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

Abstract

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.

1. Introduction

The classical Robin boundary conditions on a smooth domain Ω of ℝ^𝑁 (𝑁≥0) is giving by 𝜕𝑢𝜕𝜈+𝛽𝑢=0on𝜕Ω, (1.1) 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 𝐶³-domains since the third considered bounded domains with Lipschitz boundary, and the study of [4] was concerned with 𝐶³-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 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 𝒫^𝜇_𝑡𝑓(𝑥)=𝐸_𝑥𝑓𝑋_𝑡𝑒^{−∫𝑡0𝛽(𝑋𝑠)𝑑𝐿𝑠}, (1.2) 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 𝐿²(Ω) by ℰ(𝑢,𝑣)=_Ω∇𝑢∇𝑣𝑑𝑥,∀𝑢,𝑣∈ℱ=𝐻¹(Ω), (1.3) 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=𝐻¹(Ω), the closure of 𝐻¹(Ω)∩𝐶_𝑐(Ω) in 𝐻¹(Ω). The domain 𝐻¹(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 𝐿²(Ω) by ℰ^𝜇(𝑢,𝑣)=_Ω∇𝑢∇𝑣𝑑𝑥+_𝜕Ω𝑢𝑣𝑑𝜇,∀𝑢,𝑣∈ℱ^𝜇=𝐻¹(Ω)∩𝐿²𝜕Ω,𝑑𝜇. (1.4) 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,𝐻¹(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.

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 [17], particularly, chapter 2 and 5, and the paper in [11]. Throughout [17], the form (ℰ,ℱ) is a regular Dirichlet form on 𝐿²(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 𝐿²(Ω)=𝐿²(𝑋,ℬ(𝑋),𝑚), and we define a regular Dirichlet form (ℰ,ℱ) on 𝐿²(Ω) by ℰ(𝑢,𝑣)=_Ω∇𝑢∇𝑣𝑑𝑥,ℱ=𝐻¹(Ω), (2.1) where 𝐻¹(Ω)=𝐻¹(Ω)∩𝐶_𝑐(Ω)^𝐻1(Ω). The domain 𝐻¹(Ω) is so defined to insure the Dirichlet form (ℰ,ℱ) to be regular, instead of ℱ=𝐻¹(Ω) which make the form not regular in general, but if Ω is bounded open set with Lipschiz boundary, then 𝐻¹(Ω)=𝐻¹(Ω).

We denote for any 𝛼>0∶ℰ_𝛼(𝑢,𝑣)=ℰ(𝑢,𝑣)+𝛼(𝑢,𝑣)_𝑚, for all 𝑢,𝑣∈ℱ.

2.1. 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)∶=infℰ₁(𝑢,𝑢)∶𝑢∈𝐻¹(Ω)∶𝑢≥1a.e.on𝐴. (2.2)

And for arbitrary 𝐴⊂Ω, we set Cap_Ω(𝐴)∶=infCap_Ω(𝐵)∶𝐵relativelyopen𝐴⊂𝐵⊂Ω. (2.3)

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 𝐻¹(Ω) as defined on Ω, and we call a function 𝑢∶Ω→ℝ relatively quasicontinuous (r.q.c.) if for every 𝜖>0 there exists a relatively open set 𝐺⊂Ω such that Cap_Ω(𝐺)<𝜖 and 𝑢|_Ω⧵𝐺 is continuous.

It follows [13] that for each 𝑢∈𝐻¹(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 𝐻¹₀(Ω) in term of relative capacity is also given, namely, 𝐻¹₀(Ω)=𝑢∈𝐻¹(Ω)∶𝑢(𝑥)=0r.q.e.on𝜕Ω. (2.4)

2.2. 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 𝑆₀, 𝑆₀₀, 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 Ω⊂ℝ^𝑁 be open. A positive Radon measure 𝜇 on 𝜕Ω is said to be of finite energy integral if _𝜕Ω|𝑣(𝑥)|𝜇(𝑑𝑥)≤𝐶ℰ₁(𝑣,𝑣),𝑣∈ℱ∩𝐶_𝑐Ω (2.5) 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_𝛼𝜇∈ℱ such that ℰ_𝛼𝑈_𝛼𝜇,𝑣=_𝜕Ω𝑣(𝑥)𝜇(𝑑𝑥). (2.6) We call 𝑈_𝛼𝜇 an 𝛼-potential.

