities (and hence, of exceptional sets) used in the textbooks [FOT11] and [MR92] are introduced in different ways, but that the defintions coincide in our setting (see [MR92, Chap. III, Remark 2.9 and Exercise 2.10]). (T t ) t>0 denotes the sub-Markovian strongly continuous contraction semigroup on L 2 (Ω; µ) corresponding to (E, D(E)).
Proposition 3.14. Suppose that Condition 3.12 is satisfied. Then, C 2 (Ω) ⊂ D(L) and Lf =Lf := 1 2 Proof. Let f ∈ C 2 (Ω) and g ∈ D = C 1 (Ω). Then we get by the divergence theorem on Ω and (2.1): By density of D in D(E) with respect to the E 1 -norm, the claim follows.
We can define the operator L Ω and the boundary operator L Γ by for f ∈ C 2 (Ω). Then the generator L has the representation Lf = 1 Ω L Ω f + 1 Γ L Γ f . The associated Cauchy problem for g ∈ C 2 (Ω) has the form (3. 2) The condition in (3.2) is called Wentzell boundary condition. Note that if we multiply (3.2) for δ = 0 by β and then set β to zero, the equation reduces to the Neumann boundary condition.

Solution to the martingale problem and SDE
Theorem 3.15. The diffusion process M from Theorem 3.9 is up to µ-equivalence the unique diffusion process having µ as symmetrizing measure and solving the martingale problem for (L, D(L)), i.e., for all g ∈ D(L) By the explicit calculation of L given in Proposition 3.14 and the notation in (3.5), we obtain the following corollary: Corollary 3.16. Assume that Condition 3.12 is fulfilled. Let g ∈ C 2 (Ω) and let M be the diffusion process from Theorem 3.9. Then is an F t -martingale under P x for quasi every x ∈ Ω, where A and b are defined as in (3.3) and (3.4).
Due to the connection of martingale problems and SDEs we get for the coefficients given by A and b as defined above the following (see [Kal97,Theorem 18.7]): for quasi every starting point x ∈ Ω, where (B t ) t≥0 is a d-dimensional standard Brownian motion, i.e., almost surely under P x for quasi every x ∈ Ω.
Remark 3.18. A Fukushima decomposition of M (see [FOT11,Chap. 5]) yields the same result as in Theorem 3.17. We would like to mention that the argument used here in order to get a solution to the SDE (1.1) does not work for reflecting (Neumann) boundary conditions, since in this case the reflection is not given by a drift term. However, a Fukushima decomposition is still valid (see e.g. [Tru03]), because in this case it is also possible to assign an additive functional to the surface measure σ. The advantage in our situation is that we are able to express the boundary behavior in terms of the generator.

Ergodicity and occupation time
Throughout this section we assume that Condition 3.1 and Condition 3.3 are fulfilled and denote by M the process constructed in Theorem 3.9. Given the process M, we can define via its transition semigroup (p t ) t≥0 a Dirichlet form and by construction of M this form is (E, D(E)) again. Recall that the sub-Markovian strongly continuous contraction semigroup on L 2 (Ω; µ) of (E, D(E)) is denoted by (T t ) t≥0 . We use the results provided in [FOT11,Chap. 4.7] in order to prove an ergodic theorem for M. To do this, we restrict to invariant subsets of Ω and show the part of the form (E, D(E)) on the invariant set is irreducible recurrent. This allows to determine the occupation time of the process on Γ and, as a consequence, to show that the boundary behavior is indeed sticky. The main result of this section is Theorem 3.23. In order to avoid confusion, we label the capacity of a set by the underlying Dirichlet form. For the sake of convenience, we state all proofs for the case δ = 1, which can easily be modified to hold for δ = 0.
First, we define the notion of parts of Dirichlet forms: Definition 3.19 (part of a Dirichlet form). Let (G, D(G)) be an arbitrary regular Dirichlet form on some locally compact, separable metric space X, m a positive Radon measure on X with full topological support and G an open subset of X. Then we define by Throughout this section, suppose that Condition 3.12 is satisfied and denote by the process constructed in Section 3.1. Furthermore, for an open subset G of Ω is called the part of the process M on G, where X 0 t (ω) results from X t (ω) by killing the path upon leaving G for ω ∈ Ω. By [FOT11, Theorem 4.4.2] the process M G is associated to (E G , D(E G )).
Note that Condition 3.20 implies Condition 3.3.
Lemma 3.21. Assume that Condition 3.20 is fulfillded. Then (iii) The assertion in (ii) holds accordingly for G and G Γ with respect to E Ω and E Γ respectively.
Proof. (i) Note that D(E) is a subset of D(E Ω ) and D(E Γ ) by restriction and E Ω,1 , E Γ,1 ≤ E 1 on this set. Let ε > 0. Then there exists an open set U in Ω which contains Ξ such that cap E (U ) < ε. By definition of the capacity, we get also cap EΩ (U ) < ε and cap EΓ (U ∩Γ) < ε.
