Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains

Using a capacity approach, and the theory of measure's perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain $\Omega$, involving smooth measures, which give arise to a new process obtained by killing the general reflecting Brownian motion at a random time. We obtain some properties of the semigroup directly from its probabilistic representation, and some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary.


Introduction
The classical Robin boundary conditions on a smooth domain Ω of R N N ≥ 0 is giving by ∂u ∂ν βu 0 on ∂Ω, where ν is the outward normal vector field on the boundary ∂Ω and β a positive bounded Borel measurable function defined on ∂Ω.
The probabilistic treatment of Robin boundary value problems has been considered by many authors 1-4 . The first two authors considered bounded C 3 -domains since the third considered bounded domains with Lipschitz boundary, and the study of 4 was concerned with C 3 -domains but with smooth measures instead of β. If one wants to generalize the probabilistic treatment to a general domain, a difficulty arise when we try to get a diffusion process representing Neumann's boundary conditions.
In fact, the Robin boundary conditions 1.1 are nothing but a perturbation of ∂/∂ν, which represent Neumann's boundary conditions, by the measure μ β · σ, where σ is 2 International Journal of Stochastic Analysis the surface measure. Consequently, the associated diffusion process is the reflecting Brownian motion killed by a certain additive functional, and the semigroup generated by the Laplacian with classical Robin boundary conditions is then giving by where X t t≥0 is a reflecting Brownian motion RBM and L t is the boundary local time, which corresponds to σ by Revuz correspondence. It is clear that the smoothness of the domain Ω in classical Robin boundary value problem follows the smoothness of the domains where RBM is constructed see 5-10 and references therein for more details about RBM . In 6 , the RBM is defined to be the Hunt process associated with the form E, F defined on L 2 Ω by where Ω is assumed to be bounded with Lipschitz boundary so that the Dirichlet form E, F is regular. If Ω is an arbitrary domain, then the Dirichlet form needs not to be regular, and to not to lose the generality we consider F H 1 Ω , the closure of The domain H 1 Ω is so defined to insure the Dirichlet form E, F to be regular. Now, if we perturb the Neumann boundary conditions by Borel's positive measure 11-13 , we get the Dirichlet form E μ , F μ defined on L 2 Ω by In the case of μ β ·σ Ω bounded with Lipschitz boundary , 1.4 is the form associated with Laplacian with classical Robin boundary conditions and 1.2 gives the associated semigroup.
In the case of an arbitrary domain Ω, we make use of the theory of the measure's perturbation of the Dirichlet forms, see, for example, 14-23 . More specifically, we adapt the potential theory and associated stochastic analysis to our context, which is the subject of Section 2. In Section 3, we focus on the diffusion process X t t≥0 associated with the regular Dirichlet form E, H 1 Ω . We apply a decomposition theorem of additive functionals to write X t in the form X t x B t N t , we prove that the additive functional N t is supported by ∂Ω, and we investigate when it is of bounded variations.
In Section 3, we get the probabilistic representation of the semigroup associated with 1.4 , and we prove that it is sandwiched between the semigroup generated by the Laplacian with the Dirichlet boundary conditions and that of the Neumann ones. In addition, we prove some convergence theorems, and we give a probabilistic interpretation of the phenomena occurring on the boundary.

Preliminaries and Notations
The aim of this section is to adapt the potential theory and the stochastic analysis for application to our problem. More precisely, it concerns the notion of relative capacity, smooth International Journal of Stochastic Analysis 3 measures, and its corresponding additive functionals. This section relies heavily on the book of Fukushima 17 , particularly, chapter 2 and 5, and the paper in 11 . Throughout 17 , the form E, F is a regular Dirichlet form on L 2 X, m , where X is a locally compact separable metric space, and m a positive Radon measure on X with supp m X. For our purposes, we take X Ω, where Ω is an Euclidean domain of R N , and the measure m on the σ-algebra B X is given by m A λ A ∩ Ω for all A ∈ B X with λ the Lebesgue measure; it follows that L 2 Ω L 2 X, B X , m , and we define a regular Dirichlet form E, F on L 2 Ω by