We denote by 𝑆₀(𝜕Ω) the family of all positive Radon measures of finite energy integral.

Lemma 2.1. Each measure in 𝑆₀(𝜕Ω) charges no set of zero relative capacity.

Let us consider a subset 𝑆₀₀(𝜕Ω) of 𝑆₀ defined by 𝑆₀₀(𝜕Ω)=𝜇∈𝑆₀(𝜕Ω)∶𝜇(𝜕Ω)<∞,𝑈₁𝜇_∞<∞. (2.7)

Lemma 2.2. For any 𝜇∈𝑆₀(𝜕Ω), there exist an increasing sequence (𝐹_𝑛)_𝑛≥0 of compact sets of 𝜕Ω such that 1_𝐹𝑛⋅𝜇∈𝑆₀₀(𝜕Ω),𝑛=1,2,…,𝐶𝑎𝑝_Ω𝐾⧵𝐹_𝑛→0,𝑛→+∞𝑓𝑜𝑟𝑎𝑛𝑦𝑐𝑜𝑚𝑝𝑎𝑐𝑡𝑠𝑒𝑡𝐾⊂𝜕Ω. (2.8)

We note that 𝜇∈𝑆₀(𝜕Ω) vanishes on 𝜕Ω⧵∪_𝑛𝐹_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 𝑆₀(𝜕Ω). Let us call a (positive) Borel measure 𝜇 on 𝜕Ω smooth if it satisfies the following conditions:(i)𝜇charges no set of zero relative capacity;(ii)there exist an increasing sequence (𝐹_𝑛)_𝑛≥0 of closed sets of 𝜕Ω such that 𝜇𝐹_𝑛<∞,𝑛=1,2,…,lim_𝑛→+∞Cap_Ω𝐾⧵𝐹_𝑛=0foranycompact𝐾⊂𝜕Ω. (2.9)(2.10)Let us note that 𝜇 then satisfies 𝜇𝜕Ω⧵∪_𝑛𝐹_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 𝑆₀(𝜕Ω) and in 𝑆₀₀(𝜕Ω) as well.

Theorem 2.3. The following conditions are equivalent for a positive Borel measure 𝜇 on 𝜕Ω.(i)𝜇∈𝑆(𝜕Ω).(ii)There exists a generalized nest (𝐹_𝑛) satisfying (2.11) and 1_𝐹𝑛⋅𝜇∈𝑆₀(𝜕Ω) for each 𝑛.(iii)There exists a generalized compact nest (𝐹_𝑛) satisfying (2.11) and 1_𝐹𝑛⋅𝜇∈𝑆₀₀(𝜕Ω) for each 𝑛.

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 𝜕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.

Definition 2.4. A function 𝐴∶[0,+∞[×Ξ→[−∞,+∞] is said to be an additive functional (AF) if(1)𝐴_𝑡 is ℱ_𝑡-measurable,(2)there exist a defining set Λ∈ℱ_∞ and an exceptional set 𝑁⊂𝜕Ω with cap_Ω(𝑁)=0 such that 𝑃_𝑥(Λ)=1, for all 𝑥∈𝜕Ω⧵𝑁, 𝜃_𝑡Λ⊂Λ, for all 𝑡>0; for all 𝜔∈Λ, 𝐴₀(𝜔)=0; |𝐴_𝑡(𝜔)|<∞ for 𝑡<𝜉. 𝐴(𝜔) is right continuous and has left limit, and 𝐴_𝑡+𝑠(𝜔)=𝐴_𝑡(𝜔)+𝐴_𝑠(𝜃_𝑡𝜔)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 𝐴¹ and 𝐴² are said to be equivalent if for each 𝑡>0, 𝑃_𝑥(𝐴¹_𝑡=𝐴²_𝑡)=1r.q.e𝑥∈Ω.