Hence, the assertion holds true.
Hence, G is quasi closed.
(iii) Note that G and G Γ are open in Ω and Γ respectively. The remaining part of the statement follows by (i) with the same arguments as in (ii).
(i) Due to [FOT11, Lemma 4. 6.3], the preceding lemma implies that there exists a (ii) It is possible that G Γ = G ∩ Γ is not connected in Γ\Ξ. Therefore, we denote by C G the set containing all connected components of G Γ . In particular, G∈C C G is the set of all connected components of Γ\Ξ.
k } and define γ k with respect to σ.
(iv) By a similar argument as in (iii), L p -norms on K with respect to the measures µ and λ (or σ) respectively are equivalent for some compact set K properly contained in some G (or A G ).
Theorem 3.23. Suppose that Condition 3.20 is fulfilled. Then for all G ∈ C and f ∈ L 1 (G; µ) it holds almost surely under P x for quasi all x ∈ G.
Proof. Fix G ∈ C. Due to [FOT11, Theorem 4.7.3(iii)], the definition of M G and Remark 3.22 (i) it is sufficient to show that (E G , D(E G )) is irreducible recurrent. In order to deduce recurrence of (E, D(E)), by [FOT11, Theorem 1.6.3] it is enough to observe that This implies recurrence of (E G , D(E G )) by [FOT11, Theorem 1.6.3]. Taking into account that the considered form is recurrent, irreducibility is equivalent to the condition that [FOT11,Theorem 4.4.3] and thus, it is irreducible. Indeed, the closure of the pre-Dirichlet form EJP 22 (2017), paper 7. on L 2 (A G ; σ) yields reflecting Brownian motion which is irreducible (see e.g. [CF11,p.128]). Hence, the closure of the form defined for functions in Then the restriction to Γ is by definition E Γ -Cauchy and converges to the restriction of f in L 2 (Γ; βσ). Therefore, the convergence holds also in D(E Γ ). An analogous statement holds in D(E Ω ). Thus, Therefore, E G (f, f ) = 0 implies that each summand on the right hand side vanishes and Choose a C ∞ -cutoff function η defined on Ω which is constantly one near z and has support properly contained in U . Then it is easy to see that ηf ∈ D(E G ) and (ηf k ) k∈N is an approximation for ηf whenever (f k ) k∈N is a sequence of C 1 (G)-functions which approximates f in D(E G ). In particular, this implies convergence in L 2 (U ∩ Γ; σ) and even in L 2 (∂(U ∩ Ω); σ). Since ηc G is the unique continuous extension of f | U ∩Ω to U , it is clear that ηf ∈ H 1,2 (U ∩ Ω) ∩ C(U ∩ Ω) and Tr(ηf ) = ηc G , where Tr : H 1,2 (U ∩Ω) → L 2 (U ∩Γ; σ) is the (restricted) trace operator. Thus, Corollary 3.24. Suppose that Condition 3.20 is fulfilled. Fix a component G of Ω 1 which intersects Γ. Then almost surely under P x for quasi all x ∈ G.
Remark 3.25. Note that the right hand side of (3.7) is strictly positive if µ(G ∩ Γ) > 0 and there exists always some G ∈ C such that µ(G ∩ Γ) > 0, since µ(Γ) > 0. This implies that the process sojourns arbitrarily long on Γ.
For the subsequent proposition and example we need the notion of a strongly regular Dirichlet form (see also [Stu94] and [Stu95]): Definition 3.26 (strong regularity). A regular Dirichlet form (G, D(G)) on L 2 (X; m), where X is a connected, locally compact, separable Hausdorff space and m is a positive Radon measure with full support, is called strongly regular, if the topology induced by the intrinsic metric coincides with the original topology on X. Here ν f ≤ m means that the so-called energy measure of f is absolutely continuous with respect to m and its Radon-Nikodym derivative dν f dm is almost everywhere less or equal than one.
Lemma 3.27. Let ∅ = U be an open subset of Ω\Ξ such that U ⊂ Ω\Ξ. Then the restric- Proof. By continuity of α and β, there exist constants 0 < − and Hence, i 1 : D → H 1,2 (U ∩ Ω) and i 1 : D → H 1,2 (U ∩ Ω) are well-defined and continuous. Therefore, the maps admit a continuous extension to D(E). Let f ∈ D(E). Then the image of f is simply the restriction of f to the respective set (see also Remark 3.22 (iv)) and thus, the restriction is an element of the corresponding Sobolev space. The last Lemma 3.28. Let f ∈ D(E) ∩ C(Ω) and choose a sequence (f k ) k∈N in D whiches converges to f with respect to E 1 . Then and |∇ Γ f k | 2 = |∇f k | 2 − |nn t ∇f k | 2 for each k ∈ N . Moreover, (∇f k ) k∈N has the limit ∇f in L 2 (Ω; αλ) and similarly, (∇ Γ f k ) k∈N has the limit ∇ Γ f ∈ L 2 (Γ; βσ). In particular the convergence holds in L 2 loc (Ω\Ξ; λ) and L 2 loc (Γ\Ξ; σ). The energy measure of f is given by . Then the result follows by uniqueness of ν f . Since also D is dense in C(Ω) with respect to · sup , it is enough to restrict to functions g ∈ D. In this case, EJP 22 (2017), paper 7. Hence, Then each G j fulfills the assumptions of Lemma 3.27 and G j ↑ Ω\Ξ as j → ∞. This yields a weak gradient ∇f and ∇ Γ f on each set G j and G j ∩ Γ respectively. Therfore, we can define ∇f and ∇ Γ f globally outside Ξ and since the last inequality holds for fixed j ∈ N. The statement holds similarly for ∇ Γ f .