Relative Capacity
The relative capacity is introduced in a first time in 11 to study the Laplacian with general Robin boundary conditions on arbitrary domains. It is a special case of the capacity associated with a regular Dirichlet form as described in chapter 2 of 17 . It seems to be an efficient tool to analyse the phenomena occurring on the boundary ∂Ω of Ω.
The relative capacity which we denote by Cap Ω is defined on a subsets of Ω by the following: for A ⊂ Ω relatively open i.e., open with respect to the topology of Ω we set Cap Ω A : inf E 1 u, u : u ∈ H 1 Ω : u ≥ 1 a.e. on A .

2.2
And for arbitrary A ⊂ Ω, we set A set N ⊂ Ω is called a relatively polar if Cap Ω N 0. The relative capacity just as a cap has the properties of a capacity as described in 17 . In particular, Cap Ω is also an outer measure but not a Borel measure and a Choquet capacity.
A statement depending on x ∈ A ⊂ Ω is said to hold relatively quasieverywhere r.q.e. on A, if there exist a relatively polar set N ⊂ A such that the statement is true for every x ∈ A \ N.
Now we may consider functions in H 1 Ω as defined on Ω, and we call a function u : Ω → R relatively quasicontinuous r.q.c. if for every > 0 there exists a relatively open set G ⊂ Ω such that Cap Ω G < and u| Ω\G is continuous.

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It follows 13 that for each u ∈ H 1 Ω there exists a relatively quasicontinuous function u : Ω → R such that u x u x m-a.e. This function is unique relatively quasieverywhere. We call u the relatively quasicontinuous representative of u.
For more details, we refer the reader to 11, 13 , where the relative capacity is investigated, as well as its relation to the classical one. A description of the space H 1 0 Ω in term of relative capacity is also given, namely,

Smooth Measures
All families of measures on ∂Ω defined in this subsection were originally defined on X 17 , and then in our settings on X Ω, as a special case. We reproduce the same definitions, and most of their properties on ∂Ω, as we deal with measures concentrated on the boundary of Ω for our approach to the Robin boundary conditions involving measures. There is three families of measures, as we will see in the sequel: the familes S 0 , S 00 , and S. We put ∂Ω between brackets to recall our context, and we keep in mind that the same things are valid if we put Ω or Ω instead of ∂Ω. Let for some positive constant C. A positive Radon measure on ∂Ω is of finite energy integral if and only if there exists, for each α > 0, a unique function U α μ ∈ F such that We call U α μ an α-potential. We denote by S 0 ∂Ω the family of all positive Radon measures of finite energy integral.

Lemma 2.1. Each measure in S 0 ∂Ω charges no set of zero relative capacity.
Let us consider a subset S 00 ∂Ω of S 0 defined by Lemma 2.2. For any μ ∈ S 0 ∂Ω , there exist an increasing sequence F n n≥0 of compact sets of ∂Ω such that 1 F n · μ ∈ S 00 ∂Ω , n 1, 2, . . . ,

2.8
International Journal of Stochastic Analysis 5 We note that μ ∈ S 0 ∂Ω vanishes on ∂Ω \ ∪ n F n for the sets F n of Lemma 2.2, because of Lemma 2.1.
We now turn to a class of measures S ∂Ω larger than S 0 ∂Ω . Let us call a positive Borel measure μ on ∂Ω smooth if it satisfies the following conditions: i μ charges no set of zero relative capacity; ii there exist an increasing sequence F n n≥0 of closed sets of ∂Ω such that Let us note that μ then satisfies μ ∂Ω \ ∪ n F n 0.

