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

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1 Dedicated to Professor Nicu Boboc on the occasion of his 8 th birthday 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 equation of distorted Brownian motion with respect to a certain discontinuous Muckenhoupt A -weight ψ. The discontinuities of ψ typically take place on a sequence of level sets of the Euclidean norm D k := {x R d x = d k }, k Z, where d k k Z, may have accumulation points and each level set D k plays the role of a permeable membrane. AMS 1 Subject Classication: 6J6, 6J55, 31C5, 31C15, 35J5. 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 X t = x + W t + t ρ ρ X s ds + t ν a X s dl a s X ηda, x R d, where W is a d-dimensional standard Brownian motion, ρ is typically a Muckenhoupt A -weight, ν a is the unit outward normal on the boundary of the Euclidean ball of radius a about zero, l a X is the symmetric semimartingale local time at a, of X, η = k Z α k 1δ dk with α k k Z, 1 is a sum of Dirac measures at a sequence d k k Z, with exactly two accumulation points in [,, one is zero and the other is m >. More accumulation points could be allowed. For a discussion on this point we refer to [1, Remark.6ii]. The absolutely continuous component of drift ρ ρ is typically unbounded and discontinuous. For an interpretation of the equation we refer to Theorem.6 and Remark.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 , 1, 15717

2 158 Jiyong Shin and Gerald Trutnau [8, 9, 17, 18]. In particular, 1 is a multidimensional analogue of an equation that was thoroughly studied in [1] and both equations share a lot of similarities. For instance the way to determine γ k k Z and γ k k Z in below, when α k k Z in 1 is given, is the same here as in [1, Proposition.11]. The way to obtain α k k Z from γ k k Z and γ k k Z is described in Remark.7 cf. also [1, Remark.4ii and Theorem.5]. See further [1, Remark.7] for another similarity and [1, 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, and ρ is typically a Muckenhoupt A -weight see Remark.1iii below. For the precise conditions we refer to H1H3 in Section. The logarithmic derivative, which is the drift of the distorted Brownian motion, is then informally given by dρφ ρφ = dρ ρ + dφ φ cf. [1, Remark.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 for ways to construct such a process, we refer to [13], see also Remark.3. Under the conditions H1H4, 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.6. Finally one could think of generalizing 1, or more precisely the process of Theorem.6i 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 [1, Section 5]. In case of skew reection at the boundary of a single C 1,λ -domain, λ, 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.6.

3 3 Pointwise weak existence of distorted skew Brownian motion 159. FORMULATION OF THE MAIN THEOREM Let m, and l k k Z, m, < l k < l k+1 < m, be a sequence converging to as k and converging to m as k. Let r k k Z m,, m < r k < r k+1 <, be a sequence converging to m as k and tending to innity as k. Let φ := γ k 1 Ak + γ k 1Âk, k Z where γ k, γ k,, A k := B lk \ B lk 1, Â k := B rk \ B rk 1, B r := {x R d x < r} for any r >, B r denotes the closure of B r, 1 A the indicator function of a set A, and x the Euclidean norm of x R d. We denote by dσ r the surface measure on the boundary B r of B r, r >. Let d 3, and let C Rd denote the set of all innitely dierentiable functions with compact support in R d. Let f := 1 f,..., d f where j f is the j-th partial derivative of f, and f := d j=1 jjf and jj f := j j f, 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 Rd, µ := {f f 1 U 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 functions f L q Ω, dx such that j f L q Ω, dx, j = 1,..., d, and H 1,q loc Rd, 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 BX we denote the set of all BX-measurable f : X R which are bounded, or nonnegative by B b X, B + X respectively. We further denote the set of continuous functions on X, the set of continuous bounded functions on X by CX, C b X respectively. A function ψ BR d with ψ > dx-a.e. is said to be a Muckenhoupt A -weight in notation ψ A, if there exists a positive constant A such that, for every ball B R d, B ψdx ψ 1 dx A 1 dx. B B For more on Muckenhoupt weights, we refer to []. Throughout. we shall assume H1 k Z γ k+1 γ k + k γ k+1 γ k < and for all r > there exists δ r > such that φ δ r dx-a.e. on B r. H ρ φ A.