Applying this to f − f k finishes the proof.
Proof. We show that the intrinsic metric d is equivalent to the euclidean metric d euc . First and the last expression is locally bounded by d euc . Indeed, by the proof of [Alt06, Satz 8.5] every f ∈ H 1,∞ (Ω) has a unique continuous version in C 0,1 (Ω) and there is some Example 3.30. Assume additionally to Condition 3.1 that α, β ∈ C(Ω) and the following property: Then, as a consequence of strong regularity, cap E (Ξ) = 0 by [Stu95, Theorem 3] and therefore, Theorem 3.23 applies.

L p -strong Feller properties
The diffusion process constructed in Section 3.2.2 has the drawback that the main result given in Theorem 3.17 only holds for quasi every starting point x ∈ Ω and it is not explicitly known how this set of admissible starting points looks like. In the following, we prove regularity properties of the associated L p -resolvent and conclude that the results More precisely, we show the sufficient conditions given in [BGS13, Condition 1.3] in use of a regularity result from [Nit11] for δ = 0 and from [War13] (see also [War12]) for δ = 1. Then, we apply [BGS13,Theorem 1.4]. Moreover, we use the connection of parts of processes and parts of Dirichlet forms in order to identify the Dirichlet form of the new process with state space Ω 1 . This allows to proceed again as in Section 3.2.2, but now without a set of starting points we have to exclude. Note that Ω 1 is not closed in R d if Ξ = ∅. We use this notation in order to be consistent with [BGS13].
Assume that Condition 3.12 is fulfilled. In order to prove the required regularity result we assume additionally the following property: Condition 3.31. There exists p ≥ 2 with p > d 2 and p > d if δ = 0 such that |∇α| α ∈ L p loc (Ω\Ξ; αλ) and additionally In the following, we assume Condition 3.12, Condition 3.31 and again that (i) cap E (Ξ) = 0 (i.e., Condition 3.20), which is e.g. fulfilled under the condition (3.8) given in Example 3.30. We prove that (ii) there exists p > 1 such that D(L p ) → C(Ω 1 ) and the embedding is locally continuous, i.e., for x ∈ Ω 1 there exists a Ω 1 -neighborhood U and a constant C = C(U ) < ∞ such that whereũ denotes the continuous version of u (on Ω 1 ), (iii) for each point x ∈ Ω 1 exists a sequence of functions (u n ) n∈N in D(L p ) such that for every y = x, y ∈ Ω 1 , exists a u n with u n (y) = 0 and u n (x) = 1.
We say that a sequence (u n ) n∈N as in (iii) is point separating in x.
with state space Ω which leaves Ω 1 P x -a.s., x ∈ Ω 1 , invariant. The Dirichlet form associated to M is given by (E, D(E)) and the transition semigroup (p t ) t>0 of M is L pstrong Feller, i.e., p t (L p (Ω; µ)) ⊂ C(Ω 1 ). Moreover, it solves the (L p , D(L p )) martingale problem for every point x ∈ Ω 1 . is fulfilled, the statement holds even for every f ∈ C 2 (Ω).
Proof. The statement for p = 2 has been proven in Proposition 3.14. Then, the general statement follows by the assumptions on α and β similar to [BG14, Lemma 2.3], since f and Lf are elements of L p (Ω; µ) for f ∈ C 2 c (Ω 1 ). Under the additional condition we even have f, Lf ∈ L p (Ω; µ) for f ∈ C 2 (Ω) and the statement extends to the larger class.
In a similar way as in the case of Neumann boundary conditions (see [BG14, Section 4]) we get the following: Theorem 3.33. Assume that Condition 3.12 is fulfilled. Let U be an open subset of Ω in the subspace topology. The following holds: (ii) Assume additionally that U ⊂ Ω 1 . The restriction maps i Ω and i Γ (supposed that δ = 1 and U ∩ Γ = ∅), which restrict functions from Ω to U ∩ Ω and U ∩ Γ respectively, are continuous mappings from D(E) to H 1,2 (U ∩ Ω) and H 1,2 (U ∩ Γ) respectively.