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 lim_𝑡↘01𝑡𝐸_ℎ⋅𝑚^𝑡₀𝑓𝑋_𝑠𝑑𝐴_𝑠=_𝜕Ωℎ(𝑥)𝑓⋅𝜇(𝑑𝑥). (2.12) 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] lim_𝑡↘01𝑡𝐸_ℎ⋅𝑚^𝑡₀𝑓𝑋_𝑠𝑑𝐿_𝑠=12_𝜕Ωℎ(𝑥)𝑓(𝑥)𝜎(𝑑𝑥), (2.13) 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 𝑈^𝛼_𝐴𝑓(𝑥)=𝐸_𝑥^∞₀𝑒^−𝛼𝑡𝑓𝑋_𝑡𝑑𝐴_𝑡,𝑅^𝐴_𝛼𝑓(𝑥)=𝐸_𝑥^∞₀𝑒^−𝛼𝑡𝑒^−𝐴𝑡𝑓𝑋_𝑡,𝑅𝑑𝑡_𝛼𝑓(𝑥)=𝐸_𝑥^∞₀𝑒^−𝛼𝑡𝑓𝑋_𝑡𝑑𝑡. (2.14)

Proposition 2.6. Let 𝜇∈𝑆₀(𝜕Ω) and 𝐴∈𝒜⁺_𝑐(𝜕Ω), the corresponding PCAF. For 𝛼>0, 𝑓∈ℬ⁺_𝑏, 𝑈^𝛼_𝐴 is a relatively quasicontinuous version of 𝑈_𝛼(𝑓⋅𝜇).

Proposition 2.7. Let 𝐴∈𝒜⁺_𝑐(𝜕Ω), and 𝑓∈ℬ⁺_𝑏, then 𝑅^𝐴_𝛼 is relatively quasicontinuous and 𝑅^𝐴_𝛼𝑓−𝑅_𝛼𝑓+𝑈^𝛼_𝐴𝑅^𝐴_𝛼𝑓=0. (2.15)

In general, the support of an AF 𝐴 is defined by supp[𝐴]=𝑥∈𝑋⧵𝑁∶𝑃_𝑥(𝑅=0)=1, (2.16) where 𝑅(𝜔)=inf{𝑡>0∶𝐴_𝑡(𝜔)≠0}.

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(𝑋₀) can be expressed uniquely as 𝑢𝑋_𝑡−𝑢𝑋₀=𝑀^[𝑢]+𝑁^[𝑢], (2.17) where 𝑀_𝑡^[𝑢] is a martingale additive functional of finite energy and 𝑁_𝑡^[𝑢] 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∩𝒞₀(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: 𝑃_𝑥𝑁_𝑡^[𝑢]=0∶∀𝑡<𝜎_𝐹=1𝑟.𝑞.𝑒𝑥∈𝑋. (2.18)

3. General Reflecting Brownian Motion

Now we turn our attention to the process associated with the regular Dirichlet form (ℰ,ℱ) on 𝐿²(Ω) defined by ℰ(𝑢,𝑣)=_Ω∇𝑢∇𝑣𝑑𝑥,ℱ=𝐻¹(Ω). (3.1)

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_𝑡𝑓(𝑥)=𝐸_𝑥[𝑓(𝑋_𝑡)], 𝑥∈Ω is a version of the 𝐿²-semigroup 𝒫_𝑡𝑓 generated by ℰ for any nonnegative 𝐿²-function 𝑓.

𝑀 is unique up to a set of zero relative capacity.

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,𝐻¹(Ω)). Indeed, when Ω is bounded with Lipschitz boundary, we have that 𝐻¹(Ω)=𝐻¹(Ω) and by [6] the reflecting Brownian motion 𝑋_𝑡 admits the following Skorohod representation: 𝑋_𝑡=𝑥+𝑊_𝑡+12^𝑡₀𝜈𝑋_𝑠𝑑𝐿_𝑠, (3.2) 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,𝐻¹(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(𝑋₀) can be decomposed as follows: 𝑢𝑋_𝑡−𝑢𝑋₀=𝑀_𝑡^[𝑢]+𝑁_𝑡^[𝑢], (3.3) where 𝑀_𝑡^[𝑢] is a martingale additive functional of finite energy and 𝑁_𝑡^[𝑢] is a continuous additive functional of zero energy.