2.11
An increasing sequence F n of closed sets satisfying condition 2.10 will be called a generalized nest; if further each F n is compact, we call it a generalized compact nest.
We denote by S ∂Ω the family of all smooth measures. The class S ∂Ω is quiet large and it contains all positive Radon measures on ∂Ω charging no set of zero relative capacity. There exist also, by 15, Theorem 1.1 , a smooth measure μ on ∂Ω hence singular with respect to m "nowhere Radon" in the sense that μ G ∞ for all nonempty relatively open subset G of ∂Ω see 15, Example 1.6 .
The following Theorem, says that, any measure in S ∂Ω can be approximated by measures in S 0 ∂Ω and in S 00 ∂Ω as well. ii There exists a generalized nest F n satisfying 2.11 and 1 F n · μ ∈ S 0 ∂Ω for each n.
iii There exists a generalized compact nest F n satisfying 2.11 and 1 F n · μ ∈ S 00 ∂Ω for each n.

Additive Functionals
Now we turn our attention to the correspondence between smooth measures and additive functionals, known as Revuz correspondence. As the support of an additive functional is the quasisupport of its Revuz measure, we restrict our attention, as for smooth measures, to additive functionals supported by ∂Ω. Recall that as the Dirichlet form E, F is regular, then there exists a Hunt process M Ξ, X t , ξ, P x on Ω which is m-symmetric and associated with it. 6 International Journal of Stochastic Analysis A ω is right continuous and has left limit, and A t s ω A t ω A s θ t ω s, t ≥ 0.
An additive functional is called positive continuous PCAF if, in addition, A t ω is nonnegative and continuous for each ω ∈ Λ. The set of all PCAF's on ∂Ω is denoted A c ∂Ω .
Two additive functionals A 1 and A 2 are said to be equivalent if for each t > 0, We say that A ∈ A c ∂Ω and μ ∈ S ∂Ω are in the Revuz correspondence, if they satisfy, for all γ-excessive function h, and f ∈ B Ω , the relation The family of all equivalence classes of A c ∂Ω and the family S ∂Ω are in one to one correspondence under the Revuz correspondence. In this case, μ ∈ S ∂Ω is called the Revuz measure of A.
Example 2.5. We suppose Ω to be bounded with Lipschitz boundary. We have 2 where L t is the boundary local time of the reflecting Brownian motion on Ω. It follows that 1/2 σ is the Revuz measure of L t . In the following we give some facts useful in the proofs of our main results. We set In general, the support of an AF A is defined by Theorem 2.8. The support of A ∈ A c ∂Ω is the relative quasisupport of its Revuz measure.
In the following, we give a well-known theorem of decomposition of additive functionals of finite energy. We will apply it to get a decomposition of the diffusion process associated with E, F . Theorem 2.9. For any u ∈ F, the AF A u u X t − u X 0 can be expressed uniquely as where M u t is a martingale additive functional of finite energy and N u t is a continuous additive functional of zero energy.
A set σ u is called the 0 -spectrum of u ∈ F, if σ u is the complement of the largest open set G such that E u, v vanishes for any v ∈ F ∩ C 0 X with supp v ⊂ G. The following Theorem means that supp N u ⊂ σ u , for all u ∈ F. Theorem 2.10. For any u ∈ F, the CAF N u vanishes on the complement of the spectrum F σ u of u in the following sense:

General Reflecting Brownian Motion
Now we turn our attention to the process associated with the regular Dirichlet form E, F on Due to the theorem of Fukushima 1975 , there is a Hunt process X t t≥0 associated with it. In addition, E, F is local, thus the Hunt process is in fact a diffusion process i.e., A strong Markov process with continuous sample paths . The diffusion process M X t , P x on Ω is associated with the form E in the sense that the transition semigroup p t f x E x f X t , x ∈ Ω is a version of the L 2 -semigroup P t f generated by E for any nonnegative L 2 -function f.
M is unique up to a set of zero relative capacity.

