For a discussion on this point we refer to [10, Remark 2.6(ii

POINTWISE WEAK EXISTENCE OF DISTORTED SKEW BROWNIAN MOTION WITH RESPECT TO DISCONTINUOUS

MUCKENHOUPT WEIGHTS

JIYONG SHIN and GERALD TRUTNAU

For any starting point in R^d,d≥3, we identify the stochastic dierential equa- tion of distorted Brownian motion with respect to a certain discontinuous Muck- enhouptA2-weightψ. The discontinuities ofψtypically take place on a sequence of level sets of the Euclidean norm Dk :={x∈R^d| kxk=dk}, k∈ Z, where (dk)k∈Z⊂(0,∞)may have accumulation points and each level set Dkplays the role of a permeable membrane.

AMS 2010 Subject Classication: 60J60, 60J55, 31C25, 31C15, 35J25.

Key words: skew Brownian motion, distorted Brownian motion, Dirichlet forms, diusion processes, permeable membranes, multidimensional local time.

1. INTRODUCTION

In this note we are concerned with the construction of a weak solution to the stochastic dierential equation

(1) Xt=x+Wt+ Z t

0

∇ρ

2ρ(Xs) ds+ Z ∞

0

Z t 0

νa(Xs)d`<sup>a</sup><sub>s</sub>(kXk)η(da), x∈R<sup>d</sup>, whereW is ad-dimensional standard Brownian motion,ρ is typically a Muck- enhouptA2-weight,νa is the unit outward normal on the boundary of the Eu- clidean ball of radius a about zero, `^a(kXk) is the symmetric semimartingale local time at a∈(0,∞) of kXk,η =P

k∈Z(2α_k−1)δ_dk with (α_k)_k∈Z ⊂(0,1) is a sum of Dirac measures at a sequence (d_k)k∈Z ⊂ (0,∞) with exactly two accumulation points in[0,∞), one is zero and the other ism₀>0. More accu- mulation points could be allowed. For a discussion on this point we refer to [10, Remark 2.6(ii)]. The absolutely continuous component of drift ^∇ρ_2ρ is typically unbounded and discontinuous. For an interpretation of the equation we refer to Theorem 2.6 and Remark 2.7. Variants of (1) with reection on hyperplanes, instead of balls, but without accumulation points and Lipschitz drift appear in

REV. ROUMAINE MATH. PURES APPL. 59 (2014), 1, 157170

[8, 9, 17, 18]. In particular, (1) is a multidimensional analogue of an equation that was thoroughly studied in [10] and both equations share a lot of similarities.

For instance the way to determine (γk)k∈Z and (γ_k)k∈Z in (2) below, when (α_k)k∈Z in (1) is given, is the same here as in [10, Proposition 2.11]. The way to obtain (α_k)k∈Z from (γ_k)k∈Z and (γ_k)k∈Z is described in Remark 2.7 (cf. also [10, Remark 2.4(ii) and Theorem 2.5]). See further [10, Remark 2.7]

for another similarity and [10, Section 3.3] as well as references therein for a possible application to models with countably many permeable membranes that accumulate. For the construction of a solution to (1) for any starting point x∈R^d the key point is to identify (1) as distorted Brownian motion (see [1, 4]

for an introduction to distorted Brownian motion). Then, one needs to show that the absolute continuity condition [5, p.165] is satised for the underlying Dirichlet form and that the strict Fukushima decomposition [5, Theorem 5.5.5]

is applicable. In order to identify (1) as distorted Brownian motion we proceed informally as follows. We consider the density ρφ for the underlying Dirichlet form in (3), where φ is a step function on annuli in R^d, see (2), and ρ is typically a MuckenhouptA₂-weight (see Remark 2.1(iii) below). For the precise conditions we refer to (H1)(H3) in Section 2. The logarithmic derivative, which is the drift of the distorted Brownian motion, is then informally given by

d(ρφ) ρφ = dρ

ρ +dφ φ

(cf. [10, Remark 2.6]). This is rigorously performed through an integration by parts formula in Proposition 3.1 below. By results on Muckenhoupt weights in [15, 16], the existence of a jointly continuous transition kernel density for the semigroup associated to the Dirichlet form given in (3) is obtained. A Hunt process with the given transition kernel density is implicitly assumed to exist in condition (H4) of Section 2 (for ways to construct such a process, we refer to [13], see also Remark 2.3). Under the conditions (H1)(H4), we then show in a series of statements in Sections 4 and 5, that the strict Fukushima decomposition can be applied to obtain a solution to (1) (see main Theorem 2.6).

