Scholarly article on topic 'Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains'

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Academic research paper on topic "Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains"

Hindawi Publishing Corporation International Journal of Stochastic Analysis Volume 2012, Article ID 163096,17 pages doi:10.1155/2012/163096

Research Article

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

Khalid Akhlil

Institut fur Angewandte Analysis, Universitat Ulm, 89069 Ulm, Germany Correspondence should be addressed to Khalid Akhlil, akhlil.khalid@gmail.com Received 13 April 2012; Revised 30 August 2012; Accepted 5 December 2012 Academic Editor: Donal O'Regan

Copyright © 2012 Khalid Akhlil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Q, 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 Q of RN (N > 0) is giving by

ÎU + 6u = 0 on dQ, (1.1)

where v is the outward normal vector field on the boundary dQ and ¡5 a positive bounded Borel measurable function defined on dQ.

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 ¡¡. 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 d/dv, which represent Neumann's boundary conditions, by the measure f = 5 ■ a, where a 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 (Xt)t>0 is a reflecting Brownian motion (RBM) and Lt is the boundary local time, which corresponds to a by Revuz correspondence. It is clear that the smoothness of the domain Q 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(Q) by

where Q is assumed to be bounded with Lipschitz boundary so that the Dirichlet form (E, F) is regular. If Q 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(Q), the closure of H 1(Q) n Cc(Q) in H 1(Q). The domain H 1(Q) 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 (Ep, Fp) defined on L2(Q) by

In the case of p = ¡5 ■ a (Q 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 Q, 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 (Xt)t>0 associated with the regular Dirichlet form (E,H!(Q)). We apply a decomposition theorem of additive functionals to write Xt in the form Xt = x + Bt + Nt, we prove that the additive functional Nt is supported by dQ, 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 = Q, where Q is an Euclidean domain of RN, and the measure m on the c-algebra B(X) is given by m(A) = X(A n Q) for all A e B(X) with X the Lebesgue measure; it follows that L2(Q) = L2(X, B(X),m), and we define a regular Dirichlet form (E, F) on L2 (Q) by

E(u,v)=f VuVvdx, F = H1 (Q), (2.1)

___=-h1 (q) _

where H!(Q) = H!(Q) n Cc(Q) . The domain H!(Q) is so defined to insure the Dirichlet form (E, F) to be regular, instead of F = H!(Q) which make the form not regular in general, but if Q is bounded open set with Lipschiz boundary, then H1 (Q) = H!(Q).

We denote for any a> 0 : Ea(u, v) = E(u,v) + a(u,v)m, for all u,v e 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 dQ of Q.

The relative capacity which we denote by Capq is defined on a subsets of Q by the following: for A c Q relatively open (i.e., open with respect to the topology of Q) we set

CapQ(A) := inf|Ei(u,u) : u e H!(Q) : u > 1 a.e. on (2.2)

And for arbitrary A c Q, we set

CapQ(A) := inf jCapQ(B) : B relatively open A c B c Q. (2.3)

A set N c Q is called a relatively polar if Capq-(N) = 0.

The relative capacity (just as a cap) has the properties of a capacity as described in [17]. In particular, Capq is also an outer measure (but not a Borel measure) and a Choquet capacity.

A statement depending on x e A c Q is said to hold relatively quasieverywhere (r.q.e.) on A, if there exist a relatively polar set N c A such that the statement is true for every x e A \ N.

Now we may consider functions in H 1(Q) as defined on Q, and we call a function u : Q ^ R relatively quasicontinuous (r.q.c.) if for every e > 0 there exists a relatively open set G c Q such that Capq-(G) < e and u|q\G is continuous.

It follows [13] that for each u e H!(Q) there exists a relatively quasicontinuous function u : Q ^ 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 Hq1 (Q) in term of relative capacity is also given, namely,

HQ(Q) = {u e H!(Q) : u(x) = 0 r.q.e. on dQ}. (2.4)

2.2. Smooth Measures

All families of measures on dQ defined in this subsection were originally defined on X [17], and then in our settings on X = Q, as a special case. We reproduce the same definitions, and most of their properties on dQ, as we deal with measures concentrated on the boundary of Q 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 dQ between brackets to recall our context, and we keep in mind that the same things are valid if we put Q or Q instead of dQ.