Since (𝑋_𝑡)_𝑡≥0 has continuous sample paths, 𝑀_𝑡^[𝑢] is a continuous martingale whose quadratic variation process is 𝑀^[𝑢],𝑀^[𝑢]_𝑡=^𝑡₀|∇𝑢|²𝑋_𝑠𝑑𝑠. (3.4)

Instead of 𝑢, we take coordinate function 𝜙_𝑖(𝑥)=𝑥_𝑖. We have 𝑋_𝑡=𝑋₀+𝑀_𝑡+𝑁_𝑡. (3.5)

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 𝑀^[𝜙𝑖],𝑀^[𝜙𝑖]=𝛿_𝑖𝑗𝑡. (3.6)

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 (0)-spectrum of 𝜙, which means the complement of the largest open set 𝐺 such that ℰ(𝜙_𝑖,𝑣)=0 for all 𝑣∈ℱ∩𝐶_𝑐(Ω) with supp[𝑣]⊂𝐺.
Step 1. If Ω is smooth (Bounded with Lipschitz boundary, e.g.), then we have ℰ𝜙_𝑖,𝑣=−_𝜕Ω𝑣⋅𝑛_𝑖𝑑𝜎. (3.7)
Then, ℰ(𝜙_𝑖,𝑣)=0 for all 𝑣∈ℱ∩𝐶_𝑐(Ω) with supp[𝑣]⊂Ω. 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=0Ω_𝑛=Ω. Define the family of Dirichlet forms (ℰ_Ω𝑛,ℱ_Ω𝑛) 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_Ω𝑛⊂ℱ and ℰ_Ω𝑛=ℰ on ℱ_Ω𝑛×ℱ_Ω𝑛. We have that Ω_𝑛 is the largest open set such that ℰ_Ω𝑛(𝜙_𝑖,𝑣)=0 for all 𝑣∈ℱ_Ω𝑛∩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, |𝑁|_𝑡=sup^𝑛−1_𝑖=1𝑁_𝑡𝑖−𝑁_𝑡𝑖−1, (3.8) where the supremum is taken over all finite partition 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛=𝑡, and |⋅| denote the Euclidian distance. If |𝑁| is bounded, then we have the following expression: 𝑁_𝑡=^𝑡₀𝜈_𝑠𝑑|𝑁|_𝑠, (3.9) 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: _Ω𝜕𝑣𝜕𝑥_𝑖𝑑𝑥≤𝐶‖𝑣‖_∞,∀𝑣∈𝐻¹(Ω)∩𝐶_𝑏Ω, (3.10) for some constant 𝐶. Then, 𝑁_𝑡 is of bounded variation.

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 _Ω𝜕𝑣𝜕𝑥_𝑖𝑑𝑥=−_𝜕Ω𝑣𝑑𝜇,∀𝑣∈𝐻¹(Ω)∩𝐶_𝑏Ω. (3.11) And 𝜈=𝜈¹−𝜈² is associated with the CAF −𝑁_𝑡=−𝐴¹_𝑡+𝐴²_𝑡 with the Revuz correspondence. Consequently 𝜈 charges no set of zero relative capacity.

To get a Skorohod type representation, we set 𝜈=^𝑁_𝑖=1𝜇_𝑖,𝜙_𝑖=𝑑𝜇_𝑖𝑑𝜈𝑖=1,…,𝑁. (3.12) We define the measure 𝜎 on 𝜕Ω by 𝜎(𝑑𝑥)=2^𝑁_𝑖=1𝜙_𝑖(𝑥)^21/2𝜈(𝑑𝑥) (3.13) and the vector of length 1 at 𝑥∈𝜕Ω by 𝑛_𝑖(𝑥)=𝜙_𝑖(𝑥)∑^𝑁_𝑖=1𝜙_𝑖(𝑥)^21/2if∑^𝑁_𝑖=1𝜙_𝑖(𝑥)²>0;0if∑^𝑁_𝑖=1𝜙_𝑖(𝑥)²=0. (3.14)

Thus, 𝜇_𝑖(𝑑𝑥)=(1/2)𝑛_𝑖(𝑥)𝜎(𝑑𝑥),  𝑖=1,…,𝑁.

Then 𝑁_𝑡=^𝑡₀𝑛𝑋_𝑠𝑑𝐿_𝑠, (3.15) where 𝐿 is the PCAF associated with (1/2)𝜎.

Theorem 3.5. If Ω is a Caccioppoli set, then for r.q.e 𝑥∈Ω, one has 𝑋_𝑡=𝑥+𝐵_𝑡+^𝑡₀𝑛𝑋_𝑠𝑑𝐿_𝑠, (3.16) where 𝐵 is an 𝑁-dimensional Brownian motion, and 𝐿 is a PCAF associated by the Revuz correspondence to the measure (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.