4 16 Jiyong Shin and Gerald Trutnau 4 Remark.1. i H1 implies that φ L 1 loc Rd, dx and that γ :=lim k γ k, γ := lim k γ k exist and γ >, γ >. In particular, φ is locally bounded above and locally bounded away from zero. ii H1 and H imply ρ > dx-a.e. iii Let c >. If c 1 φ c and ρ A, or if ρ = 1 and c 1 ψ φ c ψ for some ψ A, then ρ φ A. Furthermore, we shall throughout assume the following condition. H3 ρ = ξ for some ξ H 1, loc Rd, dx. Remark.. H3 implies that ρ H 1,1 loc Rd, dx and by H1 ρ ρ L loc Rd, ρφ dx since φ is locally bounded above dx-a.e. We consider the symmetric positive denite bilinear form 3 Ef, g := 1 f g ρ φ dx, f, g C R d. R d Since ρ φ A, we have 1 ρ φ L1 loc Rd, dx, and the latter implies that 3 is closable in L R d, ρφ dx see [7, II. a]. The closure E, DE of 3 is a strongly local, regular, symmetric Dirichlet form cf. e.g. [16, p. 74]. As usual we dene E 1 f, g := Ef, g + f, g L R d, ρφ dx for f, g DE and f DE := E 1 f, f 1/, f DE. Let T t t and G α α> be the L R d, ρφ dx-semigroup and resolvent associated to E, DE and L, DL be the corresponding generator see [7, Diagram 3, p. 39]. From [15, p. 33 Proposition.3] and [16, p. 86 A] we know that there exists a jointly continuous transition kernel density p t x, y such that P t fx := p t x, y fy ρy φy dy, t >, x R d, R d f B b R d, is a ρφdy-version of T t f if f L R d, ρ φ dx B b R d. Furthermore, taking the Laplace transform of p t x, y there exists a resolvent kernel density r α x, y such that R α fx := r α x, y fy ρy φy dy, α >, x R d, R d f B b R d, is a ρφdy-version of G α f if f L R d, ρ φ dx B b R d. Accordingly, for a signed Radon measure µ, let us dene R α µx = r α x, y µdy, α >, x R d, 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 is strong

5 5 Pointwise weak existence of distorted skew Brownian motion 161 Feller, i.e. P t B b R d C b R d. For details, we refer to [13]. In particular R 1 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, ζ, X t t, P x x R d { }, with state space R d { } and life time ζ, which has P t t as transition function, R 1 f is continuous for any f L R d, ρφdx with compact support, and if φ const., we additionally assume R 1 1 G ρdσ r is continuous for any G R d relatively compact open, r >. In H4, is the cemetery point and as usual any function f : R d R is extended to { } by setting f :=. Remark.3. A Hunt process associated with P t t as in H4 can be constructed by the same methods as used in [, section 4]. The method used in [, section 4] is applicable, if one can nd enough nice functions in DL cf. [, proof of Lemma 4.6]. A Hunt process with transition function P t t as in H4 can also be constructed by showing that P t t denes a classical Feller semigroup. For details and concrete examples we refer to [13]. Under H4, M satises in particular the absolute continuity condition as stated in [5, p. 165]. Proposition.4. Let a Dirichlet form be given as the closure of 1 f g ψ dx, f, g C R d R d in L R d, ψ dx where ψ A. Then it is conservative. Proof. By [, Proposition 1..7] ψ dx is volume doubling. Hence, by [6, Proposition 5.1, Proposition 5.] c 1 r α ψ dxb r c r α, r 1, where c 1, c, α, α > are constants. In particular, 1 r log dr =. ψ dxb r Hence, conservativeness follows by [14, Theorem 4]. Remark.5. Proposition.4 holds more generally for A p -weights, p [1, see [, Denitoin 1..] for the denition of A p by the same arguments as in the proof of Proposition.4.