Finally one could think of generalizing (1), or more precisely the process of Theorem 2.6(i) with reections on boundaries of Lipschitz domains (instead of smooth Euclidean balls). The main ingredient to obtain this generalization would be [19, Theorem 5.1] (see [21, Section 5]). In case of skew reection at the boundary of a single C^1,λ-domain, λ∈(0,1],ρ ≡1, and smooth diusion coecient, a weak solution has been constructed in [11, III. Ÿ3 and Ÿ4], see also references therein. However, the reection term is dened as generalized drift and not explicitly as in Theorem 2.6.

2. FORMULATION OF THE MAIN THEOREM

Let m0 ∈ (0,∞) and (lk)k∈Z ⊂ (0, m0), 0 < lk < lk+1 < m0, be a sequence converging to 0 as k → −∞ and converging to m0 as k → ∞. Let (r_k)k∈Z ⊂(m₀,∞), m₀ < r_k < r_k+1 <∞, be a sequence converging to m₀ as k→ −∞and tending to innity ask→ ∞. Let

φ:=X

k∈Z

γ_k·1_Ak+γ_k·1_Aˆ

k

(2) ,

where γk , γ_k ∈ (0,∞), Ak := Blk \Blk−1 , Aˆk := Brk \Brk−1 , Br := {x ∈ R^d| kxk < r} for any r > 0, Br denotes the closure of Br, 1A the indicator function of a setA, and kxk the Euclidean norm ofx∈R^d. We denote bydσr

the surface measure on the boundary∂B_r of B_r,r >0.

Let d≥ 3, and let C₀^∞(R^d) denote the set of all innitely dierentiable functions with compact support inR^d. Let∇f := (∂1f, . . . , ∂_df) where ∂jf is the j-th partial derivative of f, and ∆f := Pd

j=1∂jjf and ∂jjf := ∂j(∂jf), j = 1, . . . , d. As usual dx is the Lebesgue measure on R^d and L^q(R^d, µ), q ≥ 1, are the usual L^q-spaces with respect to the measure µ on R^d, and L^q_loc(R^d, µ) := {f| f ·1U ∈ L^q(R^d, µ),∀U ⊂ R^d, U relatively compact open}. For any open set Ω ⊂ R^d, H^1,q(Ω, dx), q ≥ 1 is dened to be the set of all functionsf ∈L^q(Ω, dx) such that∂_jf ∈L^q(Ω, dx),j= 1, . . . , d, and

H_loc^1,q(R^d, dx) :={f|f·1_U ∈H^1,q(U, dx),∀U ⊂R^d, U relatively compact open}.

For a topological space X ⊂ R^d with Borel σ-algebra B(X) we denote the set of allB(X)-measurable f :X→ Rwhich are bounded, or nonnegative by B_b(X),B⁺(X) respectively. We further denote the set of continuous func- tions on X, the set of continuous bounded functions on X by C(X), Cb(X) respectively.

A function ψ ∈ B(R^d) with ψ > 0 dx-a.e. is said to be a Muckenhoupt A2-weight (in notationψ∈A2), if there exists a positive constantA such that, for every ball B⊂R^d,

Z

B

ψdx Z

B

ψ⁻¹dx

≤A Z

B

1dx 2

. For more on Muckenhoupt weights, we refer to [22].

Throughout. we shall assume (H1) P

k∈Z|γ_k+1−γ_k|+P

k≤0|γ_k+1−γ_k|<∞ and for allr >0there exists δr >0 such thatφ≥δr dx-a.e. onBr.

(H2) ρ φ∈A₂.

Remark 2.1. (i) (H1) implies thatφ∈L¹_loc(R^d, dx)and thatγ:= limk→∞γ_k, γ := limk→−∞γ_k exist and γ > 0, γ >0. In particular, φ is locally bounded above and locally bounded away from zero.

(ii) (H1) and (H2) imply ρ >0 dx-a.e.

(iii) Letc >0. Ifc⁻¹ ≤φ≤candρ∈A₂, or ifρ= 1andc⁻¹ψ≤φ≤c ψ for some ψ∈A2, thenρ φ∈A2.

Furthermore, we shall throughout assume the following condition.

(H3) ρ=ξ² for someξ ∈H_loc^1,2(R^d, dx).

Remark 2.2. (H3) implies that ρ ∈ H_loc^1,1(R^d, dx) and by (H1) ^k∇ρk_ρ ∈ L²_loc(R^d, ρφ dx)since φ is locally bounded above dx-a.e.

We consider the symmetric positive denite bilinear form (3) E(f, g) := 1

2 Z

R^d

∇f · ∇g (ρ φ) dx, f, g∈C₀^∞(R^d).

Since ρ φ ∈ A2, we have _{ρ φ}¹ ∈ L¹_loc(R^d, dx), and the latter implies that (3) is closable in L²(R^d, ρφ dx) (see [7, II.2 a)]). The closure (E, D(E)) of (3) is a strongly local, regular, symmetric Dirichlet form (cf. e.g. [16, p.