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 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 𝐿²(Ω) defined by ℱ_𝜇=ℱ∩𝐿²,ℰ𝜕Ω,𝜇_𝜇(𝑢,𝑣)=ℰ(𝑢,𝑣)+_𝜕Ω𝑢𝑣𝑑𝜇𝑢,𝑣∈ℱ_𝜇. (4.1)

We will see in the following theorem that the transition function 𝒫^𝜇_𝑡𝑓(𝑥)=𝐸_𝑥𝑓𝑋_𝑡𝑒^{−𝐴𝜇𝑡} (4.2) is associated with (ℰ_𝜇,ℱ_𝜇), where 𝐴^𝜇_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.

Proposition 4.1. 𝒫^𝜇_𝑡 is a strongly continuous semigroup on 𝐿²(Ω).

Proof. The proof of the above proposition can be found in [14].

Theorem 4.2. Let 𝜇 be a positive Radon measure on 𝜕Ω charging no set of zero relative capacity and (𝐴^𝜇_𝑡)_𝑡≥0 be its associated PCAF of (𝑋_𝑡)_𝑡≥0. Then 𝒫^𝜇_𝑡 is the strongly continuous semigroup associated with the Dirichlet form (ℰ_𝜇,ℱ_𝜇) on 𝐿²(Ω).

Proof. To prove that 𝒫^𝜇_𝑡 is associated with the Dirichlet form (ℰ_𝜇,ℱ_𝜇) on 𝐿²(Ω), it suffices to prove the assertion 𝑅^𝐴_𝛼𝑓∈ℱ^𝜇,ℰ^𝜇_𝛼𝑅^𝐴_𝛼,𝑢=𝑓,𝑢,𝑓∈𝐿²(Ω,𝑚),𝑢∈ℱ^𝜇. (4.3) Since ||𝑅^𝐴_𝛼𝑓||_𝐿²_(Ω)≤||𝑅_𝛼𝑓||_𝐿²_(Ω)≤(1/𝛼)||𝑓||_𝐿²_(Ω), we need to prove (4.7) only for bounded 𝑓∈𝐿²(Ω). We first prove that (4.7) is valid when 𝜇∈𝑆₀₀(𝜕Ω). According to Proposition 2.7 we have 𝑅^𝐴_𝛼𝑓−𝑅_𝛼𝑓+𝑈^𝛼_𝐴𝑅^𝐴_𝛼𝑓=0,𝛼>0,𝑓∈ℬ⁺Ω. (4.4) If 𝜇∈𝑆₀₀(𝜕Ω), and if 𝑓 is bounded function in 𝐿²(Ω), then ||𝑅_𝛼𝑓||<∞, and 𝑈^𝛼_𝐴𝑅^𝐴_𝛼𝑓 is a relative quasicontinuous version of the 𝛼-potential 𝑈_𝛼(𝑅^𝐴_𝛼𝑓⋅𝜇)∈ℱ by Proposition 2.6. Since ||𝑈_𝛼(𝑅^𝐴_𝛼𝑓⋅𝜇)||_∞≤||𝑅^𝐴_𝛼𝑓||_∞||𝑈_𝛼𝜇||_∞<∞ and 𝜇(𝜕Ω)<∞, we have that 𝑅^𝐴_𝛼𝑓=𝑅_𝛼𝑓−𝑈^𝛼_𝐴𝑅^𝐴_𝛼𝑓∈ℱ^𝜇 (4.5) and that ℰ_𝛼𝑅^𝐴_𝛼𝑓,𝑢=ℰ_𝛼𝑅_𝛼𝑓,𝑢−ℰ_𝛼𝑈^𝛼_𝐴𝑅^𝐴_𝛼=−𝑅𝑓,𝑢𝑓,𝑢^𝐴_𝛼𝑓,𝑢_𝜇,𝑢∈ℱ^𝜇, (4.6)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_𝐹𝑛⋅𝜇∈𝑆₀₀(𝜕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_𝐹𝑛⋅𝐴. Then 𝐴_𝑛 is a PCAF of 𝑋_𝑡 with Revuz measure 𝜇_𝑛. Since 𝜇_𝑛∈𝑆₀₀(𝜕Ω) we have for 𝑓∈𝐿²(Ω): 𝑅^𝐴𝑛_𝛼𝑓∈ℱ^𝜇𝑛,ℰ^𝜇𝑛_𝛼𝑅^𝐴𝑛_𝛼,𝑢=𝑓,𝑢,𝑓∈𝐿²(Ω,𝑚),𝑢∈ℱ^𝜇𝑛. (4.7) Clearly |𝑅^𝐴𝑛_𝛼𝑓|≤𝑅_𝛼|𝑓|<∞ r.q.e, and hence lim_𝑛→+∞𝑅^𝐴𝑛_𝛼𝑓(𝑥)=𝑅^𝐴_𝛼𝑓(𝑥) for r.q.e 𝑥∈Ω. For 𝑛<𝑚, we get from (4.8) ℰ^𝜇𝑛_𝛼𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓,𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓≤𝑓,𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓, (4.8) which converges to zero as 𝑛,𝑚→+∞. Therefore, (𝑅^𝐴𝑛_𝛼𝑓)_𝑛 is ℰ₁-convergent in ℱ and the limit function 𝑅^𝐴_𝛼𝑓 is in ℱ. On the other hand, we also get from (4.8) 𝑅^𝐴𝑛_𝛼𝑓_{𝐿2(𝜕Ω,𝜇)}≤𝑓,𝑅^𝐴𝑛_𝛼𝑓_𝐿2(Ω)≤1𝛼𝑓_𝐿2(Ω). (4.9) And by Fatou’s lemma: ||𝑅^𝐴_𝛼𝑓||_𝐿²_(Ω)≤(1/√𝛼)||𝑓||_𝐿²_(Ω), getting 𝑅^𝐴_𝛼𝑓∈ℱ^𝜇. Finally, observe the estimate 𝑅^𝐴𝑛_𝛼𝑓,𝑢_𝜇𝑛−𝑅^𝐴_𝛼𝑓,𝑢_𝜇≤𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴_𝛼𝑓_{𝐿2(𝜕Ω,𝜇𝑛)}‖𝑢‖_𝐿²_{(𝜕Ω,𝜇)}+𝑅_𝛼𝑓,𝑢_{𝜇−𝜇𝑛} (4.10) holding for 𝑢∈𝐿²(𝜕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^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓||_𝐿²_{(𝜕Ω,𝜇𝑛)}≤(𝑓,𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓), 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 𝑆₀(𝜕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.