6 16 Jiyong Shin and Gerald Trutnau 6 It follows from Proposition.4 and the strong Feller property of P t t that under H4 4 P x 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.6. Suppose H1-H4 hold. Then: i The process M satises 5 X t = x + W t + t ρ ρ X s ds + N t, t, P x -a.s. for any x R d, where W is a standard d-dimensional Brownian motion starting from zero and N t := k Z γk+1 γ k γ k+1 + γ k t + γ γ γ + γ ν lk X s dl l k s t + γ k+1 γ k γ k+1 + γ k ν m X s dl m s, t ν rk X s dl r k s where ν r = ν 1 r,..., ν d r, r > is the unit outward normal vector on B r and l l k, l r k and l m are boundary local times of X, i.e. they are positive continuous additive functionals of X in the strict sense associated via the Revuz correspondence cf. [5, Theorem 5.1.3] with the weighted surface measures γ k+1+γ k ρ dσ lk on B lk, γ k+1 +γ k ρ dσ rk on B rk, and γ+γ ρ dσ m on B m, respectively, and related via the formulas E x [ E x [ E x [ ] e t dl l k t ] e t dl r k t e t dl m t which all hold for any x R d, k Z. ] = R 1 γk+1 + γ k ρdσ lk x, γk+1 + γ = R k 1 ρdσ rk x, γ + γ = R 1 ρdσ m x, ii X t t, P x is a continuous semimartingale for any x R d and P x l a t = l a t X = 1, x R d, t, a {m, l k, r k : k Z}, where l a t X is the symmetric semimartingale local time of X at a, as dened in [1, VI.1.5].

7 7 Pointwise weak existence of distorted skew Brownian motion 163 Remark.7. In view of Theorem.6, the non absolutely continuous drift component N in 5 may be interpreted as follows. Dene Then α k, 1 and and so the drift component α k := γ k+1 γ k+1 + γ k, k Z. γ k+1 γ k γ k+1 + γ k = α k 1 = α k 1 α k. γ k+1 γ k γ k+1 + γ k t ν lk X s dl l k s X corresponds to an outward normal reection with probability α k and an inward normal reection with probability 1 α k when X t hits B lk cf. [1]. Analogous interpretations hold for the other reection terms. Thus, B lk, B rk and B m 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 C Rd 1 ρ Ef, g = f + f g ρ φ dx + γ γ f ν m g ρ dσ m R d ρ B m + γ k+1 γ k f ν lk g ρ dσ lk + γ k+1 γ k f ν rk g ρ dσ rk. k Z B lk B rk The last summation is in particular only over nitely many r k, k 1, since f has compact support. Proof. For f, g C Rd Ef, g = 1 d j f j g ρ φ dx j=1 R d = 1 d γ k jj f + j f jρ g ρ dx + γ ρ k + 1 j=1 k Z j=1 k Z A k d γ k j j f g ρ dx + A k  k  k jj f + j f jρ g ρ dx ρ γ k j j f g ρ dx.

8 164 Jiyong Shin and Gerald Trutnau 8 The rst term equals 1 R d f + f ρ g ρ φ dx, ρ and the second term equals 1 d γ k j f ν j l k g ρ dσ lk γ k j f ν j l k 1 g ρ dσ lk 1 j=1 k Z B lk B lk d γ k j f νr j k g ρ dσ rk γ k j f νr j k 1 g ρ dσ rk 1 B rk B rk 1 = 1 j=1 k Z lim k lim k B lk 1 γ k f ν lk 1 g ρ dσ lk 1 + k Z B lk γ k+1 γ k f ν lk g ρ dσ lk γ k+1 f ν lk+1 g ρ dσ lk+1 B lk+1 1 lim γ k f ν rk 1 g ρ dσ rk 1 + γ k+1 γ k f ν rk g ρ dσ rk k B rk 1 k Z B rk = 1 γ k+1 γ k f ν lk g ρ dσ lk γ f ν m g ρ dσ m k Z B lk B m 1 γ f ν m g ρ dσ m + γ k+1 γ k f ν rk g ρ dσ rk, B m k Z B rk because lim k B lk 1 γ k f ν lk 1 g ρ dσ lk 1 = lim d k j=1 B lk 1 γ k j j f g ρ dx =, by H1 and Lebesgue, since j j f g ρ L 1 loc Rd. Similarly d lim γ k+1 f ν lk+1 g ρ dσ lk+1 = lim γ k+1 j j f g ρ dx k B lk+1 k B lk+1 and = j=1 lim k j=1 d γ j j f g ρ dx = B m γ f ν m g ρ dσ m, B m γ k f ν rk 1 g ρ dσ rk 1 = γ f ν m g ρ dσ m. B rk 1 B m