Moreover, it holds
and there exists a constant C 2 = C 2 (α, β, d, G) < ∞ such that for u ∈ D(E) and v ∈ C 1 c (U ).
Proof. (i) is clear. The first part of (ii) and inequality (3.10) hold by Lemma 3.27 (the result for δ = 0 holds similarly).
is continuous on D(E) (or rather on the space obtained by restricting functions to U ) with respect to the norm given by u 2 H 1,2 (U ∩Ω) + δ u 2 H 1,2 (U ∩Γ) , since α and β are bounded from above and from below away from zero on U (by continuity). Thus, it is also continuous with respect to the E 1 2 1 -norm in view of (3.10) and therefore, F has to coincide with E(·, v) by uniqueness, since the equality holds on the dense subset D. Therefore, (3.9) is established.
Next, we prove (iii). Let R and R 0 be as stated. First, we show (3.12) for p = 2. Choose a cutoff function η which is constantly one in B R (x) for some R 0 > R > R and has compact support in B R0 (x). For f ∈ L 2 (Ω; µ) we have u := G 2 γ f ∈ D(E) and it is easy to see that also ηu ∈ D(E), since ηu n converges to ηu as n → ∞ if (u n ) n∈N approximates u in D(E). As in (ii) it can be shown that for fixed v ∈ D(E) holds Note that η 2 is again a cutoff function with the properties we supposed for η. We have by calculation (3.14) We get with the inequality ab ≤ ε 2 b 2 + 1 2ε a 2 for ε > 0, a, b ≥ 0: EJP 22 (2017), paper 7.
for suitable constants K 1 ≤ K 2 ≤ K 3 < ∞. Similarly, we get (by eventually increasing For the two corresponding terms in (3.14) on U 0 ∩ Γ, the similar statement follows by the same arguments. Moreover, we have for a constant K 4 < ∞. Together with (3.10) and (3.14) follows that there exists a constant K 5 < ∞ such that For arbitrary p ≥ 2 note that for W := L 1 (Ω; µ) ∩ L ∞ (Ω; µ) ⊂ L 2 (Ω; µ) ∩ L p (Ω; µ), W is dense in L p (Ω; µ) and G p γ f = G 2 γ f for f ∈ W . For f ∈ W inequality (3.12) applies, since the L 2 -norm on U 0 can be estimated by the L p -norm. Then (3.12) holds also for each f ∈ L p (Ω; µ) by a density argument and continuity of G p γ . (3.13) is a direct consequence of (3.12) and the fact that G p γ is a contraction.
It rests to prove (3.11). For f ∈ W and v ∈ C 1 Fix v ∈ C 1 c (U ) and let f ∈ L p (Ω; µ). Then we can approximate f in L p (Ω; µ) by functions from W due to density. Using (3.13) and continuity of the considered functionals, this proves (3.11).
Corollary 3.34. Assume that Condition 3.12 is fulfilled. Let 2 ≤ p < ∞, γ > 0. Furthermore, let x ∈ Ω and U := B R (x) = {y ∈ Ω| d euc (x, y) < R} be an open ball around x in Ω with radius R > 0 such that U ⊂ Ω 1 . For u := G p γ f holds for all v ∈ K, where K is defined as the closure of C 1 c (U ) with respect to the norm given by Remark 3.35. We want to deduce from (3.15) that G p γ f is continuous on Ω 1 for p as in Condition 3.31. Note that for interior points x ∈ Ω\Ξ = Ω\{α = 0} it is possible to choose R small enough such that B R (x) ∩ Γ = ∅. In this case, (3.15) reduces to for all v ∈ H 1,2 0 (U ) with f ∈ L p (G; λ), i.e., this is the weak formulation of an elliptic PDE on G with Dirichlet boundary conditions. Then it is well-known by the theory of DeGiorgi-Nash-Moser that u is Hölder continuous near x for p > d 2 (see e.g. [GT01], [Sta63] or [HL97]). Thus, x ∈ Γ\Ξ is the case of main interest.
Proof. For x ∈ Ω 1 let U be an open ball in Ω around x such that U ⊂ Ω 1 . This is possible, since Ω 1 is open in Ω by Condition 3.12. u solves (3.15) and therefore, u possesses a Hölder continuous versionũ on U in view of [Nit11, Theorem 3.14] for δ = 0 and [War13, Theorem 3.2] for δ = 1. Moreover, the aforementioned results yield the stated norm estimate, since u ∈ L p (Ω; µ), G p γ is a contraction, α, β are bounded and the L p (U )-norm as well as the L p (U ∩ Γ)-norm can be estimated by the L p (Ω; µ)-norm.