Example 4.3. We give some particular examples of 𝒫^𝜇_𝑡.(1) If 𝜇=0, then 𝒫⁰_𝑡𝑓(𝑥)=𝐸_𝑥𝑓𝑋_𝑡, (4.11) the semigroup generated by Laplacian with, Neumann boundary conditions.(2) If 𝜇 is locally infinite (nowhere Radon) on 𝜕Ω, then 𝒫^∞_𝑡𝑓(𝑥)=𝐸_𝑥𝑓𝐵_𝑡1_{𝑡<𝜏}, (4.12) the semigroup generated by the Laplacian with Dirichlet boundary conditions (see [13, Proposition 3.2.1]).(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=∫^𝑡₀𝛽(𝑋_𝑠)𝑑𝐿_𝑠, where 𝐿_𝑡 is a boundary local time. Consequently 𝒫^𝜇_𝑡𝑓(𝑥)=𝐸_𝑥𝑓𝑋_𝑡exp−^𝑡₀𝛽𝑋_𝑠𝑑𝐿_𝑠 (4.13) 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 [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.

Proposition 4.4. 𝒫^𝜇_𝑡 is sub-Markovian, that is, 𝒫^𝜇_𝑡≥0 for all 𝑡≥0, and 𝒫^𝜇_𝑡𝑓_∞≤𝑓_∞(𝑡≥0). (4.14)

Proof. It is clear that if 𝑓∈𝐿²(Ω)₊, then 𝒫^𝜇_𝑡𝑓≥0 for all 𝑡≥0. In addition we have |𝒫^𝜇_𝑡𝑓(𝑥)|≤𝐸_𝑥[|𝑓|(𝑋_𝑡)], and then ||𝒫^𝜇_𝑡𝑓||_∞≤||𝑓||_∞(𝑡≥0).