9 9 Pointwise weak existence of distorted skew Brownian motion 165 Remark 3.. The integration by parts formula in Proposition 3.1 extends to fx = x a, a R, and the coordinate projections. 4. STRICT FUKUSHIMA DECOMPOSITION A positive Radon measure µ on R d is said to be of nite energy integral if R d fx µdx C E 1 f, f, f DE C R d, where C is some constant independent of f, and C R d is the set of compactly supported continuous functions on R d. A positive Radon measure µ on R d is of nite energy integral if and only if there exists a unique function U 1 µ DE such that E 1 U 1 µ, f = fx µdx, R d for all f DE C R d. U 1 µ is called 1-potential of µ. In particular, R 1 µ is a ρφ dx-version of U 1 µ. The measures of nite energy integral are denoted by S. Let further S := {µ S µr d <, U 1 µ < }, where f := inf{c > R d 1 { f >c } ρφ dx = } and dene for l, C B l := { f : B l R g C R d with f = g on B l }. Lemma 4.1. Suppose H1H3 hold. Then for l, and f C B l, 1/ f ρ dσ l Cl f ρ φ dx + f ρ φ dx, B l B l B l where C :, R is an increasing function. In particular, for any f DE B l f ρ dσ l Cl f DE. Proof. Since B l has Lipschitz boundary actually C -boundary, we can see from [3, Theorem 1 in section 4.3] and [19, Theorem 5.1 i] that there exists a constant independent of l, such that for f C B l f ρ dσ l f ρ + ξ f ξ + f ρ dx B l B l [ 1/ f ρ dx ρ dxbl 1/ 1/ 1/ + f ρ dx ξ dx B l B l B l 1/ + f ρ dx ρ dxbl 1/] B l

10 166 Jiyong Shin and Gerald Trutnau 1 8 ρ dxbl 1/ 1/ + ξ L δ B l,dx f ρ φ dx + f ρ φ dx. l B l B l This is the rst statement. Let f C Rd. Then, by the above, since f restricted to B l is in C B l, 1/ f ρ dσ l Cl f ρ φ dx + f ρ φ dx Cl f DE. B l B l B l Since C Rd is dense in DE, the second statement follows. A positive Borel measure µ on R 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 S for each k and P x lim k σ R d \E k ζ = 1, x R d. Here σ B := inf{t > X t B} for B BR d. The totality of the smooth measures in the strict sense is denoted by S 1 see [5]. Lemma 4.. Suppose H1-H4 hold. Let l,. Then, for any relatively compact open set G, 1 G ρdσ l S. In particular, ρdσ l S 1. Proof. Let l,. By Lemma 4.1, ρ dσ l S. Let us rst show that ρ dσ l S 1 with respect to an increasing sequence of open sets E k k 1. Choose ϕ, ϕ L 1 b Rd, ρφ dx, < ϕ, ϕ 1 ρφ dx-a.e. By assumption H4 R 1 ρdσ l is continuous. Since furthermore R 1 ϕ is continuous and R 1 ϕ is strictly positive, it follows that E k := { R 1 ρdσ l < k R 1 ϕ }, k 1, are open sets that increase to R d. Choosing the constant function 1 C B l in Lemma 4.1 we see that ρdσ l is nite. Then, clearly 1 Ek ρdσ l is also nite for all k 1. So, it remains to show that the corresponding 1-potentials U 1 1 Ek ρdσ l are ρφ dx-essentially bounded. Let G 1 ϕ Ek be the 1-reduced [ ] function of G 1 ϕ on E k as dened in []. Then E σ Ek e t ϕx t dt is a ρφ dx-version of G 1 ϕ Ek. We have for intermediate steps see [, p. 416] ϕu 1 1 Ek ρdσ l ρφ dx = R 1 ϕ 1 Ek ρdσ l R d B l [ ] = E e t ϕx t dt 1 Ek ρdσ l = E 1 G 1 ϕ Ek, U 1 1 Ek ρdσ l B l σ Ek = E 1 G 1 ϕ Ek, U 1 1 Ek ρdσ l k G 1 ϕ E 1 G1 ϕ, U 1 1 Ek ρdσ l k G 1 ϕ = ϕ U 1 1 Ek ρdσ l k G 1 ϕ ρφ dx. R d