Let Q c Rn be open. A positive Radon measure p on dQ is said to be of finite energy integral if

|v(x) |p(dx) < C\] ¿\(v,v), v e?n Cc (q) (2.5)

for some positive constant C. A positive Radon measure on dQ is of finite energy integral if and only if there exists, for each a> 0, a unique function Uap e F such that

¿a(Uap,v) = v(x)p(dx). (2.6)

We call Uap an a-potential.

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

Lemma 2.1. Each measure in S0(dQ) charges no set of zero relative capacity. Let us consider a subset S00 (dQ) of S0 defined by

Sqq(dQ) = {p e So(dQ) : p(dQ) < ||U1p||M < »}. (2.7)

Lemma 2.2. For any p e S0 (dQ), there exist an increasing sequence (Fn)„>0 of compact sets of dQ such that

1f„ • p e Soo(dQ), „ = 1,2,...,

CapQ(K \ F„) —> 0, „ —> for any compact set K c dQ.

We note that ¡1 e S0(dQ) vanishes on dQ \ UnFn for the sets Fn of Lemma 2.2, because of Lemma 2.1.

We now turn to a class of measures S(dQ) larger than S0(dQ). Let us call a (positive) Borel measure ¡ on dQ smooth if it satisfies the following conditions:

(i) ¡ charges no set of zero relative capacity;

(ii) there exist an increasing sequence (Fn)n>0 of closed sets of dQ such that

¡(Fn) < go, n = 1,2,..., (2.9)

lim CapQ-(K \ Fn) = 0 for any compact K c dQ. (2.10)

Let us note that ¡ then satisfies

¡(dQ \UnFn) = 0. (2.11)

An increasing sequence (Fn) of closed sets satisfying condition (2.10) will be called a generalized nest; if further each Fn is compact, we call it a generalized compact nest.

We denote by S(dQ) the family of all smooth measures. The class S(dQ) is quiet large and it contains all positive Radon measures on dQ charging no set of zero relative capacity. There exist also, by [ 15, Theorem 1.1 ], a smooth measure ¡i on dQ (hence singular with respect to m) "nowhere Radon" in the sense that ¡(G) = g for all nonempty relatively open subset G of dQ (see [15, Example 1.6]).

The following Theorem, says that, any measure in S(dQ) can be approximated by measures in S0(dQ) and in S00(dQ) as well.

Theorem 2.3. The following conditions are equivalent for a positive Borel measure ¡i on dQ.

(i) ¡i e S(dQ).

(ii) There exists a generalized nest (Fn) satisfying (2.11) and 1Fn ■ ¡i e S0(dQ) for each n.

(iii) There exists a generalized compact nest (Fn) satisfying (2.11) and 1Fn ■ ¡i e S00(dQ) 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 dQ. Recall that as the Dirichlet form (E, F) is regular, then there exists a Hunt process M = (S, Xt, I, Px) on Q which is m-symmetric and associated with it.

Definition 2.4. A function A : [0, +to[xS ^ [-to, +to] is said to be an additive functional (AF) if

(1) At is Fi-measurable,

(2) there exist a defining set A e Fto and an exceptional set N c dQ with cap^ (N) = 0 such that Px(A) = 1, for all x e dQ \ N, 6tA c A, for all i > 0; for all w e A, A0(w) = 0; |At(w)| < to for t < A(w) is right continuous and has left limit, and At+s(w) = At(w) + As(6tw)s, t > 0.

An additive functional is called positive continuous (PCAF) if, in addition, At(w) is nonnegative and continuous for each w e A. The set of all PCAF's on dQ is denoted A+(dQ).

Two additive functionals A1 and A2 are said to be equivalent if for each t> 0, Px(Al = A2) = 1 r.q.e x e Q.

We say that A e A+(dQ) and p e S(dQ) are in the Revuz correspondence, if they satisfy, for all y-excessive function h, and f e B+(Q), the relation

lim- Ehm

{ f (Xs)dAs = f h(x)(/ • n)(dx)

Jo J da

(2.12)

The family of all equivalence classes of A+(dQ) and the family S(dQ) are in one to one correspondence under the Revuz correspondence. In this case, p e S(dQ) is called the Revuz measure of A.