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Definition 3.1. We call the diffusion process on Ω associated with E, F the general reflecting Brownian motion.
The process X t is so named to recall the standard reflecting Brownian motion in the case of bounded smooth Ω, as the process associated with E, H 1 Ω . Indeed, when Ω is bounded with Lipschitz boundary, we have that H 1 Ω H 1 Ω and by 6 the reflecting Brownian motion X t admits the following Skorohod representation: where W is a standard N-dimensional Brownian motion, L is the boundary local continuous additive functional associated with surface measure σ on ∂Ω, and ν is the inward unit normal vector field on the boundary. For a general domain, the form E, H 1 Ω needs not to be regular. Fukushima 9 constructed the reflecting Brownian motion on a special compactification of Ω, the so-called Kuramuchi compactification. In 6 , it is shown that if Ω is a bounded Lipschitz domain, then the Kuramochi compactification of Ω is the same as the Euclidean compactification. Thus for such domains, the reflecting Brownian motion is a continuous process who does live on the set Ω. Now, we apply a general decomposition theorem of additive functionals to our process M, in the same way as in 6 . According to Theorem 2.9, the continuous additive functional u X t − u X 0 can be decomposed as follows: where M u t is a martingale additive functional of finite energy and N u t is a continuous additive functional of zero energy.
Since X t t≥0 has continuous sample paths, M u t is a continuous martingale whose quadratic variation process is 3.4 Instead of u, we take coordinate function φ i x x i . We have We claim that M t is a Brownian motion with respect to the filtration of X t . To see that, we use Lévys criterion. This follows immediately from 3.2 , which became in the case of coordinate function International Journal of Stochastic Analysis 9 Now we turn our attention to the additive functional N t . Two natural questions need to be answered. The first is, where is the support of N t located and the second concern the boundedness of its total variation.
For the first question we claim the following.

Proposition 3.2. The additive functional N t is supported by ∂Ω.
Proof. Following Theorem 2.10, we have that supp N t ⊂ σ φ , where σ φ is the 0spectrum of φ, which means the complement of the largest open set G such that Step 1. If Ω is smooth Bounded with Lipschitz boundary, e.g. , then we have We can then see that the largest G is Ω. Consequently σ φ Ω \ Ω and then σ φ ∂Ω.
Step 2. If Ω is arbitrary, then we take an increasing sequence of subset of Ω such that ∞ n 0 Ω n Ω. Define the family of Dirichlet forms E Ω n , F Ω n to be the parts of the form E, F on each Ω n as defined in Section 4.4 of 17 . By Theorem 4.4.5 in the same section, we have that F Ω n ⊂ F and E Ω n E on F Ω n × F Ω n . We have that Ω n is the largest open set such that E Ω n φ i , v 0 for all v ∈ F Ω n ∩ C c Ω n . By limit, we get the result.
The interest of the question of boundedness of total variation of N t appears when one needs to study the semimartingale property and the Skorohod equation of the process X t of type 3.2. Let |N| be the total variation of N t , that is, where the supremum is taken over all finite partition 0 t 0 < t 1 < · · · < t n t, and | · | denote the Euclidian distance. If |N| is bounded, then we have the following expression: where ν is a process such that |ν| s 1 for |N|-almost all s. According to § 5.4. in 17 , we have the following result.

Theorem 3.3.
Assume that Ω is bounded and that the following inequality is satisfied: for some constant C. Then, N t is of bounded variation.

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A bounded set verifying 2.10 is called strong Caccioppoli set. This notion is introduced in 8 , and is a purely measured theoretic notion. An example of this type of sets is bounded sets with Lipschitz boundary.

Theorem 3.4.
If Ω is a Caccioppoli set, then there exist a finite signed smooth measure ν such that

3.11
And ν ν 1 − ν 2 is associated with the CAF −N t −A 1 t A 2 t with the Revuz correspondence. Consequently ν charges no set of zero relative capacity.
To get a Skorohod type representation, we set

3.12
We define the measure σ on ∂Ω by and the vector of length 1 at x ∈ ∂Ω by 3.14 Thus, μ i dx 1/2 n i x σ dx , i 1, . . . , N. Then where L is the PCAF associated with 1/2 σ. Remark 3.6. The above theorem can be found in 9, 24 . In particular, Fukushima proves an equivalence between the property of Caccioppoli sets and the Skorohod representation.