Indeed, [War13, Theorem 3.2] is proven directly by a generalization of de Girogi's method to the Wentzell setting and is formulated for the special case that α is strictly positive and C 1 as well as β is a positive constant. Nevertheless, by the ideas of the proof of [GT01, Theorem 8.24] the proof generalizes to our setting, since the densities α and β are assumed to be continuous and therefore, they are locally on Ω 1 bounded from below away from zero. Moreover, the localization in Section 4.3 of [War13] shows that it is sufficient to consider test functions in K, i.e., functions vanishing on ∂U ∩ Ω, in order to obtain the required regularity result. Hence, the claim follows by (3.15). Note that [War13, Theorem 3.2] applies directly in use of the localization to U if we assume additionally to Condition 3.12 that α ∈ C 1 (Ω) and β is a positive constant.
In [Nit11, Theorem 3.14] the problem is reduced to the interior case mentioned in Remark 3.35 and [GT01, Theorem 8.24] is used. Thus, this approach also includes a localization which implies that it is sufficient to consider test functions in C 1 c (U ) in order to get a local result. Hence, the claim follows again by (3.15).
Lemma 3.37. For each point x ∈ Ω 1 exists a sequence (u n ) n∈N in C ∞ c (Ω 1 ) ⊂ D(L p ), p > 1, that is point separating in x.
Proof. Fix x ∈ Ω 1 and n ∈ N. Then it is clear that we can find a functionũ n in C ∞ c (R d ) such thatũ n (x) = 1 and supp(ũ n ) ⊂ B 1 n (x). Define u n :=ũ n+m | Ω for m large enough.
, that the embedding is locally continuous and the point separating property. It holds D(L p ) = G p γ L p (Ω; µ) and hence, we have D(L p ) → C(Ω 1 ) and moreover, The existence of a point spearating sequence for each point x ∈ Ω 1 follows by Lemma 3.32 and Lemma 3.37. This assures the existence of a process M with state space Ω as stated at the beginning of this section such that Ω 1 is invariant for all starting points in Ω 1 and its transition semigroup is L p -strong Feller. In particular, the process M solves the (L p , D(L p )) martingale problem and L p is given as in Proposition 3.14 for functions in C 2 c (Ω 1 ) (see also Lemma 3.32). Since B b (Ω) ⊂ L p (Ω; µ), it follows that the process is also strong Feller in the sense that the transition semigroup maps B b (Ω) into C(Ω 1 ). By admitting only starting points in Ω 1 and invariance, we obtain a process M as stated.
Then, this process is (L p -)strong Feller. In particular, the absolute continuity condition given in [FOT11, (4.2.9)] is fulfilled. The associated Dirichlet form is given by (E, D(E)) on L 2 (Ω 1 ; µ) by Definition 3.19 and the following remark on parts of processes. Similarly, the generator is also given by (L, D(L)) considered as an operator on L 2 (Ω 1 ; µ). For this reason, M solves the (L, D(L))-martingale problem under P x for every x ∈ Ω 1 . By Proposition 3.14 we even get that C 2 (Ω) ⊂ D(L). Hence, by the same arguments as in Theorem 3.17 we can conclude that M solves the given SDE and furthermore, the ergodicity result holds accordingly for every starting point x ∈ Ω 1 , since the required properties directly transfer from the L 2 (Ω; µ) to the L 2 (Ω 1 ; µ) setting.
Remark 3.40. Note that Γ is only assumed to be Lipschitz continuous in Theorem 3.38 for δ = 0 and Corollary 3.39. This weak assumption is possible, since we do not need to assume an additional boundary condition in order to identify elements in the domain of the L p -generator in the sticky boundary setting. The Wentzell boundary condition is rather contained in the measure µ in terms of the surface measure σ. Therefore, we can specify a point separating sequence in Lemma 3.37. As a consequence, the outward normal direction n(x) is only well-defined for σ-a.e. x ∈ Γ. Nevertheless, it is reasonable that the constructed processes do not hit such non-smooth boundary points for t > 0 (set of Hausdorff dimension d − 2) and starting in a non-smooth boundary point does not require to define a normal direction. A similar result to Corollary 3.39 can e.g. be found in [FT95]. Nevertheless, as far as we know Corollary 3.39 extends previous results, since we provide an additional singular drift resulting from α.
dµ j for f, g ∈ D := C 1 (Λ). Using the definition of the form G yields In particular, Λ i,Ω∪ Λ i,Γ = Λ for every i = 1, . . . , N . Here, the subindex i = 1, . . . , N in ∇ i and ∇ Γ,i refers to the gradient with respect to the i-th component in x = (x 1 , . . . , x N ) ∈ Λ with x i ∈ Ω. In the same way, we use the notation ∆ i , i = 1, . . . , N , for the Laplacian with respect to the i-th component.
Note that the case δ = 0 corresponds to the setting of a system of particles which has a sticky but static boundary behavior. Then, the bilinear form (E, D) can be written in the simpler form dµ j for f, g ∈ D.