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 Δ_𝜇 be the self-adjoint operator on 𝐿²(Ω) generator of the semigroup 𝒫^𝜇_𝑡, we write 𝒫^𝜇_𝑡𝑓(𝑥)=𝑒^{−𝑡Δ𝜇}𝑓(𝑥). (4.15) Following [13], we know that Δ_𝜇 is a realization of the Laplacian. Then we call Δ_𝜇 the Laplacian with General Robin boundary conditions.

Theorem 4.6. Let 𝜇∈𝑆(𝜕Ω), then the semigroup 𝒫^𝜇_𝑡 is sandwiched between the semigroup of Neumann Laplacian, and the semigroup of Dirichlet Laplacian. That is 0≤𝑒^{−𝑡Δ𝐷}≤𝒫^𝜇_𝑡≤𝑒^{−𝑡Δ𝑁} (4.16) for all 𝑡≥0, in the sense of positive operators.

Proof. Let 𝑓∈𝐿²(Ω)₊. Since 𝐴^𝜇_𝑡≥0 we get easily the following: 𝒫^𝜇_𝑡𝑓(𝑥)≤𝐸_𝑥[𝑓(𝑋_𝑡)] for any 𝑥∈Ω. On the other hand, we have 𝒫^𝜇_𝑡𝑓(𝑥)≥𝐸_𝑥[𝑓(𝑋_𝑡)𝑒^{−𝐴𝜇𝑡}1_{{𝑡<𝜎𝜕Ω}}], where 𝜎_𝜕Ω is the first hitting time of 𝜕Ω. Since the relative quasisupport of 𝐴^𝜇_𝑡 and 𝑁_𝑡 are in 𝜕Ω, then in {𝑡<𝜎_𝜕Ω}, 𝑁_𝑡 and 𝐴^𝜇_𝑡 vanishes. Consequently, 𝑋_𝑡=𝐵_𝑡 in {𝑡<𝜎_𝜕Ω} and 𝒫^𝜇_𝑡𝑓(𝑥)≥𝐸_𝑥[𝑓(𝐵_𝑡)1_{{𝑡<𝜎𝜕Ω}}]. The theorem follows.

Remark 4.7. The fact that the semigroup 𝒫^𝜇_𝑡 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 𝜇,𝜈∈𝑆(𝜕Ω) such that 𝜈≤𝜇 (i.e., 𝜈(𝐴)≤𝜇(𝐴),  for all 𝐴∈ℬ(𝜕Ω)), then 0≤𝑒^{−𝑡Δ𝐷}≤𝒫^𝜇_𝑡≤𝒫^𝜈_𝑡≤𝑒^{−𝑡Δ𝑁} (4.17) for all 𝑡≥0, in the sense of positive operators.

Proof. It follows from the remark that if 𝜈≤𝜇, then 𝐴^𝜈_𝑡≤𝐴^𝜇_𝑡, which means that (𝐴^𝜇_𝑡)_𝜇 is increasing, and then (𝒫^𝜇_𝑡)_𝜇 is decreasing.

There exist a canonical Hunt process 𝑋^𝐴_𝑡 possessing the transition function 𝒫^𝜇_𝑡 which is directly constructed from 𝑋_𝑡 by killing the paths with rate −𝑑𝐿_𝑡, where 𝐿_𝑡=𝑒^−𝐴𝑡.

To construct the process associated with 𝒫^𝜇_𝑡, we follow A.2 of [17], so we need a nonnegative random variable 𝑍(𝜔) on (Ξ,ℳ,𝑃_𝑥) which is of an exponential distribution with mean 1, independent of (𝑋_𝑡)_𝑡≥0 under 𝑃_𝑥 for every 𝑥∈Ω satisfying 𝑍(𝜃_𝑠(𝜔))=(𝑍(𝜔)−𝑠)∨0. Introducing now a Random time 𝜉^𝐴 defined by 𝜉^𝐴=inf𝑡≥0∶𝐴_𝑡≥𝑍. (4.18)

We define the process (𝑋^𝐴_𝑡)_𝑡≥0 by 𝑋^𝐴_𝑡=𝑋_𝑡if𝑡<𝜉^𝐴;Δif𝑡≥𝜉^𝐴, (4.19) where Δ is a one-point compactification.