Example 2.5. We suppose Q to be bounded with Lipschitz boundary. We have [2]

lim- Ehm

f (Xs)dLs

1 I h(x)f (x)a (dx),

2 J da

(2.13)

where Lt is the boundary local time of the reflecting Brownian motion on Q. It follows that (1/2)c is the Revuz measure of Lt.

In the following we give some facts useful in the proofs of our main results. We set

UAf (x) = E rA/ (x) = Ex

Raf (x) = Ex

■ sœ

e-atf (Xt)dAt

e-ate-Atf (Xt)dt

e-atf (Xt)dt

(2.14)

Proposition 2.6. Let p e S0(da) and A e A^(da), the corresponding PCAF. For a > 0, f eBJ, UA is a relatively quasicontinuous version of Ua(f • p).

Proposition 2.7. Let A e A+(dQ), and / e B+, then RA is relatively quasicontinuous and

RAf - Ra/ + UARA/ = 0. (2.15)

In general, the support of an AF A is defined by

supp[A] = {x e X \ N : Px(R = 0) = 1}, (2.16)

where R(w) = inf{t > 0: At(w) = 0}.

Theorem 2.8. The support of A e A+(dQ) is the relative quasisupport of its Revuz measure.

In the following, we give a well-known theorem of decomposition of additive func-tionals 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 eF, the AF A[u] = u(Xt) - u(X0) can be expressed uniquely as

u(Xt) - u(X0) = M[u] + N[u], (2.17)

where M[u] is a martingale additive functional of finite energy and N[u] is a continuous additive functional of zero energy.

A set a (u) is called the (0)-spectrum of u eF, if o(u) is the complement of the largest open set G such that E(u, v) vanishes for any v e Fn C0(X) with supp[v] c G. The following Theorem means that supp[N[u] ] c a (u), for all u eF.

Theorem 2.10. For any u eF, the CAF N[u] vanishes on the complement of the spectrum F = a (u) of u in the following sense:

Px(n[u = 0 : Vt<af) = 1 r.q.e x e 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(Q) defined by

E(u,v) = i VuVvdx, F = H1 (Q). (3.1)

Due to the theorem of Fukushima (1975), there is a Hunt process (Xt)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 = (Xt, Px) on Q is associated with the form E in the sense that the transition semigroup ptf (x) = Ex[f (Xt)], x e Q is a version of the L2-semigroup Ptf 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 Q associated with (¿, F) the general reflecting Brownian motion.

The process Xt is so named to recall the standard reflecting Brownian motion in the case of bounded smooth Q, as the process associated with (¿,H1 (Q)). Indeed, when Q is bounded with Lipschitz boundary, we have that H!(Q) = H!(Q) and by [6] the reflecting Brownian motion Xt admits the following Skorohod representation:

Xt = x + Wt + - v(Xs)dLs, (3.2)

where W is a standard N-dimensional Brownian motion, L is the boundary local (continuous additive functional) associated with surface measure a on dQ, and v is the inward unit normal vector field on the boundary.

For a general domain, the form (¿,H1 (Q)) needs not to be regular. Fukushima [9] constructed the reflecting Brownian motion on a special compactification of Q, the so-called Kuramuchi compactification. In [6], it is shown that if Q is a bounded Lipschitz domain, then the Kuramochi compactification of Q 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 Q.

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(Xt) - u(X0) can be decomposed as follows:

u(Xt) - u(Xo) = M[u] + Nt[u], (3.3)

where M[u] is a martingale additive functional of finite energy and Nt[u] is a continuous additive functional of zero energy.

Since (Xt)t>0 has continuous sample paths, M[u] is a continuous martingale whose quadratic variation process is

(m[u],mM^ = J |Vu|2(Xs)ds. (3.4)

Instead of u, we take coordinate function ^¿(x) = xi. We have

Xt = Xo + Mt + Nt. (3.5)

We claim that Mt is a Brownian motion with respect to the filtration of Xt. To see that, we use Levys criterion. This follows immediately from (3.2), which became in the case of coordinate function

^M^ ],mM)= Sijt.

Now we turn our attention to the additive functional Nt. Two natural questions need to be answered. The first is, where is the support of Nt 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 Nt is supported by dQ.

