Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains
Using a capacity approach and the theory of the measure’s perturbation of the Dirichlet forms, we give the probabilistic representation of the general Robin boundary value problems on an arbitrary domain Ω, involving smooth measures, which give rise 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, some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary.
The classical Robin boundary conditions on a smooth domain of () is giving by where is the outward normal vector field on the boundary and a positive bounded Borel measurable function defined on .
The probabilistic treatment of Robin boundary value problems has been considered by many authors [1–4]. The first two authors considered bounded -domains since the third considered bounded domains with Lipschitz boundary, and the study of  was concerned with -domains but with smooth measures instead of . 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 , which represent Neumann’s boundary conditions, by the measure , where is 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 is a reflecting Brownian motion (RBM) and is the boundary local time, which corresponds to by Revuz correspondence. It is clear that the smoothness of the domain 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 , the RBM is defined to be the Hunt process associated with the form defined on by where is assumed to be bounded with Lipschitz boundary so that the Dirichlet form is regular. If is an arbitrary domain, then the Dirichlet form needs not to be regular, and to not to lose the generality we consider , the closure of in . The domain is so defined to insure the Dirichlet form to be regular.
Now, if we perturb the Neumann boundary conditions by Borel’s positive measure [11–13], we get the Dirichlet form defined on by In the case of ( 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 , 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 associated with the regular Dirichlet form . We apply a decomposition theorem of additive functionals to write in the form , we prove that the additive functional is supported by , 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.
2. 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 measures, and its corresponding additive functionals. This section relies heavily on the book of Fukushima , particularly, chapter 2 and 5, and the paper in . Throughout , the form is a regular Dirichlet form on , where is a locally compact separable metric space, and a positive Radon measure on with .
For our purposes, we take , where is an Euclidean domain of , and the measure on the -algebra is given by for all with the Lebesgue measure; it follows that , and we define a regular Dirichlet form on by where . The domain is so defined to insure the Dirichlet form to be regular, instead of which make the form not regular in general, but if is bounded open set with Lipschiz boundary, then .
We denote for any , for all .
2.1. Relative Capacity
The relative capacity is introduced in a first time in  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 . It seems to be an efficient tool to analyse the phenomena occurring on the boundary of .
The relative capacity which we denote by is defined on a subsets of by the following: for relatively open (i.e., open with respect to the topology of ) we set
And for arbitrary , we set
A set is called a relatively polar if .
The relative capacity (just as a cap) has the properties of a capacity as described in . In particular, is also an outer measure (but not a Borel measure) and a Choquet capacity.
A statement depending on is said to hold relatively quasieverywhere (r.q.e.) on , if there exist a relatively polar set such that the statement is true for every .
Now we may consider functions in as defined on , and we call a function relatively quasicontinuous (r.q.c.) if for every there exists a relatively open set such that and is continuous.
It follows  that for each there exists a relatively quasicontinuous function such that —a.e. This function is unique relatively quasieverywhere. We call the relatively quasicontinuous representative of .
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 in term of relative capacity is also given, namely,
2.2. Smooth Measures
All families of measures on defined in this subsection were originally defined on , and then in our settings on , as a special case. We reproduce the same definitions, and most of their properties on , as we deal with measures concentrated on the boundary of 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 , , and . We put between brackets to recall our context, and we keep in mind that the same things are valid if we put or instead of .
Let be open. A positive Radon measure on is said to be of finite energy integral if for some positive constant . A positive Radon measure on is of finite energy integral if and only if there exists, for each , a unique function such that We call an -potential.
We denote by the family of all positive Radon measures of finite energy integral.
Lemma 2.1. Each measure in charges no set of zero relative capacity.
Let us consider a subset of defined by
Lemma 2.2. For any , there exist an increasing sequence of compact sets of such that
We note that vanishes on for the sets of Lemma 2.2, because of Lemma 2.1.
We now turn to a class of measures larger than . Let us call a (positive) Borel measure on smooth if it satisfies the following conditions:(i)charges no set of zero relative capacity;(ii)there exist an increasing sequence of closed sets of such that Let us note that then satisfies
An increasing sequence of closed sets satisfying condition (2.10) will be called a generalized nest; if further each is compact, we call it a generalized compact nest.
We denote by the family of all smooth measures. The class is quiet large and it contains all positive Radon measures on charging no set of zero relative capacity. There exist also, by [15, Theorem 1.1], a smooth measure on (hence singular with respect to ) “nowhere Radon” in the sense that for all nonempty relatively open subset of (see [15, Example 1.6]).
The following Theorem, says that, any measure in can be approximated by measures in and in as well.
Theorem 2.3. The following conditions are equivalent for a positive Borel measure on .(i).(ii)There exists a generalized nest satisfying (2.11) and for each .(iii)There exists a generalized compact nest satisfying (2.11) and for each .
2.3. 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 . Recall that as the Dirichlet form is regular, then there exists a Hunt process on which is -symmetric and associated with it.
Definition 2.4. A function is said to be an additive functional (AF) if(1) is -measurable,(2)there exist a defining set and an exceptional set with such that , for all , , for all ; for all , ; for . is right continuous and has left limit, and , .