274]). As usual we dene E₁(f, g) :=E(f, g) + (f, g)_L2(R^d, ρφ dx) for f, g∈D(E) and kfk_D(E) := E₁(f, f)^1/2, f ∈ D(E). Let (T_t)t≥0 and (G_α)_α>0 be the L²(R^d, ρφ dx)-semigroup and resolvent associated to (E, D(E)) and (L, D(L)) be the corresponding generator (see [7, Diagram 3, p. 39]). From [15, p. 303 Proposition 2.3] and [16, p. 286 A)] we know that there exists a jointly contin- uous transition kernel densityp_t(x, y)such that

P_tf(x) :=

Z

R^d

p_t(x, y)f(y)ρ(y)φ(y)dy , t >0, x∈R^d,

f ∈ B_b(R^d), is a ρφdy-version of T_tf if f ∈ L²(R^d, ρ φ dx)∩ B_b(R^d). Further- more, taking the Laplace transform of pt(x, y) there exists a resolvent kernel densityrα(x, y) such that

Rαf(x) :=

Z

R^d

rα(x, y)f(y)ρ(y)φ(y)dy , α > 0, x∈R^d,

f ∈ B_b(R^d), is a ρφdy-version of Gαf if f ∈L²(R^d, ρ φ dx)∩ B_b(R^d). Accord- ingly, for a signed Radon measure µ, let us dene

R_αµ(x) = Z

R^d

r_α(x, y)µ(dy), α >0, x∈R^d,

whenever this makes sense. Since p_t(·,·) is jointly continuous and p_t(x, y) has exponential decay in y for xed t and x in a compact set, (P_t)t≥0 is strong

Feller, i.e. P_t(B_b(R^d)) ⊂ C_b(R^d). For details, we refer to [13]. In particular R₁(B_b(R^d))⊂C_b(R^d).

We consider further the following condition (H4) There exists a Hunt process

M= (Ω,F,(F_t)t≥0, ζ,(Xt)t≥0,(Px)_x∈R^d_∪{∆}),

with state spaceR^d∪ {∆}and life timeζ, which has(Pt)t≥0 as transition function,R₁f is continuous for anyf ∈L²(R^d, ρφdx)with compact sup- port, and ifφ6≡const., we additionally assumeR1(1Gρdσr)is continuous for any G⊂R^d relatively compact open, r >0.

In (H4), ∆is the cemetery point and as usual any functionf :R^d→Ris extended to{∆} by settingf(∆) := 0.

Remark 2.3. A Hunt process associated with (P_t)t≥0 as in (H4) can be constructed by the same methods as used in [2, section 4]. The method used in [2, section 4] is applicable, if one can nd enough nice functions inD(L)(cf.

[2, proof of Lemma 4.6]). A Hunt process with transition function(P_t)t≥0 as in (H4) can also be constructed by showing that (P_t)t≥0 denes a classical Feller semigroup. For details and concrete examples we refer to [13].

Under (H4), Msatises in particular the absolute continuity condition as stated in [5, p. 165].

Proposition 2.4. Let a Dirichlet form be given as the closure of 1

2 Z

R^d

∇f· ∇g ψdx, f, g∈C₀^∞(R^d) in L²(R^d, ψ dx) where ψ∈A2. Then it is conservative.

Proof. By [22, Proposition 1.2.7] ψ dx is volume doubling. Hence, by [6, Proposition 5.1, Proposition 5.2]

c1r^α0≤ψ dx(Br)≤c2r^α, ∀r ≥1, wherec₁, c₂, α, α0>0are constants. In particular,

Z ∞ 1

r

log ψ dx(B_r)dr=∞.

Hence, conservativeness follows by [14, Theorem 4].

Remark 2.5. Proposition 2.4 holds more generally for Ap-weights, p ∈ [1,∞)(see [22, Denitoin 1.2.2] for the denition ofA_p) by the same arguments as in the proof of Proposition 2.4.

It follows from Proposition 2.4 and the strong Feller property of (P_t)t≥0

that under (H4)

(4) Px(ζ =∞) = 1, ∀x∈R^d.

We will refer to [5] from now on till the end, hence, some of its standard notations may be adopted below without denition. The following theorem is the main result of our paper. It will be proved in section 5.