Proof. Following Theorem 2.10, we have that supp[Nt] c a($), where a($) is the (0)-spectrum of which means the complement of the largest open set G such that ¿(fa,v) = 0 for all v Cc (Q) with supp[v] c G.

Step 1. If Q is smooth (Bounded with Lipschitz boundary, e.g.), then we have

¿($i,v) = - v • Uida. (3.7)

Then, ¿(fa,v) = 0 for all v e F n Cc(Q) with supp[v] c Q. We can then see that the largest G is Q. Consequently a($) = Q \ Q and then a($) = dQ.

Step 2. If Q is arbitrary, then we take an increasing sequence of subset of Q such that UiT=o Qn = Q. Define the family of Dirichlet forms (¿qn, FQn) to be the parts of the form (¿, F) on each Qn as defined in Section 4.4 of [17]. By Theorem 4.4.5 in the same section, we have that FQn c F and ¿qn = E on fq x FQn. We have that Qn is the largest open set such that ¿qn (fy, v) = 0 for all v e FQn n Cc(Qn). By limit, we get the result. □

The interest of the question of boundedness of total variation of Nt appears when one needs to study the semimartingale property and the Skorohod equation of the process Xt of type 3.2. Let |N| be the total variation of Nt, that is,

|N|t = sup £|Nti — Nt—11, (3.8)

where the supremum is taken over all finite partition 0 = t0 < t\ < ••• <tn = t, and | • | denote the Euclidian distance. If |N| is bounded, then we have the following expression:

Nt = (vsd\N\s, (3.9)

where v is a process such that | v|s = 1 for |N|-almost all s.

According to § 5.4. in [17], we have the following result.

Theorem 3.3. Assume that Q is bounded and that the following inequality is satisfied:

^-dx jq dx-,

< CIML, Vv e H!(Q) n C^Q), (3.10)

for some constant C. Then, Nt 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 Q is a Caccioppoli set, then there exist a finite signed smooth measure v such that

dvdx = -f vdu, Vv e H1(Q) n cJQ) . (3.11)

Jq dxi jdq v '

And v = v1 - v2 is associated with the CAF -Nt = -Aj + A2 with the Revuz correspondence. Consequently v charges no set of zero relative capacity.

To get a Skorohod type representation, we set

v = X i=1

♦ ■ |i -1.....-

We define the measure a on ÖQ by

and the vector of length 1 at x e ÖQ by

ni(x) ■

Thus,¡ii(dx) = (1/2)ni(x)a(dx), i = 1,...,N. Then

where L is the PCAF associated with (1/2)a.

Theorem 3.5. If Q is a Caccioppoli set, then for r.q.e x e Q, one has

(3.12)

a (dx) ■ 2( £ | ♦i (x) |2 ) v(dx) (3.13)

7-if El^i(x)l2 > 0;

(S- ♦ (x)|2) i-1 (3.14)

0 if£|&(x)|2 ■ 0.

Nt ■ I" n(Xs)dLs, (3.15)

Xt ■ x + Bt + f n(Xs)dLs, (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)a.

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 ¡i be a positive Radon measure on dQ charging no set of zeo relative capacity. Consider the perturbed Dirichlet form (¿¡, Ff) on L2(Q) defined by

F = ?nl2( dQ,f,

¿f(u, v) = ¿(u, v) + uvdf u,v eFf.

We will see in the following theorem that the transition function

Pdf (x) = Exf (Xt)e

is associated with (¿¡, Ff), where Af 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. Pi is a strongly continuous semigroup on L2(Q).

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

Theorem 4.2. Let f be a positive Radon measure on dQ charging no set of zero relative capacity and (Ai) t>0 be its associated PCAF of (Xt)t>0. Then Pf is the strongly continuous semigroup associated with the Dirichlet form (¿¡, F^) on L2(Q).