Remark 4.2. In the case of immediately reflecting (Neumann) boundary condition the invariant measure for the corresponding diffusion on Ω (for N = 1) is given by λ. For this reason, the invariant measure for an interacting N particle system, N ∈ N, is absolutely continuous with respect to the Lebesgue measure on Ω N . Thus, the cases N = 1 and N > 1 can be unified. In the case of a sticky boundary behavior, this is not possible anymore, since the surface measure σ is involved.
By the fact that µ is a Borel measure on Λ we get again the following result: Although (E, D) is given in (4.3) by a square field operator, it is sometimes useful to rewrite (E, D) as sum of bilinear forms. Define Λ B := {x ∈ Λ| x i ∈ Ω for i ∈ B, x i ∈ Γ for i ∈ I\B} and ν B := i∈B λ i i∈I\B σ i . Then In this terms it holds Moreover, define for x ∈ Γ N −|B| , B = ∅, and ∈ L 1 (Λ; The dependence of x is given in the sense that the variables of given by the index set I\B are fixed by the components of x. Since is an element of L 1 (Λ B ; µ B ), R Ω (B, x) is only defined for i∈I\B σ i -a.e. x ∈ Γ N −|B| . Similarly, for y ∈ Ω |B| let R Γ (B, y) be given by In this case, the variables of given by the index set B are fixed by the components of y and R Γ (B, y) is only defined for i∈B λ i -a.e. y ∈ Ω |B| . Note that in both cases B determines the components which are not at the boundary. = 0 i∈B λ i -a.e. on Ω |B| \R Ω (B, x) for i∈I\B σ i -a.e. x ∈ Γ N −|B| for every ∅ = B ⊂ I and if δ = 1 additionally (H2) = 0 i∈I\B σ i -a.e. on Γ N −|B| \R Γ (B, y) for i∈B λ i -a.e. y ∈ Ω |B| for every B I.  Proof. Let (f k ) k∈N be an E-Cauchy sequence in D such that f k → 0 in L 2 (Λ; µ) as k → ∞.
In particular, (f k ) k∈N is E B -Cauchy and converges to 0 in L 2 (Λ B ; µ B ) for every ∅ = B ⊂ I. Thus, by definition of E B we have that (∂ j f k ) k∈N is Cauchy in L 2 (Λ B ; µ B ) for every j = d(i − 1) + l, where i ∈ B and l ∈ {1, . . . , d} and hence, ∂ j f k → h j ∈ L 2 (Λ B ; µ B ) as k → ∞. In other words, Therefore, it exists a subsequence (∂ j f k l ) l∈N such that ∂ j f k l → h j as l → ∞ in the space L 2 (Ω |B| ; i∈B λ i ) i∈I\B σ i -a.e. and similarly, f k l → 0 as l → ∞ in L 2 (Ω |B| ; i∈B λ i ) EJP 22 (2017), paper 7. i∈I\B σ i -a.e.. This implies that h j = 0 on Ω |B| i∈B λ i -a.e. i∈I\B σ i -a.e. by (H1) of Condition 4.4 (see [MR92, Chapter II, Section 2a)]) and hence, h j = 0 µ B -a.e. on Λ B . In the case δ = 1, we obtain a similar statement for the components of ∇ Γ,i f k , i ∈ I\B, k ∈ N, by considering the term in analogy to (4.4) and integrating first with respect to i∈I\B σ i and afterwards with respect to i∈B λ i . By Fatou's lemma holds We denote the closure of (E, D) on L 2 (Λ; µ) by (E, D(E)).  We summarize the preceding results in the following theorem:  D) is densely defined and closable on L 2 (Λ; µ). Its closure (E, D(E)) is a recurrent, strongly local, regular, symmetric Dirichlet form on L 2 (Λ; µ).
In analogy to Theorem 3.9 we can conclude the following: with state space Λ which is properly associated with (E, D(E)), i.e., for all (µ-versions of) f ∈ B b (Λ) ⊂ L 2 (Λ; µ) and all t > 0 the function is a quasi continuous version of T t f . M is up to µ-equivalence unique. In particular, M is µ-symmetric (µ is stationary), i.e., Remark 4.12. Note that M is canonical, i.e., Ω = C(R + , Λ) and X t (ω) = ω(t), ω ∈ Ω. For each t ≥ 0 we denote by Θ t : Ω → Ω the shift operator defined by Θ t (ω) = ω(· + t) for ω ∈ Ω such that X s • Θ t = X s+t for all s ≥ 0. We take into account to extend the setting to C(R + , R N d ) by neglecting paths leaving Λ.

Densities with product structure
We introduce a special case of the setting given in Section 4.1.1 which will be of particular importance later on.
. . , N are given by where n i is the outward normal for the i-th particle.
Remark 4.19. Note that the drift in normal direction increases if the factor αi βi increases.