An additive functional is called positive continuous (PCAF) if, in addition, is nonnegative and continuous for each . The set of all PCAF’s on is denoted .
Two additive functionals and are said to be equivalent if for each , .
We say that and are in the Revuz correspondence, if they satisfy, for all -excessive function , and , the relation The family of all equivalence classes of and the family are in one to one correspondence under the Revuz correspondence. In this case, is called the Revuz measure of .
Example 2.5. We suppose to be bounded with Lipschitz boundary. We have 
where is the boundary local time of the reflecting Brownian motion on . It follows that is the Revuz measure of .
In the following we give some facts useful in the proofs of our main results. We set
Proposition 2.6. Let and , the corresponding PCAF. For , , is a relatively quasicontinuous version of .
Proposition 2.7. Let , and , then is relatively quasicontinuous and
In general, the support of an AF is defined by where .
Theorem 2.8. The support of 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 .
Theorem 2.9. For any , the AF can be expressed uniquely as where is a martingale additive functional of finite energy and is a continuous additive functional of zero energy.
A set is called the -spectrum of , if is the complement of the largest open set such that vanishes for any with . The following Theorem means that , for all .
Theorem 2.10. For any , the CAF vanishes on the complement of the spectrum of in the following sense:
3. General Reflecting Brownian Motion
Now we turn our attention to the process associated with the regular Dirichlet form on defined by
Due to the theorem of Fukushima (1975), there is a Hunt process associated with it. In addition, 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 on is associated with the form in the sense that the transition semigroup , is a version of the -semigroup generated by for any nonnegative -function .
is unique up to a set of zero relative capacity.
Definition 3.1. We call the diffusion process on associated with the general reflecting Brownian motion.
The process is so named to recall the standard reflecting Brownian motion in the case of bounded smooth , as the process associated with . Indeed, when is bounded with Lipschitz boundary, we have that and by  the reflecting Brownian motion admits the following Skorohod representation: where is a standard -dimensional Brownian motion, is the boundary local (continuous additive functional) associated with surface measure on , and is the inward unit normal vector field on the boundary.
For a general domain, the form needs not to be regular. Fukushima  constructed the reflecting Brownian motion on a special compactification of , the so-called Kuramuchi compactification. In , it is shown that if is a bounded Lipschitz domain, then the Kuramochi compactification of 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 .
Now, we apply a general decomposition theorem of additive functionals to our process , in the same way as in . According to Theorem 2.9, the continuous additive functional can be decomposed as follows: where is a martingale additive functional of finite energy and is a continuous additive functional of zero energy.
Since has continuous sample paths, is a continuous martingale whose quadratic variation process is
Instead of , we take coordinate function . We have
We claim that is a Brownian motion with respect to the filtration of . To see that, we use Lévys criterion. This follows immediately from (3.2), which became in the case of coordinate function
Now we turn our attention to the additive functional . Two natural questions need to be answered. The first is, where is the support of 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 is supported by .
Proof. Following Theorem 2.10, we have that , where is the -spectrum of , which means the complement of the largest open set such that for all with .
Step 1. If is smooth (Bounded with Lipschitz boundary, e.g.), then we have
Then, for all with . We can then see that the largest is . Consequently and then .
Step . If is arbitrary, then we take an increasing sequence of subset of such that . Define the family of Dirichlet forms to be the parts of the form on each as defined in Section 4.4 of . By Theorem 4.4.5 in the same section, we have that and on . We have that is the largest open set such that for all . By limit, we get the result.
The interest of the question of boundedness of total variation of appears when one needs to study the semimartingale property and the Skorohod equation of the process of type 3.2. Let be the total variation of , that is, where the supremum is taken over all finite partition , and denote the Euclidian distance. If is bounded, then we have the following expression: where is a process such that for -almost all .
According to 5.4. in , we have the following result.
Theorem 3.3. Assume that is bounded and that the following inequality is satisfied: for some constant . Then, is of bounded variation.
A bounded set verifying (2.10) is called strong Caccioppoli set. This notion is introduced in , and is a purely measured theoretic notion. An example of this type of sets is bounded sets with Lipschitz boundary.
Theorem 3.4. If is a Caccioppoli set, then there exist a finite signed smooth measure such that And is associated with the CAF with the Revuz correspondence. Consequently charges no set of zero relative capacity.
To get a Skorohod type representation, we set We define the measure on by and the vector of length 1 at by
Thus, , .
Then where is the PCAF associated with .
Theorem 3.5. If is a Caccioppoli set, then for r.q.e , one has where is an -dimensional Brownian motion, and is a PCAF associated by the Revuz correspondence to the measure .
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.
4. 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 be a positive Radon measure on charging no set of zeo relative capacity. Consider the perturbed Dirichlet form on defined by
We will see in the following theorem that the transition function is associated with , where is a positive additive functional whose Revuz measure is ; note that the support of the AF is the same as the relative quasisupport of its Revuz measure.
Proposition 4.1. is a strongly continuous semigroup on .
Proof. The proof of the above proposition can be found in .