Proof. To prove that Pf is associated with the Dirichlet form (¿¡, F^) on L2(Q), it suffices to prove the assertion

RAf e ¿f(RA,u) = (f,u), f e L2(Q,m), u e (4.3)

Since WR^f ||L2(q) < \\Raf ||l2(q) < (1/a)\\f ||l2(q), we need to prove (4.7) only for bounded f e L2(Q). We first prove that (4.7) is valid when p e Soo(ôQ). According to Proposition 2.7 we have

RAf - Raf + UARAf = 0, a > 0, f eB+(â). (4.4)

If p e Soo(dQ), and if f is bounded function in L2(Q), then \\Raf || < to, and UaARAf is a relative quasicontinuous version of the a-potential Ua(RAf ■ p) eF by Proposition 2.6. Since \\Ua(RAf ■ p)\u < \\RAf\U\UaP\L < to and p(dQ) < to, we have that

RAf = Raf - UARAf e Fp (4.5)

and that

¿a{RAf,u) = Ea(Raf,u) - ¿a(uaAR£f,u) = (f,u) - (RAf,u) , u eFp,

then (4.7) follows.

For general positive measure p charging no set of zero relative capacity, we can take by virtue of Theorem 2.3 and Lemma 2.2 an increasing sequence (Fn) of generalized nest of dQ, and pn = 1Fn ■pe S00(5Q). Since p charges no set of zero relative capacity, pn(B) increases to p(B) for any B e B(dQ).

Let An = 1Fn ■ A. Then An is a PCAF of Xt with Revuz measure pn. Since pn e S00(5Q) we have for f e L2(Q):

RAnf e Fpn, EPn(RAan,u) = (f,u), f e L2(Q,m), u e Fpn. (4.7)

Clearly \RAnf \ < Ra\f \ < to r.q.e, and hence limn^ +TORAnf (x) = RAf (x) for r.q.e x e Q. For n < m, we get from (4.8)

Epn(RA"f - RAAmf, R^f - RAmf) < f, R^f - RAmf), (4.8)

which converges to zero as n,m ^ +to. Therefore, (RaAnf)n is Ei-convergent in F and the limit function RAf is in F. On the other hand, we also get from (4.8)

Kf iU) < (f,RAnf )l2(q) <( a) »f hl2(q). (4.9)

And by Fatou's lemma: \ \RAf \\L2(Q) < (1/vO)\\f \\L2(Q), getting RAf e Fp. Finally, observe the estimate

(RA"f,u) - (.RAf,u)

L2(dQ.,iin)

llL2(dq,p )

(4.10)

holding for u € L2(dQ,f). The second term of the right-hand side tends to zero as n ^ +<x>. The first term also tends to zero because we have from (4.8)\\RAnf-RAmf WmdQrHn,) < (f,RAnf-RAnf), and it suffices to let first m ^ +<x> and then n ^ +<x>. By letting n ^ +<x> 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 = Q, and working just with measures on S0(dQ), the proof still the same and works also for any smooth measure concentrated on dQ. Consequently, the theorem is still verified for smooth measures "nowhere Radon," that is, measures locally infinite on dQ.

Example 4.3. We give some particular examples of Pf. (1) If f = 0, then

P0f (x) = Ex[f (Xt)], (4.11)

the semigroup generated by Laplacian with, Neumann boundary conditions. (2) If f is locally infinite (nowhere Radon) on dQ, then

PTf (x) = Ex [f (Bt)1[t<T} ], (4.12)

the semigroup generated by the Laplacian with Dirichlet boundary conditions (see [13, Proposition 3.2.1]).

(3) Let Q be a bounded and enough smooth to insure the existence of the surface measure a, and f = ¡5 ■ a, with ¡5 is a measurable bounded function on dQ, then Af = J0 ¡5(Xs)dLs, where Lt is a boundary local time. Consequently

Pff (x) = Ex

f (Xt) expf -Jq№)dL£

(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. pf is sub-Markovian, that is, pf > 0 for all t > 0, and

||Pff ||m< ||f L (t > 0). (4.14)

Proof. It is clear that if f € L2(Q)+, then Pff > 0 for all t > 0. In addition we have |Pff (x)\ < Ex[\f\(Xt)], and then\\Pff\\„ <\\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 Af be the self-adjoint operator on L2(Q) generator of the semigroup Pf, we write

Pff (x) = e-tAf (x). (4.15)

Following [13], we know that Af is a realization of the Laplacian. Then we call Af the Lapla-cian with General Robin boundary conditions.

Theorem 4.6. Let f e S(ôQ), then the semigroup Pf is sandwiched between the semigroup of Neumann Laplacian, and the semigroup ofDirichlet Laplacian. That is

0 < e-tAD <pf < e-tAN (4.16)

for all t > 0, in the sense of positive operators.