Probabilistic Solution to General Robin Boundary Value Problem
This section is concerned with the probabilistic representation to the semigroup generated by the Laplacian with general Robin boundary conditions, which is, actually, obtained by perturbing the Neumann boundary conditions by a measure. We start with the regular Dirichlet form defined by 3.1 , which we call always as the Dirichlet form associated with the Laplacian with Neumann boundary conditions. Let μ be a positive Radon measure on ∂Ω charging no set of zeo relative capacity. Consider the perturbed Dirichlet form E μ , F μ on L 2 Ω defined by

4.1
We will see in the following theorem that the transition function t is a positive additive functional whose Revuz measure is μ; note that the support of the AF is the same as the relative quasisupport of its Revuz measure.  Proof. To prove that P μ t is associated with the Dirichlet form E μ , F μ on L 2 Ω , it suffices to prove the assertion Since ||R A α f|| L 2 Ω ≤ ||R α f|| L 2 Ω ≤ 1/α ||f|| L 2 Ω , we need to prove 4.7 only for bounded f ∈ L 2 Ω . We first prove that 4.7 is valid when μ ∈ S 00 ∂Ω . According to Proposition 2.7 we have If μ ∈ S 00 ∂Ω , and if f is bounded function in L 2 Ω , then ||R α f|| < ∞, and U α A R A α f is a relative quasicontinuous version of the α-potential U α R A α f · μ ∈ F by Proposition 2.6. Since and that then 4.7 follows. For general positive measure μ charging no set of zero relative capacity, we can take by virtue of Theorem 2.3 and Lemma 2.2 an increasing sequence F n of generalized nest of ∂Ω, and μ n 1 F n · μ ∈ S 00 ∂Ω . Since μ charges no set of zero relative capacity, μ n B increases to μ B for any B ∈ B ∂Ω .
Let A n 1 F n · A. Then A n is a PCAF of X t with Revuz measure μ n . Since μ n ∈ S 00 ∂Ω we have for f ∈ L 2 Ω : Clearly |R A n α f| ≤ R α |f| < ∞ r.q.e, and hence lim n → ∞ R A n α f x R A α f x for r.q.e x ∈ Ω. For n < m, we get from 4.8 which converges to zero as n, m → ∞. Therefore, R A n α f n is E 1 -convergent in F and the limit function R A α f is in F. On the other hand, we also get from 4.8 And by Fatou's lemma: ||R A α f|| L 2 Ω ≤ 1/ √ α ||f|| L 2 Ω , getting R A α f ∈ F μ . Finally, observe the estimate International Journal of Stochastic Analysis 13 holding for u ∈ L 2 ∂Ω, μ . The second term of the right-hand side tends to zero as n → ∞.
The first term also tends to zero because we have from 4.8 ||R A n α f −R A m α f|| L 2 ∂Ω,μ n ≤ f, R A n α f − R A m α f , and it suffices to let first m → ∞ and then n → ∞. By letting n → ∞ in 4.7 , we arrive to desired equation 4.3 .
The proof of Theorem 4.2 is similar to 17, Theorem 6.1.1 which was formulated in the first time by Albeverio and Ma 14 for general smooth measures in the context of general X, m . In the case of X Ω, and working just with measures on S 0 ∂Ω , the proof still the same and works also for any smooth measure concentrated on ∂Ω. Consequently, the theorem is still verified for smooth measures "nowhere Radon," that is, measures locally infinite on ∂Ω.  3 Let Ω be a bounded and enough smooth to insure the existence of the surface measure σ, and μ β · σ, with β is a measurable bounded function on ∂Ω, then A μ t t 0 β X s dL s , where L t is a boundary local time. Consequently The setting of the problem from the stochastic point of view and the stochastic representation of the solution of the problem studied are important on themselves and are new. In fact before there was always additional hypothesis on the domain or on the class of measures. Even if our approach is inspired by the works 14, 15 and Chapter 6 of 17 , the link is not obvious and give as rise to a new approach to the Robin boundary conditions. As a consequence, the proof of many propositions and properties become obvious and direct. The advantage of the stochastic approach is, then, to give explicitly the representation of the semigroup and an easy access of it.