Theorem 4.2. Let be a positive Radon measure on charging no set of zero relative capacity and be its associated PCAF of . Then is the strongly continuous semigroup associated with the Dirichlet form on .
Proof. To prove that is associated with the Dirichlet form on , it suffices to prove the assertion
Since , we need to prove (4.7) only for bounded . We first prove that (4.7) is valid when . According to Proposition 2.7 we have
If , and if is bounded function in , then , and is a relative quasicontinuous version of the -potential by Proposition 2.6. Since and , we have that
then (4.7) follows.
For general positive measure charging no set of zero relative capacity, we can take by virtue of Theorem 2.3 and Lemma 2.2 an increasing sequence of generalized nest of , and . Since charges no set of zero relative capacity, increases to for any .
Let . Then is a PCAF of with Revuz measure . Since we have for : Clearly r.q.e, and hence for r.q.e . For , we get from (4.8) which converges to zero as . Therefore, is -convergent in and the limit function is in . On the other hand, we also get from (4.8) And by Fatou’s lemma: , getting . Finally, observe the estimate holding for . The second term of the right-hand side tends to zero as . The first term also tends to zero because we have from (4.8) , and it suffices to let first and then . By letting 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  for general smooth measures in the context of general . In the case of , and working just with measures on , the proof still the same and works also for any smooth measure concentrated on . Consequently, the theorem is still verified for smooth measures “nowhere Radon,” that is, measures locally infinite on .
Example 4.3. We give some particular examples of . If , then the semigroup generated by Laplacian with, Neumann boundary conditions. If is locally infinite (nowhere Radon) on , then the semigroup generated by the Laplacian with Dirichlet boundary conditions (see [13, Proposition 3.2.1]).(3) Let be a bounded and enough smooth to insure the existence of the surface measure , and , with is a measurable bounded function on , then , where is a boundary local time. Consequently is the semigroup generated by the Laplacian with (classical) Robin boundary conditions given by (1.1).
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 , 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.
Proposition 4.4. is sub-Markovian, that is, for all , and
Proof. It is clear that if , then for all . In addition we have , and then .
Remark 4.5. The analytic proof needs the first and the second BeurlingDeny criterion [11, Proposition 3.10] while our proof is obvious and direct.
Let be the self-adjoint operator on generator of the semigroup , we write Following , we know that is a realization of the Laplacian. Then we call the Laplacian with General Robin boundary conditions.
Theorem 4.6. Let , then the semigroup is sandwiched between the semigroup of Neumann Laplacian, and the semigroup of Dirichlet Laplacian. That is for all , in the sense of positive operators.
Proof. Let . Since we get easily the following: for any . On the other hand, we have , where is the first hitting time of . Since the relative quasisupport of and are in , then in , and vanishes. Consequently, in and . The theorem follows.
Remark 4.7. The fact that the semigroup is sandwiched between the Neumann semigroup and the Dirichlet one as proved in [13, Theorem 3.4.1] is not obvious and needs a result characterizing the domination of positive semigroups due to Ouhabaz, while our proof is simple and direct.
Proposition 4.8. Let such that (i.e., , for all ), then for all , in the sense of positive operators.
Proof. It follows from the remark that if , then , which means that is increasing, and then is decreasing.
There exist a canonical Hunt process possessing the transition function which is directly constructed from by killing the paths with rate , where .
To construct the process associated with , we follow A.2 of , so we need a nonnegative random variable on which is of an exponential distribution with mean , independent of under for every satisfying . Introducing now a Random time defined by
We define the process by where is a one-point compactification.
And, the admissible filtration of the process is defined by
Since , we may and will assume that .
Now, we can write
The Hunt process is called the canonical subprocess of relative to the multiplicative functional . In fact, is a Diffusion process as is local.
In the literature, the Diffusion process is called a partially reflected Brownian motion , in the sense that, the paths of are reflected on the boundary since they will be killed (absorbed) at the random time with rate .
Theorem 4.9. Let such that is monotone and converges setwise to , that is, converges to for any , then converges to in strongly resolvent sense.
Proof. We prove the theorem for increasing, the proof of the decreasing case is similar. Let (resp. ) be the additive functional associated to (resp. ) by the Revuz correspondence. Similarly to the second part of the proof of Theorem 4.2, we have for r.q.e . Consequently . For , we have , and then which converges to zero as . Therefore, is -convergent in and the limit function is in . The result follows.
Corollary 4.10. Let finite and let . We defined for : then in the strong resolvent sense.
Intuitively speaking, when the measure is infinity (locally infinite on the boundary), the semigroup is the Dirichlet semigroup as said in the example 2 in section 4, which means that the boundary became “completely absorbing,” and any other additive functional in the boundary cannot influence this phenomena, which explain why does not appear yet in the decomposition of , which means that the reflecting phenomena disappear, and so any path of is immediately killed when it arrives to the boundary.
When is null on the boundary, then the semigroup is the Neumann one, and then the boundary became completely reflecting, but for a general measure the paths are reflected many times before they are absorbed at a random time.
This work was the result of a project that got the DAAD fellowship for a research stay in Germany. The author would like also to thank his thesis advisor Prof. Dr Omar El-Mennaoui for his guidance.
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