Proof. Let f € L2(Q)+. Since Af > 0 we get easily the following: Pff (x) < Ex[f (Xt)] for any x e Q. On the other hand, we have Pff (x) > Ex[f(Xt)e-Af 1{t<„dQ}], where adQ is the first hitting time of dQ. Since the relative quasisupport of Af and Nt are in dQ, then in {t < adQ}, Nt and Af vanishes. Consequently, Xt = Bt in {t < adQ} and Pff (x) > Ex[f (Bt)1{t<adQ}]. The theorem follows. □

Remark 4.7. The fact that the semigroup Pf 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 f,v € S(5Q) such that v < f (i.e., v(A) < f(A), for all A € B(dQ)), then

0 < e~tA° <Pf < Pvt < e^tAN (4.17)

for all t > 0, in the sense of positive operators.

Proof. It follows from the remark that if v < f, then Avt < Af, which means that (Af)f is increasing, and then (Pf)f is decreasing. □

There exist a canonical Hunt process Xf possessing the transition function Pf which is directly constructed from Xt by killing the paths with rate -dLt, where Lt = e-At.

To construct the process associated with Pp, we follow A.2 of [17], so we need a nonnegative random variable Z(w) on (S, M, Px) which is of an exponential distribution with mean 1, independent of (Xt)t>0 under Px for every x e Q satisfying Z(ds(w)) = (Z(w) - s) V0. Introducing now a Random time lA defined by

¿A = inf{t > 0: At > Z}. (4.18)

We define the process (XA)t>0 by

XA = X if ^ (4.19)

t 'A if t > lA, K '

where A is a one-point compactification.

And, the admissible filtration of the process (XA)t>0 is defined by

FA = (A eFoo : an {At <Z} = At n {At < Z}, 3A e Ft}- (4.20)

Since (At < Z}n (At = œ} = 0, we may and will assume that At d (At = œ}. Now, we can write

Exlf(.XA)\ = Ex[f(Xt) : t<A = Ex f (Xt) : At<Z]

(4.21)

= Exf (Xt)e-At

= Pf (x).

The Hunt process (XtA)f>0 is called the canonical subprocess of (Xt)t>0 relative to the multiplicative functional Lt. In fact, (Xf)t>0 is a Diffusion process as (Ep, Fp) is local.

In the literature, the Diffusion process Xf is called a partially reflected Brownian motion [25], in the sense that, the paths of Xt are reflected on the boundary since they will be killed (absorbed) at the random time lA with rate -dLt.

Theorem 4.9. Let p,pn e S(dQ) such that pn is monotone and converges setwise to p, that is, pn(B) converges to p(B) for any B e B(dQ), then Apn converges to Ap in strongly resolvent sense.

Proof. We prove the theorem for pn increasing, the proof of the decreasing case is similar. Let An (resp. A) be the additive functional associated to pn (resp. p) by the Revuz correspondence. Similarly to the second part of the proof of Theorem 4.2, we have lim RA"f (x) = RAf (x) for r.q.e x e Q. Consequently limn ^ +TO Fpm c Fpn, and then

Eafâf - RAmf, RA"f - RAmf ) < f, RA"f - RAmf), (4.22)

which converges to zero as n,m ^ +<x>. Therefore, (RAnf )n is ¿1-convergent in F and the limit function RAf is in F. The result follows. □

Corollary 4.10. Let f € S(5Q) finite and let k € N*. We defined for u,v € Ff

¿fk(u,v) = ( VuVvdx + 1 f uvdf, (4.23)

jq k j dq

then Afk ^ An in the strong resolvent sense.

Intuitively speaking, when the measure f is infinity (locally infinite on the boundary), the semigroup Pf 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 Nt does not appear yet in the decomposition of Xt, which means that the reflecting phenomena disappear, and so any path of Xt is immediately killed when it arrives to the boundary.

When f is null on the boundary, then the semigroup Pf is the Neumann one, and then the boundary became completely reflecting, but for a general measure f 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.

References

[1] K. Sato and T. Ueno, "Multi-dimensional diffusion and the Markov process on the boundary," Journal of Mathematics of Kyoto University, vol. 4, pp. 529-605,1965.