11 11 Pointwise weak existence of distorted skew Brownian motion 167 This implies that U 1 1 Ek ρdσ l k ρφ dx-a.e. Hence, 1 Ek ρdσ l S for all k 1. Since moreover P x lim k σ R d \E k ζ = 1, x R d is easily deduced from the absolute continuity condition, we nally obtain ρdσ l S 1 with respect to E k k 1. For a relatively compact open set G, we know that there exists k N with G G E k. Hence, U 1 1 G ρ dσ l U 1 1 Ek ρ dσ l k G 1ϕ k. Therefore, 1 G ρdσ l S. By Lemma 4., we know that ρdσ r S 1 for any r,. Hence, by [5, Theorem 5.1.7] there exists a unique l r t t A + c,1 with Revuz measure ρdσ r, such that [ ] E x e t d l r t = R 1 ρdσ r x, x R d. Here, A + c,1 denotes the positive continuous additive functionals in the strict sense. Theorem 4.3. Suppose H1H4 hold. Then, for any relatively compact open set G, 1 G µ S S, where µ = γk+1 γ k ρ dσ lk + γ k+1 γ k k Z In particular µ S 1 S 1. ρ dσ rk + γ γ ρ dσ m. Proof. It suces to show that µ n S for any n N, n > m, where µ n :=1 G γ k+1 γ k ρ dσ lk + γ k+1 γ k ρ dσ rk + γ γ ρ dσ m. k Z {k Z:r k <n} First we show that µ n S. Let f C Rd. Then, by Lemma 4.1 f dµ n Cn γ k+1 γ k + γ k+1 γ k + γ γ f DE, R d k Z {k Z:r k <n} where Cn is as in Lemma 4.1. Now, we show µ n S. Let f C B n such that f = 1 on B n ɛ where ɛ > is small enough to satisfy n ɛ > max {k Z:rk <n}r k. Then, by Lemma 4.1 again µ n R d Cn γ k+1 γ k + γ k+1 γ k + γ γ k Z {k Z:r k <n} 1/ ρφ dx <. B n By the proof of Lemma 4., we can see that U 1 1 Ek ρ dσ l k, k 1,

12 168 Jiyong Shin and Gerald Trutnau 1 independently of l,. For any relatively compact open set G, there exists k N such that G G E k. Since U 1 1 G ρ dσ l U 1 1 Ek ρ dσ l for any l, we obtain for ρφ dx-a.e. x R d U 1 µ n x k Z + {k Z:r k <n} k γ k+1 γ k U 1 1 G ρ dσ lk x Therefore, µ S. γ k+1 γ k U 1 1 G ρ dσ rk x + γ γ U 1 1 G ρ dσ m x γ k+1 γ k + k Z {k Z:r k <n} γ k+1 γ k + γ γ <. Remark 4.4. Let E k, k 1, be open sets such that E k R d and let µ = µ A, µ n = µ A n S 1 w.r.t. E k k 1, A, A n A + c,1, n 1. If µ A = n 1 µ A n, then A = n 1 An, since R 1 fdµ A x = n 1 R 1fdµ A nx for any x R d, f C R d. Theorem 4.5. Suppose H1H4 hold. For any relatively compact open set G and j = 1,..., d, 1 G j ρ φ dx S S. In particular j ρ φ dx S 1 S 1, j = 1,..., d. Proof. It suces to show that 1 G j ρ φ dx S. By Remark., it is easy to see that 1 G j ρ φ dx S and that 1 G j ρ φ dxr d <. We can show that U 1 1 G j ρ φ dx < by replacing the E k in the proof of Lemma 4. with E k := { R 11 G j ρ φ dx < k R 1 ϕ }. 5. PROOF OF THE MAIN THEOREM Proof of Theorem.6. i Applying [5, Theorem 5.5.5] to E, DE and to the coordinate projections which are in DE 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., Remark 3., Theorem 4.3, Remark 4.4 and Theorem 4.5. Note that equation 5 holds for all t by 4. ii Let fx := x, x R d. Then j f is everywhere bounded by one except in zero. Thus, applying [5, Theorem 5.5.5] to f, which is in DE b,loc, we obtain similarly to i 6 X t = x + B t + t X s X s ρ ρ X s ds + N t,