Hence, it is justifiable to say that the boundary is less sticky for the i-th particle at a point x ∈ Γ if β i (x) decreases. This property can also be discovered in a similar way by Corollary 3.24, since as a consequence of this ergodicity theorem the particle spends less time on the boundary if Γ β i dσ decreases (compare also to [FGV16,Corollary 5.7]).
Define for i = 1, . . . , N where E denotes the d × d identity matrix and P i is the projection onto the tangent space for the i-th particle. Then, set (4.8) Using this notation, we get for f ∈ C 2 (Λ) the representation (4.9) Note that AA t = A 2 = A.

Solution to the martingale problem and SDE
Theorem 4.20. The diffusion process M from Theorem 4.11 is up to µ-equivalence the unique diffusion process having µ as symmetrizing measure and solving the martingale problem for (L, D(L)), i.e., for all g ∈ D(L) By Proposition 4.18 L is explicitly known on the set C 2 (Λ). Using the representation given in (4.9), we obtain the following corollary: Corollary 4.21. Assume that Condition 4.16 is fulfilled. Let g ∈ C 2 (Λ) and let M be the diffusion process from Theorem 4.11. Then is an F t -martingale under P x for quasi every x ∈ Λ, where A and b are defined in (4.8).
Remark 4.23. As before, this results can also be deduced by a Fukushima decomposition of M the argument used here in order to get a solution to the SDE (4.10) does not work in this way for reflecting (Neumann) boundary conditions.

Remark 4.24.
In the proof of Corollary 3.39 we constructed a diffusion with immediate reflection from a diffusion with sticky reflection by a random time change. For δ = 0 an evident idea would be to construct an interacting particle system with instantaneous reflection and to transform this system of SDEs to a solution of (4.10) by a random time change. However, this seems not possible. The canonical Dirichlet form is given by the closure of where λ N denotes the Lebesgue measure on Λ. For this kind of Dirichlet form we have a well-known regularity theory at hand which enables us to construct solutions to the underlying SDE even for singular drifts for every starting point in a specified set of admissible initial values (see e.g. [FG08], [BG14] and [FT95]). Usually, only starting points in { = 0} and in the corners of Λ (two or more particles at the boundary of Ω) are not admissible, since the boundary is not sufficiently smooth at these points. Nevertheless, such kind of dynamics do not diffuse on the boundary of Λ and hence, a time changed process will also not have this property. Therefore, it is not possible to construct an interacting particle system with sticky reflection via time change in use of the closure of (4.11), since a particle which reaches Γ is expected to sojourn a positive amount of time on Γ and meanwhile, the remaining particles keep on moving undelayed. This implies a diffusion on the boundary of Λ.

Solutions by Girsanov transformations
Assume that Condition 4.16 is fulfilled and set Moreover, cap E i (Ξ i ) = 0.
Define Ω i := Ω\Ξ i . Assume that Condition 4.16 and Condition 4.25 are fulfilled. According to Theorem 3.38 there exists for every i = 1, . . . , N a diffusion process with strong Feller transition semigroup (p i t ) t>0 and transition function (p i t (x, ·)) t>0 , x ∈ Ω i . The processes M i , i = 1, . . . , N , is associated to the form (E i , D(E i )) on L 2 (Ω i ; µ i ), where µ i = α i λ + β i σ. In particular, (p i t ) t>0 is absolutely continuous with respect to µ i , i.e., for every t > 0 and x ∈ Ω i , there exists a non-negative, measurable function p i t (x, y), y ∈ Ω i , such that Let M be given by . , x N ) ∈Λ. Denote by (p t ) t>0 the transition semigroup and by (p t (x, ·)) t>0 , x ∈Λ, the transition function of M. Then, it holds for every dµ i (y i ) for every A ∈ B(Λ).
As a consequence, p t (x, ·), t > 0, x ∈Λ, is absolutely continuous with respect to and the order of thep i t , i = 1, . . . , N , is arbitrary. Consider the symmetric bilinear form on L 2 (Λ; it follows by (4.12) and Lemma 4.26 the following: Proposition 4.27. The Dirichlet form associated to M is given by the closure of (Ẽ, C 1 (Λ)) on L 2 (Λ; N i=1 µ i ).
Additionally to Condition 4.16 and Condition 4.25 we assume the following: Condition 4.28. φ is strictly positive.
Under these conditions on φ it is possible to perform a Girsanov transformation of M. Consider the multiplicative functional (Z t ) t≥0 , Z t = exp(M t − M t 2 ), given by Note that ∇ ln φ(X t ) = ∇φ φ (X t ) and B t , t ≥ 0, are R N d valued and also that ∇ ln φ is bounded due to Condition 4.28.