Theorem 2.6. Suppose (H1)-(H4) hold. Then:

(i) The process M satises

(5) X_t=x+W_t+

Z t 0

∇ρ

2ρ(X_s)ds+N_t, t≥0,

Px-a.s. for anyx∈R^d, whereW is a standard d-dimensional Brownian motion starting from zero and

Nt:=X

k∈Z

γk+1−γk

γ_k+1+γ_k Z t

0

ν_lk(Xs)d`^l_s^k+γ_k+1−γ_k γ_k+1+γ_k

Z t 0

νrk(Xs)d`^r_s^k

+ γ−γ γ+γ

Z t 0

ν_m0(X_s)d`^m_s ⁰,

where ν_r = (ν_r¹, . . . , ν_r^d), r > 0 is the unit outward normal vector on ∂B_r and

`<sup>lk</sup>, `^rk and`^m0 are boundary local times ofX, i.e. they are positive continuous additive functionals ofX in the strict sense associated via the Revuz correspon- dence (cf. [5, Theorem 5.1.3]) with the weighted surface measures ^γk+1₂^+γk ρ dσ_lk on ∂B_lk, ^γk+1₂^+γk ρ dσrk on ∂Brk, and ^γ+γ₂ ρ dσm0 on ∂Bm0, respectively, and related via the formulas

Ex

Z ∞ 0

e^−td`^l_t^k

= R1

γk+1+γk

2 ρdσlk

(x), Ex

Z ∞ 0

e^−td`^r_t^k

= R1

γ_k+1+γ_k 2 ρdσr_k

(x), Ex

Z ∞ 0

e^−td`^m_t ⁰

= R₁

γ+γ 2 ρdσ_m0

(x), which all hold for any x∈R^d, k∈Z.

(ii) (kX_tk)_t≥0,Px

is a continuous semimartingale for any x∈R^d and Px `<sup>a</sup><sub>t</sub> =`^a_t(kXk)

= 1, ∀x∈R^d, t≥0, a∈ {m₀, lk, rk:k∈Z}, where`^a_t(kXk)is the symmetric semimartingale local time ofkXkata∈(0,∞) as dened in [12, VI.(1.25)].

Remark 2.7. In view of Theorem 2.6, the non absolutely continuous drift componentN in (5) may be interpreted as follows. Dene

α_k:= γ_k+1

γ_k+1+γ_k, k∈Z. Then α_k ∈(0,1)and

γ_k+1−γ_k

γ_k+1+γ_k = 2α_k−1 =α_k−(1−α_k).

and so the drift component γ_k+1−γ_k γk+1+γk

Z t 0

ν_lk(X_s)d`^l_s^k(kXk)

corresponds to an outward normal reection with probabilityα_kand an inward normal reection with probability 1−αk when Xt hits ∂Bl_k (cf. [21]). Anal- ogous interpretations hold for the other reection terms. Thus, ∂B_lk, ∂Br_k

and ∂B_m0 can be seen as boundaries where a skew reection takes place, or alternatively as permeable membranes.

3. INTEGRATION BY PARTS FORMULA

Proposition 3.1. Suppose (H1)-(H3) hold. The following integration by parts formula holds for f, g∈C₀^∞(R^d)

− E(f, g) = Z

R^d

1

2∆f+∇f ·∇ρ 2ρ

g ρ φdx+γ−γ 2

Z

∂Bm0

∇f·ν_m0g ρdσ_m0

+X

k∈Z

γ_k+1−γ_k 2

Z

∂B_lk

∇f ·ν_lkg ρdσ_lk +γ_k+1−γ_k 2

Z

∂B_rk

∇f ·ν_rkg ρdσ_rk

! . The last summation is in particular only over nitely many r_k, k ≥ 1, since f has compact support.

Proof. For f, g∈C₀^∞(R^d) E(f, g) = 1

2

d

X

j=1

Z

R^d

∂_jf ∂_jg (ρ φ) dx

= −1 2

d

X

j=1

X

k∈Z

γ_k

Z

Ak

∂_jjf+∂_jf∂jρ ρ

g ρdx+γ_k Z

Aˆk

∂_jjf +∂_jf∂jρ ρ

g ρdx

+1 2

d

X

j=1

X

k∈Z

Z

Ak

γ_k∂_j(∂_jf g ρ) dx+ Z

Aˆk

γ_k∂_j(∂_jf g ρ) dx

.

The rst term equals

−1 2

Z

R^d

∆f +∇f ·∇ρ ρ

g ρ φdx, and the second term equals

1 2

d

X

j=1

X

k∈Z

Z

∂B_lk

γ_k∂_jf ν_l^j

kg ρdσ_lk− Z

∂B_lk−1

γ_k∂_jf ν_l^j

k−1g ρdσ_lk−1

!

+1 2

d

X

j=1

X

k∈Z

Z

∂B_rk

γ_k∂_jf ν_r^j

kg ρdσ_rk− Z

∂B_rk−1

γ_k∂_jf ν_r^j_k−1g ρdσ_rk−1

!

= −1 2 lim

k→−∞

Z

∂B_lk−1

γ_k∇f·ν_lk−1g ρdσ_lk−1+X

k∈Z

Z

∂B_lk

(γ_k+1−γ_k)∇f·ν_lkg ρdσ_lk

− lim

k→∞

Z

∂B_lk+1

γ_k+1∇f ·ν_lk+1g ρdσ_lk+1

!