13 13 Pointwise weak existence of distorted skew Brownian motion 169 P x -a.s. for any x R d, t, where B is a standard one dimensional Brownian motion starting from zero and N t := γk+1 γ k l l k γ k+1 + γ t + γ k+1 γ k l r k k γ k Z k+1 + γ t + γ γ k γ + γ lm t. Therefore, the rst statement follows. In particular, we may apply the symmetric It o-tanaka formula see [1, VI. 1.5] and obtain 7 X t a = x a t + sign X s ad X s + l a t X, P x -a.s. for any x R d, t, where sign is the point symmetric sign function. Let hx := x a, a {m, l k, r k : k Z}, x R d. Then j h is everywhere bounded by one except in a. Thus, applying [5, Theorem 5.5.5] to h, which is in DE b,loc, we obtain again similarly to i 8 X t a = x a t + sign X s ad X s + l a t, P x -a.s. for any x R d, t. Comparing 7 and 8, we get the result. Remark 5.1. see [5, Theorem i, iii, and Exercise 4.7.1] If E,DE is irreducible, then for any nearly Borel non-exceptional set B, P x σ B < >, x R d. If E, DE is irreducible and recurrent, then for any nearly Borel nonexceptional set B, Px σ B θ n <, n = 1, x R d. Here θ t t is the shift operator. Moreover, in this case any excessive function is constant. In particular, the ergodic Theorem [5, Theorem iv] holds. A sucient condition for recurrence is given by r dr =, 1 ρφ dxb r 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 National Research Foundation of KoreaNRF funded by the Ministry of Education, Science and Technology MEST 1389 and Seoul National University Research Grant REFERENCES [1] S. Albeverio, R. Høegh-Krohn and L. Streit, Energy forms, Hamiltonians, and distorted Brownian paths. J. Math. Phys , 5, [] S. Albeverio, Yu. G. Kondratiev and M. R ockner, 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, 1, pp Contemp. Math. 317, Amer. Math. Soc., Providence, RI, 3.

14 17 Jiyong Shin and Gerald Trutnau 14 [3] L.C. Evans and R.F. Gariepy, Measure theory and ne properties of functions. Stud. Adv. Math., CRC Press, Boca Raton, 199. [4] M. Fukushima, On a stochastic calculus related to Dirichlet forms and distorted Brownian motions. New stochastic methods in physics. Phys. Rep , 3, 556. [5] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes. Second revised and extended edition. De Gruyter Stud. Math. 19, Walter de Gruyter, Berlin, 11. [6] A. Grigor'yan and J. Hu, Upper bounds of heat kernels on doubling spaces, Preprint, to appear in Mosc. Math. J., 57 pages 1. [7] Z.M. Ma and M. R ockner, Introduction to the Theory of Non-symmetric Dirichlet Forms. Springer, Berlin, 199. [8] Y. Oshima, On stochastic dierential equations characterizing some singular diusion processes. Proc. Japan Acad. Ser. A Math. Sci , 3, [9] Y. Oshima, Some singular diusion processes and their associated stochastic dierential equations. Z. Wahrsch. Verw. Gebiete ,, [1] Y. Ouknine, F. Russo and G. Trutnau, On countably skewed Brownian motion with accumulation point. arxiv: [11] N.I. Portenko, Generalized Diusion Processes. Naukova Dumka, Kiev, 198; Translated from the Russian, Trans. Math. Monogr. 83. Amer. Math. Soc. Providence, RI, 199. [1] D. Revuz and M. Yor, Continuous martingales and Brownian motion. Springer Verlag, 5. [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 , [15] K.T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math ,, [16] K.T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl , 3, [17] S. Takanobu, On the existence of solutions of stochastic dierential equations with singular drifts. Probab. Theory Related Fields ,, [18] M. Tomisaki, A construction of diusion processes with singular product measures. Z. Wahrsch. Verw. Gebiete , 1, 517. [19] G. Trutnau, Skorokhod decomposition of reected diusions on bounded Lipschitz domains with singular non-reection part. Probab. Theory Related Fields 17 3, 4, [] G. Trutnau, On a class of non-symmetric diusions containing fully nonsymmetric distorted Brownian motions. Forum Math. 15 3, 3, [1] G. Trutnau, Multidimensional skew reected diusions. Stochastic analysis: classical and quantum, World Sci. Publ., Hackensack, NJ, 844, 5. [] B.O. Turesson, Nonlinear potential theory and weighted Sobolev spaces. Lecture Notes in Math Springer,. Received 17 July 13 Seoul National University, Department of Mathematical Sciences and Research Institute of Mathematics, 599 Gwanak-Ro, Gwanak-Gu, Seoul , South Korea