In view of Remark 3.10 (applied to (E i , D(E i )), i = 1, . . . , N ), it holds × N i=1 Ω i = C(R + ,Λ) and ⊗ N i=1 F i = B(C(R + ,Λ)). Thus, Then, the transition function (p φ t (x, ·)) t>0 of M φ is absolutely continuous with respect to µ for every x ∈Λ. Indeed, by the previous considerations the transition function (p t (x, ·)) t>0 is absolutely continuous with respect to N i=1 µ i . Assume that A ∈ B(Λ) is given such that µ(A) = 0. Since φ is bounded from above and from below away from zero in view of Condition 4.28 and the continuity of φ, it also holds that N i=1 µ i (A) = 0 and hence, p t (x, A) = 0 for every t > 0 and x ∈Λ, i.e., × N i=1 Ω i 1 A (X t ) dP x = 0 for every t > 0 and x ∈Λ.
Therefore, we also have We summarize the results of this section in the following theorem: Theorem 4.29. M φ is a solution to the SDE for every starting point x ∈Λ, where (B t ) t≥0 , B t = (B 1 t , . . . , B N t ), is an N d-dimensional standard Brownian motion. Moreover, the Dirichlet form associated to M φ is given by (E, D(E)) on L 2 (Λ; µ) and its transition function (p φ t (x, ·)) t>0 is absolutely continuous with respect to µ for every x ∈Λ.
Proof. Due to Theorem 3.38 every M i solves the respective d-dimensional SDE for every starting point in Ω i , i = 1, . . . , N . Hence, the process M solves the SDE for N independent particles, i.e., it solves (4.13) for φ given by the indicator function on Λ. As a consequence M φ solves (4.13) by the Girsanov transformation theorem (see [IW89, Chapter IV, Section 4]). Moreover, by the same arguments as in [GV17] the Dirichlet form of the transformed process M φ is given by (E, D(E)).

Application to particle systems with singular interactions
In [Gra88] the author investigates a martingale problem with Wentzell boundary conditions in a very general form. In particular, the relation to SDEs is developed and an existence result is shown. As an application the author constructs a system of interacting particles in a domain with sticky boundary. This particle system gives a model for particles diffusing in a chromatography tube. More precisely, the considered domain is given by Θ := {x ∈ R d | x 1 > 0} and the investigated SDE on Θ reads as follows: where (X t ) t≥0 is a continuous, Θ-valued process, (C t ) t≥0 is a d-dimensional standard Brownian motion, (N t ) t≥0 is a d-dimensional continuous martingale and (K t ) t≥0 is given such that K 0 = 0, K t is increasing, dK t = 1 ∂Θ (X t )dK t , and Here, the main focus is placed on the very general form of the martingale problem and SDE as well as the assumptions on σ and a = σσ T , which is not necessarily strictly elliptic. In former results (see e.g. [IW89, Chapter IV, Section 7]), it is assumed amongst other things that a 11 ≥ c > 0. In [Gra88] it is shown that the martingale problem with the sojourn condition ρ(X t )dK t ≤ 1 ∂Θ (X t )dt has a solution if and only if the above SDE has a weak solution. Sufficient conditions are τ = 0, σ and b are uniformly Lipschitz continuous and bounded, γ = n is the inward normal vector and ρ is bounded, measurable and positive. Nevertheless, the smoothness conditions on b are rather strong. If we assume additionally that a 11 > 0 (e.g. if σ is given by the identity matrix), it holds that ρ(X t )dK t = 1 ∂Θ (X t )dt.
In the case of the identity matrix, the underlying SDE is given by n(X t )dt, where (B t ) t≥0 is a d-dimensional standrad Brownian motion. This setting corresponds to the one considered in Section 3 for δ = 0. The corresponding system of interacting particles is given by n(X i t )dt, i = 1, . . . , N, where X t = (X 1 t , . . . , X N t ). According to [Gra88] an application for this system of SDEs is a model for molecules diffusing in a chromatography tube. The particles are pushed by a flow of gas and are absorbed and released by a liquid state deposited on the boundary of the tube. Hence, it is resonable to suppose a sticky boundary behavior. However, it is physically unreasonable that two molecules are located at the same position in Θ at the same time. In order to avoid this kind of behavior it is necessary to consider a singular drift b i , i = 1, . . . , N , which causes a strong repulsion if two particles get close to each other. The construction of such kind of stochastic dynamics via Dirichlet forms has already been realized for absorbing and reflecting boundary conditions. In analogy to [FG08, Section 5], a continuous pair potential (without hard core) is a continuous function ζ : R d → R ∪ {∞} such that ζ(−x) = ζ(x) ∈ R for all x ∈ R d \{0}. ζ is said to be repulsive if there exists a continuous decreasing function η : (0, ∞) → [0, ∞) with lim t→0 η(t) = ∞ and R > 0 such that ζ(x) ≥ η(|x|) for |x| ≤ R. Note that φ(x) = 0 if there exist i, j ∈ {1, . . . , N } such that x i = x j . Let Γ be C 2 -smooth. We assume that ζ is a repulsive, continuous pair potential such that φ ∈ C 1 (Λ) and moreover, we assume that EJP 22 (2017), paper 7.