−1 2 lim

k→−∞

Z

∂B_rk−1

γ_k∇f·ν_rk−1g ρdσ_rk−1+X

k∈Z

Z

∂B_rk

(γ_k+1−γ_k)∇f·ν_rkg ρdσ_rk

!

= −1 2

X

k∈Z

Z

∂B_lk

(γ_k+1−γ_k)∇f·ν_lkg ρdσ_lk− Z

∂Bm0

γ∇f·ν_m0g ρdσ_m0

!

−1 2

Z

∂Bm0

γ∇f·ν_m0g ρdσ_m0+X

k∈Z

Z

∂B_rk

(γ_k+1−γ_k)∇f ·ν_rkg ρdσ_rk

! , because

k→−∞lim Z

∂B_lk−1

γ_k∇f·ν_lk−1g ρdσ_lk−1 = lim

k→−∞

d

X

j=1

Z

B_lk−1

γ_k∂_j(∂_jf g ρ)dx= 0, by (H1) and Lebesgue, since ∂_j(∂_jf g ρ)∈L¹_loc(R^d). Similarly

k→∞lim Z

∂B_lk+1

γk+1∇f·νlk+1g ρdσlk+1 = lim

k→∞

d

X

j=1

Z

B_lk+1

γk+1∂j(∂jf g ρ)dx

=

d

X

j=1

Z

Bm0

γ ∂j(∂jf g ρ)dx= Z

∂Bm0

γ∇f·νm0g ρdσm0, and

k→−∞lim Z

∂B_rk−1

γ_k∇f·νrk−1g ρdσrk−1 = Z

∂Bm0

γ ∇f ·νm0g ρdσm0.

Remark 3.2. The integration by parts formula in Proposition 3.1 extends to f(x) =|||x|| −a|, a∈R,and the coordinate projections.

4. STRICT FUKUSHIMA DECOMPOSITION

A positive Radon measureµonR^dis said to be of nite energy integral if Z

R^d

|f(x)|µ(dx)≤Cp

E₁(f, f), f ∈D(E)∩C0(R^d),

whereC is some constant independent off, andC₀(R^d)is the set of compactly supported continuous functions onR^d. A positive Radon measureµonR^dis of nite energy integral if and only if there exists a unique function U1µ∈D(E) such that

E₁(U₁µ, f) = Z

R^d

f(x)µ(dx),

for allf∈D(E)∩C₀(R^d).U₁µis called1-potential ofµ. In particular,R₁µis a ρφ dx-version ofU1µ. The measures of nite energy integral are denoted byS0.

Let further

S₀₀:={µ∈S₀|µ(R^d)<∞,kU₁µk∞<∞}, wherekfk_∞:= inf{c >0| R

R^d1_{{ |f|>c}}ρφ dx= 0} and dene for l∈(0,∞) C₀^∞(B_l) :={f :B_l→R| ∃g∈C₀^∞(R^d)withf =gonB_l}.

Lemma 4.1. Suppose (H1)(H3) hold. Then forl∈(0,∞)andf∈C₀^∞(B_l), Z

∂Bl

|f|ρdσ_l≤C(l) Z

Bl

k∇fk²ρ φdx+ Z

Bl

|f|²ρ φdx 1/2

,

whereC : (0,∞)→Ris an increasing function. In particular, for anyf ∈D(E) Z

∂Bl

|f|ρdσ_l≤C(l)kfk_D(E).

Proof. Since∂Blhas Lipschitz boundary (actuallyC^∞-boundary), we can see from [3, Theorem 1 in section 4.3] (and [19, Theorem 5.1 (i)]) that there exists a constant√

2 independent ofl, such that for f ∈C₀^∞(B_l) Z

∂B_l

|f|ρ dσ_l≤√ 2

Z

B_l

k ∇fkρ + 2|ξ f| k∇ξk+|f|ρ dx

≤√ 2

"

Z

Bl

k∇fk²ρdx 1/2

ρdx(Bl)1/2

+ 2 Z

Bl

|f|²ρdx 1/2Z

Bl

k ∇ξk²dx 1/2

+ Z

Bl

|f|²ρdx 1/2

ρdx(B_l)1/2

#

≤ r8

δ_l

ρdx(Bl)1/2

+||∇ξ||_L2(Bl,dx)

Z

B_l

k∇fk²ρ φdx+ Z

B_l

|f|²ρ φdx 1/2

. This is the rst statement. Let f ∈C₀^∞(R^d). Then, by the above, since f restricted to B_l is inC₀^∞(B_l),

Z

∂Bl

|f|ρdσl≤C(l) Z

Bl

k∇fk²ρ φdx+ Z

Bl

|f|²ρ φdx 1/2

≤C(l)||f||_D(E). Since C₀^∞(R^d)is dense in D(E), the second statement follows.