[2] V. G. Papanicolaou, "The probabilistic solution of the third boundary value problem for second order elliptic equations," Probability Theory and Related Fields, vol. 87, no. 1, pp. 27-77,1990.

[3] S. Ramasubramanian, "Reflecting Brownian motion in a Lipschitz domain and a conditional gauge theorem," Sankhya A, vol. 63, no. 2, pp. 178-193, 2001.

[4] S. Renming, "Probabilistic approach to the third boundary value problem," Journal ofHebei University, vol. 9, no. 1, pp. 85-100,1989.

[5] R. F. Bass, K. Burdzy, and Z.-Q. Chen, "Uniqueness for reflecting Brownian motion in lip domains," Annales de l'Institut Henri Poincare. Probabilites et Statistiques, vol. 41, no. 2, pp. 197-235, 2005.

[6] R. F. Bass and P. Hsu, "The semimartingale structure of reflecting Brownian motion," Proceedings of the American Mathematical Society, vol. 108, no. 4, pp. 1007-1010,1990.

[7] R. F. Bass and P. Hsu, "Some potential theory for reflecting Brownian motion in Holder and Lipschitz domains," The Annals of Probability, vol. 19, no. 2, pp. 486-508,1991.

[8] Z. Q. Chen, P.J. Fitzsimmons, and R. J. Williams, "Quasimartingales and strong Caccioppoli set," Potential Analysis, vol. 2, pp. 281-315, 1993.

[9] M. Fukushima, "A construction of reflecting barrier Brownian motions for bounded domains," Osaka Journal of Mathematics, vol. 4, pp. 183-215,1967.

[10] E. P. Hsu, Reflecting Brownian motion, boundary local time and the Neumann problem [Thesis], Stanford University, 1984.

[11] W. Arendt and M. Warma, "The Laplacian with Robin boundary conditions on arbitrary domains," Potential Analysis, vol. 19, no. 4, pp. 341-363, 2003.

[12] W. Arendt and M. Warma, "Dirichlet and Neumann boundary conditions: what is in between?" Journal of Evolution Equations, vol. 3, no. 1, pp. 119-135, 2003.

[13] M. Warma, The laplacian with general Robin boundary conditions [Ph.D. thesis], University of Ulm, 2002.

[14] S. Albeverio and Z. M. Ma, "Perturbation of Dirichlet forms—lower semiboundedness, closability, and form cores," Journal of Functional Analysis, vol. 99, no. 2, pp. 332-356,1991.

[15] S. Albeverio and Z. M. Ma, "Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms," Osaka Journal of Mathematics, vol. 29, no. 2, pp. 247-265,1992.

[16] Ph. Blanchard and Z. M. Ma, "New results on the Schrodinger semigroups with potentials given by signed smooth measures," in Stochastic Analysis and Related Topics, II (Silivri, 1988), vol. 1444, pp. 213-243, Springer, Berlin, Germany, 1990.

[17] M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, vol. 19, Walter de Gruyter, Berlin, Germany, 1994.

[18] T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer, Berlin, Germany, 1995.

[19] B. Simon, "Schrodinger semigroups," Bulletin of American Mathematical Society, vol. 7, no. 3, pp. 447526,1982.

[20] P. Stollmann, "Smooth perturbations of regular Dirichlet forms," Proceedings of the American Mathematical Society, vol. 116, no. 3, pp. 747-752,1992.

[21] P. Stollmann and J. Voigt, "Perturbation of Dirichlet forms by measures," Potential Analysis, vol. 5, no. 2, pp. 109-138,1996.

[22] J. Voigt, "Absorption semigroups," Journal of Operator Theory, vol. 20, no. 1, pp. 117-131,1988.

[23] J. Voigt, "Absorption semigroups, their generators, and Schrodinger semigroups," Journal of Functional Analysis, vol. 67, no. 2, pp. 167-205,1986.

[24] M. Fukushima, "Dirichlet forms, Caccioppoli sets and the Skorohod equation," in Stochastic Differential and Difference Equations, vol. 23, pp. 59-66, Birkhauser, Boston, Mass, USA, 1997.

[25] D. S. Grebenkov, "Partially reflected Brownian motion: a stochastic approach to transport phenomena," in Focus on Probability Theory, pp. 135-169, Nova Science Publishers, New York, NY, USA, 2006.

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