And, the admissible filtration of the process (𝑋^𝐴_𝑡)_𝑡≥0 is defined by ℱ^𝐴_𝑡=Λ∈ℱ_∞∶Λ∩𝐴_𝑡<𝑍=Λ_𝑡∩𝐴_𝑡<𝑍,∃Λ_𝑡∈ℱ_𝑡. (4.20)

Since {𝐴_𝑡<𝑍}∩{𝐴_𝑡=∞}=∅, we may and will assume that Λ_𝑡⊃{𝐴_𝑡=∞}.

Now, we can write 𝐸_𝑥𝑓𝑋^𝐴_𝑡=𝐸_𝑥𝑓𝑋_𝑡∶𝑡<𝜉^𝐴=𝐸_𝑥𝑓𝑋_𝑡∶𝐴_𝑡<𝑍=𝐸_𝑥𝑓𝑋_𝑡𝑒^−𝐴𝑡=𝒫^𝜇_𝑡𝑓(𝑥). (4.21)

The Hunt process (𝑋^𝐴_𝑡)_𝑓≥0 is called the canonical subprocess of (𝑋_𝑡)_𝑡≥0 relative to the multiplicative functional 𝐿_𝑡. In fact, (𝑋^𝐴_𝑡)_𝑡≥0 is a Diffusion process as (ℰ^𝜇,ℱ^𝜇) is local.

In the literature, the Diffusion process 𝑋^𝐴_𝑡 is called a partially reflected Brownian motion [25], in the sense that, the paths of 𝑋_𝑡 are reflected on the boundary since they will be killed (absorbed) at the random time 𝜉^𝐴 with rate −𝑑𝐿_𝑡.

Theorem 4.9. Let 𝜇,𝜇_𝑛∈𝑆(𝜕Ω) such that 𝜇_𝑛 is monotone and converges setwise to 𝜇, that is, 𝜇_𝑛(𝐵) converges to 𝜇(𝐵) for any 𝐵∈ℬ(𝜕Ω), then Δ_𝜇𝑛 converges to Δ_𝜇 in strongly resolvent sense.

Proof. We prove the theorem for 𝜇_𝑛 increasing, the proof of the decreasing case is similar. Let 𝐴_𝑛 (resp. 𝐴) be the additive functional associated to 𝜇_𝑛 (resp. 𝜇) by the Revuz correspondence. Similarly to the second part of the proof of Theorem 4.2, we have lim𝑅^𝐴𝑛_𝛼𝑓(𝑥)=𝑅^𝐴_𝛼𝑓(𝑥) for r.q.e 𝑥∈Ω. Consequently lim_𝑛→+∞||𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴_𝛼𝑓||_𝐿²_(Ω)=0. For 𝑛<𝑚, we have ℱ^𝜇𝑚⊂ℱ^𝜇𝑛, and then ℰ^𝜇𝑛_𝛼𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓,𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓≤𝑓,𝑅^𝐴𝑛_𝛼𝑓−𝑅^𝐴𝑚_𝛼𝑓, (4.22) which converges to zero as 𝑛,𝑚→+∞. Therefore, (𝑅^𝐴𝑛_𝛼𝑓)_𝑛 is ℰ₁-convergent in ℱ and the limit function 𝑅^𝐴_𝛼𝑓 is in ℱ. The result follows.

Corollary 4.10. Let 𝜇∈𝑆(𝜕Ω) finite and let 𝑘∈ℕ^∗. We defined for 𝑢,𝑣∈ℱ^𝜇: ℰ^𝜇𝑘(𝑢,𝑣)=_Ω1∇𝑢∇𝑣𝑑𝑥+𝑘_𝜕Ω𝑢𝑣𝑑𝜇, (4.23) then Δ_𝜇𝑘→Δ_𝑁 in the strong resolvent sense.

Intuitively speaking, when the measure 𝜇 is infinity (locally infinite on the boundary), the semigroup 𝒫^𝜇_𝑡 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 𝑁_𝑡 does not appear yet in the decomposition of 𝑋_𝑡, which means that the reflecting phenomena disappear, and so any path of 𝑋_𝑡 is immediately killed when it arrives to the boundary.

When 𝜇 is null on the boundary, then the semigroup 𝒫^𝜇_𝑡 is the Neumann one, and then the boundary became completely reflecting, but for a general measure 𝜇 the paths are reflected many times before they are absorbed at a random time.

Acknowledgments

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.