A positive Borel measure µonR^d is said to be smooth in the strict sense if there exists a sequence (E_k)k≥1 of Borel sets increasing to R^d such that 1_Ek·µ∈S₀₀ for eachk and

Px( lim

k→∞σ_Rd\E_k ≥ζ) = 1, ∀x∈R^d.

HereσB:= inf{t >0|Xt∈B}forB ∈ B(R^d). The totality of the smooth measures in the strict sense is denoted by S₁ (see [5]).

Lemma 4.2. Suppose (H1)-(H4) hold. Let l ∈ (0,∞). Then, for any relatively compact open set G, 1_G·ρdσ_l∈S₀₀. In particular, ρdσ_l ∈S₁.

Proof. Let l∈(0,∞). By Lemma4.1, ρ dσ_l∈S₀. Let us rst show that ρ dσl∈S1 with respect to an increasing sequence of open sets(Ek)k≥1. Choose ϕ, ϕ∈L¹_b(R^d, ρφ dx),0< ϕ, ϕ≤1 ρφ dx-a.e. By assumption (H4)R₁(ρdσ_l)is continuous. Since furthermore R₁ϕis continuous and R₁ϕ is strictly positive, it follows that

E_k:={R1(ρdσ_l)< k²R1ϕ}, k≥1,

are open sets that increase to R^d. Choosing the constant function1∈C₀^∞(Bl) in Lemma 4.1 we see that ρdσ_l is nite. Then, clearly 1E_k·ρdσ_l is also nite for all k ≥ 1. So, it remains to show that the corresponding 1-potentials U1(1Ek ·ρdσl) are ρφ dx-essentially bounded. Let (G1ϕ)_E

k be the 1-reduced function of G1ϕ on E_k as dened in [20]. Then E·

hR∞

σ_Ek e^−tϕ(Xt)dt i is a ρφ dx-version of(G₁ϕ)_E

k. We have (for intermediate steps see [20, p. 416]) Z

R^d

ϕU₁(1_Ek ·ρdσ_l)ρφdx= Z

∂Bl

R₁ϕ 1_Ek·ρdσ_l

= Z

∂Bl

E·

"

Z ∞ σ_Ek

e^−tϕ(Xt)dt

#

1Ek·ρdσl=E₁

(G1ϕ)_E

k, U1(1Ek·ρdσl)

=E₁

(G1ϕ)_E

k, U1(1_Ek·ρdσ_l)∧k²G1ϕ

≤ E₁ G1ϕ, U1(1E_k ·ρdσ_l)∧k²G1ϕ

= Z

R^d

ϕ U1(1E_k·ρdσ_l)∧k²G1ϕ ρφdx.

This implies that U₁(1_Ek·ρdσ_l) ≤k² ρφ dx-a.e. Hence, 1_Ek ·ρdσ_l ∈S₀₀ for all k ≥ 1. Since moreover Px(limk→∞σ_Rd\E_k ≥ ζ) = 1, ∀x ∈R^d is easily deduced from the absolute continuity condition, we nally obtain ρdσl ∈ S1

with respect to (E_k)k≥1.

For a relatively compact open set G, we know that there exists k₀ ∈ N with G ⊂G ⊂ Ek0. Hence, U1(1G·ρ dσl) ≤U1(1Ek0 ·ρ dσl) ≤k²₀G1ϕ ≤k₀². Therefore,1_G·ρdσ_l ∈S₀₀.

By Lemma4.2, we know thatρdσ_r∈S₁ for anyr ∈(0,∞). Hence, by [5, Theorem 5.1.7] there exists a unique(¯`^r_t)t≥0 ∈A⁺_c,1 with Revuz measureρdσr, such that

Ex

Z ∞ 0

e^−td`¯^r_t

=R1(ρdσr)(x), ∀x∈R^d.

Here,A⁺_c,1denotes the positive continuous additive functionals in the strict sense.

Theorem 4.3. Suppose (H1)(H4) hold. Then, for any relatively compact open set G, 1G·µ∈S00−S00, where

µ=X

k∈Z

γk+1−γk

2 ρ dσl_k+γ_k+1−γ_k 2 ρ dσrk

+γ−γ

2 ρ dσm0. In particular µ∈S1−S1.

Proof. It suces to show that µ_n∈S₀₀ for any n∈N, n > m₀, where µ_n:= 1_G·



 X

k∈Z

|γ_k+1−γ_k|ρ dσ_lk + X

{k∈Z:rk<n}

|γ_k+1−γ_k|ρ dσ_rk+|γ−γ|ρ dσ_m0



. First we show that µ_n∈S₀. Let f ∈C₀^∞(R^d). Then, by Lemma4.1 Z

R^d

|f|dµ_n≤C(n)



 X

k∈Z

|γ_k+1−γ_k|+ X

{k∈Z:rk<n}

|γ_k+1−γ_k|+|γ−γ|



||f||_D(E), where C(n) is as in Lemma 4.1. Now, we show µ_n ∈ S₀₀. Let f ∈ C₀^∞(B_n) such that f = 1 on Bn− where > 0 is small enough to satisfy (n−) >

max_{k∈Z:rk<n}r_k. Then, by Lemma 4.1 again µ_n(R^d)≤C(n)



 X

k∈Z

|γ_k+1−γ_k|+ X

{k∈Z:rk<n}

|γ_k+1−γ_k|+|γ−γ|





× Z

Bn

ρφdx 1/2

<∞.

By the proof of Lemma 4.2, we can see that U₁(1_Ek ·ρ dσ_l)≤k², k≥1,

independently ofl∈(0,∞). For any relatively compact open setG, there exists k₀∈Nsuch thatG⊂G⊂E_k0. SinceU₁(1_G·ρ dσ_l)≤U₁(1_Ek

0 ·ρ dσ_l) for any l, we obtain for ρφ dx-a.e. x∈R^d

U1µn(x)≤ X

k∈Z

|γk+1−γk|U1(1G·ρ dσl_k) (x)

+ X

{k∈Z:rk<n}

|γ_k+1−γ_k|U₁(1_G·ρ dσ_rk) (x) +|γ−γ|U₁(1_G·ρ dσ_m0) (x)

≤ k₀²



 X

k∈Z

|γ_k+1−γ_k|+ X

{k∈Z:rk<n}

|γ_k+1−γ_k|+|γ−γ|



<∞.

Therefore, µ∈S00.

Remark 4.4. Let E_k, k≥1, be open sets such thatE_k% R^d and let µ= µA, µn =µAⁿ ∈ S1 w.r.t. (Ek)k≥1, A, Aⁿ ∈ A⁺_c,1, n≥ 1. If µA =P

n≥1µAⁿ, thenA=P

n≥1Aⁿ, sinceR₁(f dµ_A)(x) =P

n≥1R₁(f dµ_Aⁿ)(x) for anyx∈R^d, f ∈C₀(R^d).

Theorem 4.5. Suppose (H1)(H4) hold. For any relatively compact open setG and j= 1, . . . , d,

1_G·∂_jρ φ dx∈S₀₀−S₀₀. In particular ∂_jρ φ dx∈S₁−S₁,j = 1, . . . , d.

Proof. It suces to show that 1_G· |∂_jρ|φ dx∈S00. By Remark 2.2, it is easy to see that 1_G· |∂_jρ|φ dx ∈S₀ and that1_G· |∂_jρ|φ dx(R^d)<∞. We can show that

kU₁(1_G· |∂_jρ|φ dx)k_∞<∞

by replacing theEkin the proof of Lemma4.2withE_k⁰ :={R1(1G·|∂_jρ|φ dx)<

k²R₁ϕ}.

5. PROOF OF THE MAIN THEOREM

Proof of Theorem 2.6. (i) Applying [5, Theorem 5.5.5] to (E, D(E)) and to the coordinate projections which are in D(E)_b,loc, the identication of the martingale part as Brownian motion is easy. Concerning the drift part the strict decomposition holds true by Lemma 4.2, Remark 3.2, Theorem 4.3, Remark 4.4 and Theorem 4.5. Note that equation (5) holds for allt≥0 by (4).

(ii) Let f(x) :=kxk,x∈R^d. Then∂jf is everywhere bounded by one (except in zero). Thus, applying [5, Theorem 5.5.5] to f, which is in D(E)_b,loc, we obtain similarly to (i)

(6) kX_tk=kxk+B_t+ Z t

0

X_s kX_sk·∇ρ

2ρ(X_s)ds+N_t,

Px-a.s. for anyx∈R^d,t≥0, whereBis a standard one dimensional Brownian motion starting from zero and

Nt:=X

k∈Z

γk+1−γk

γ_k+1+γ_k `<sup>l</sup><sub>t</sub><sup>k</sup>+γ<sub>k+1</sub>−γ<sub>k</sub> γ<sub>k+1</sub>+γ<sub>k</sub> `^r_t^k

+γ−γ γ+γ `^m_t ⁰.

Therefore, the rst statement follows. In particular, we may apply the symmetric Ito-Tanaka formula (see [12, VI. (1.25)]) and obtain

(7)

kX_tk −a =

kxk −a +

Z t 0

sign(kX_sk −a)dkX_sk+`^a_t(kXk),

Px -a.s. for anyx∈R^d,t≥0, wheresignis the point symmetric sign function.

Leth(x) :=

kxk −a

,a∈ {m₀, l_k, r_k:k∈Z}, x∈R^d. Then∂_jh is everywhere bounded by one (except in a). Thus, applying [5, Theorem 5.5.5] toh, which is inD(E)_b,loc, we obtain again similarly to (i)

(8)

kX_tk −a =

kxk −a +

Z t 0

sign(kX_sk −a)dkX_sk+`^a_t,

Px -a.s. for any x∈R^d,t≥0. Comparing (7) and (8), we get the result.

Remark 5.1. (see [5, Theorem 4.7.1 (i), (iii), and Exercise 4.7.1]) If(E,D(E)) is irreducible, then for any nearly Borel non-exceptional set B,

Px(σ_B <∞)>0, ∀x∈R^d.

If (E, D(E)) is irreducible and recurrent, then for any nearly Borel non- exceptional set B,

Px(σ_B◦θ_n<∞,∀n≥0) = 1, ∀x∈R^d.

Here (θ_t)t≥0 is the shift operator. Moreover, in this case any excessive function is constant. In particular, the ergodic Theorem [5, Theorem 4.7.3 (iv)]

holds. A sucient condition for recurrence is given by Z ∞

1

r

ρφ dx(B_r)dr =∞, see [14, Theorem 3].

Acknowledgments. The research of Jiyong Shin and of the corresponding author Gerald Trutnau was supported by Basic Science Research Program through the Na- tional Research Foundation of Korea(NRF) funded by the Ministry of Education, Sci- ence and Technology (MEST) (2013028029) and Seoul National University Research Grant 0450-20110022.

REFERENCES

[1] S. Albeverio, R. Høegh-Krohn and L. Streit, Energy forms, Hamiltonians, and distorted Brownian paths. J. Math. Phys. 18 (1977), 5, 907917.

[2] S. Albeverio, Yu. G. Kondratiev and M. Rockner, Strong Feller properties for distorted Brownian motion and applications to nite particle systems with singular interactions . Finite and innite dimensional analysis in honor of Leonard Gross (New Orleans, LA, 2001), pp. 1535. Contemp. Math. 317, Amer. Math. Soc., Providence, RI, 2003.

[3] L.C. Evans and R.F. Gariepy, Measure theory and ne properties of functions. Stud.

Adv. Math., CRC Press, Boca Raton, 1992.

[4] M. Fukushima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motions. New stochastic methods in physics. Phys. Rep. 77 (1981), 3, 255262.

[5] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov pro- cesses. Second revised and extended edition. De Gruyter Stud. Math. 19, Walter de Gruyter, Berlin, 2011.

[6] A. Grigor'yan and J. Hu, Upper bounds of heat kernels on doubling spaces, Preprint, to appear in Mosc. Math. J., 57 pages (2010).

[7] Z.M. Ma and M. Rockner, Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin, 1992.

[8] Y. Oshima, On stochastic dierential equations characterizing some singular diusion processes. Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 3, 151154.

[9] Y. Oshima, Some singular diusion processes and their associated stochastic dierential equations. Z. Wahrsch. Verw. Gebiete 59 (1982), 2, 249276.

[10] Y. Ouknine, F. Russo and G. Trutnau, On countably skewed Brownian motion with accumulation point. arXiv:1308.0441.

[11] N.I. Portenko, Generalized Diusion Processes. Naukova Dumka, Kiev, 1982; Translated from the Russian, Trans. Math. Monogr. 83. Amer. Math. Soc. Providence, RI, 1990.

[12] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer Verlag, 2005.

[13] J. Shin, G. Trutnau, On the stochastic regularity of distorted Brownian motions . Preprint.

[14] K.T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456 (1994), 173196.

[15] K.T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995), 2, 275312.

[16] K.T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality . J. Math. Pures Appl. (9) 75 (1996), 3, 273297.

[17] S. Takanobu, On the existence of solutions of stochastic dierential equations with sin- gular drifts. Probab. Theory Related Fields 74 (1987), 2, 295315.

[18] M. Tomisaki, A construction of diusion processes with singular product measures . Z. Wahrsch. Verw. Gebiete 53 (1980), 1, 5170.

[19] G. Trutnau, Skorokhod decomposition of reected diusions on bounded Lipschitz do- mains with singular non-reection part. Probab. Theory Related Fields 127 (2003), 4, 455495.

[20] G. Trutnau, On a class of non-symmetric diusions containing fully nonsymmetric dis- torted Brownian motions. Forum Math. 15 (2003), 3, 409437.

[21] G. Trutnau, Multidimensional skew reected diusions. Stochastic analysis: classical and quantum, World Sci. Publ., Hackensack, NJ, 228244, 2005.

[22] B.O. Turesson, Nonlinear potential theory and weighted Sobolev spaces . Lecture Notes in Math. 1736. Springer, 2000.

Received 17 July 2013 Seoul National University, Department of Mathematical Sciences and Research Institute of Mathematics,

599 Gwanak-Ro, Gwanak-Gu, Seoul 151-747, South Korea