One-point Reflection

Transcription

1 One-point Reflection Zhen-Qing Chen and Masatoshi Fukushima April 18, 214 Abstract We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local time, entrance law and excursion law is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh s Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh s Brownian motion, recently obtained by Fitzsimmons and Kuter in [FiK using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD). Mathematics Subject Classification (21): Primary 6J5, 31C25; Secondary 6J45, 6J6 Keywords and phrases: boundary theory, one-point reflection, excursion law, local time, Dirichlet form, Brownian motion with darning, conformal invariance, harmonic functions 1 Introduction Boundary theory for one-dimensional diffusions is now well understood thanks to the fundamental works of Feller, Itô and McKean. Much less is known for boundary theory of multi-dimensional diffusions and of Markov processes with discontinuous sample paths. Recently, satisfactory progress has been made for symmetric Markov processes, and more generally, for strong Markov processes having weak dual, with finitely many boundary points. See [FT, CFY1, CFY2, CF1, CF2, CF3 and the references therein. In these works, Markovian extensions of the minimal processes are carried out through Poisson point processes of excursions via entrance laws and exit systems and are characterized in terms of resolvent representations, and in the symmetric case, in terms of Dirichlet forms. In this paper, we examine symmetric extensions of symmetric Markov processes with one boundary point. To be more specific, let E be a locally compact separable metric space and m be a positive Radon measure on E of full support. We fix a non-isolated point a E with m{a} = and put E = E \ {a}. Let X = (Xt, ζ, P x) be an m symmetric Borel standard process on E admitting no killing inside E such that Research partially supported by NSF Grant DMS

2 (A.1) φ(x) = P x(ζ <, X ζ = a) > for any x E. We call a Borel standard process X on E a one-point extension of X to E or a one-point reflection of X at a if X is m-symmetric and of no killings on {a}, and its subprocess obtained by killing upon hitting {a} (called the part of X on E ) is identical with X in law. It is proved in [CF3, Theorem that a one-point reflection X of X at a is unique in law and the point a is regular for itself with respect to X in this case. The existence of the one-point reflection X of X at a is also shown in [CF3, Theorem for a Hunt process X satisfying additional conditions (A.2), (A.3) as well as (A.4) in non-diffusion case that are stated in 3 of the present paper. Furthermore X becomes automatically a diffusion on E if so is X on E. The purpose of this paper is four-folds. First, we work with a general Borel right process X on E and a point a E which is X-regular for itself. We clarify the scaling relation between local time at a and its corresponding entrance law (Theorem 2.1). We then give a characterization of a local time being normalized in terms of its associated excursion law in the context of one-dimensional reflecting diffusion on an interval [, r ) (Theorem 3.1) Second, we deal with in Section 4 an elastic one-point reflection X, namely, a one-point extension of a symmetric Hunt process X on E to E that allows killings at a. Such an extension X has been constructed in [CFY1 in a more general setting by modifying the excursion law in a way to allow a jump to the cemetery. In particular, we derive in Section 4 a Dirichlet form characterization (Theorem 4.1) for the elastic one-point reflection, generalizing the corresponding one for the onepoint reflection first obtained in [CF2. We also clarify in this section that the law of the one-point reflection is not affected if the canonical entrance law in the construction is replaced with its constant multiple. The third goal is to identify in Section 5 Walsh s Brownian motion X with the one-point reflection of a certain symmetric diffusion X and determine its Dirichlet form. Walsh s Brownian motion, introduced in a descriptive way in [Wal, is a continuous Markov process on the plane R 2 that lives on rays {R θ, θ < 2π} emanating from the origin. It behaves like one-dimensional Brownian motion along the ray away from the origin, and when it hits the origin, it goes in a random direction θ according to a probability measure η(dθ) on [, 2π). There have been several characterizations or constructions of it afterwards ([R, BC, S), and in particular Barlow-Pitman- Yor [BPY gave a rigorous construction of its semigroup. Let X be the Brownian motion on each ray R θ being killed upon hitting. X is a diffusion on R 2 \ {} that can be easily verified to be symmetric with respect to the product measure m = λ η on (, ) [, 2π) where λ is the Lebesgue measure on (, ). Clearly X satisfies conditions (A.1), (A.2), (A.3). So there exists a unique one-point reflection X of X at. We call X Walsh s Brownian motion. The semigroup {T t, t > } constructed in [BPY is a Feller semigroup on C (R 2 ) whose associated Hunt process Y on R 2 has X as its part on R 2 \{}. The m-symmetry of T t can be also readily seen so that Y becomes a diffusion automatically and coincides with X in law. Such a straightforward definition of Walsh s Brownian motion along with its Dirichlet form characterization given in Theorem 5.2 of this paper are not only more direct but also capture the essence of its intuitive picture. The Dirichlet form characterization together with results from [C1, CK quickly gives the analytic characterization of harmonicity and subharmonicity for Walsh s Brownian motion, which recovers the main result of a recent paper [FiK by Fitzsimmons and 2

3 Kuter. Instead of taking a point a of E and putting E = E \ {a}, we may take any compact subset K of E, put E = E \ K, and consider the topological space E = E {a } obtained from E by rendering the set K as a single point a. The restriction of m to E is denoted by m which is extended to E by setting m ({a }) =. For a given m-symmetric diffusion X on E, its part X on E is known to be m -symmetric. As will be formulated in Section 6 under certain conditions on X and X, there exists a unique one-point reflection X of X at a. This procedure of getting X from X (or from X ) is called darning a hole K. When E is a domain of the complex plain C and X is the absorbing Brownian motion on E, the diffusion X on (E \ K) {a } so obtained is called a Brownian motion with darning (BMD). In Section 6, we consider a special case where K is a connected set containing at least two points so that C \ K connected. The fourth goal of this paper is to show the conformal invariance of BMD and to identify, in this special case, BMD with the excursion reflected Brownian motion (ERBM) introduced by Lawler [L and Drenning [D. This is done by combining a quite analogous consideration to Section 5 with the conformal invariance of BMD. BMD has been formulated and characterized in a more general setting allowing many holes in E rather than a single hole K together with its Dirichlet form characterization [C2, CFR. It has played important roles in the study of Komatu-Loewner equations for multiply connected planar domains [CFR, FuK. In Section 6, we also consider a Walsh s Brownian motion with darning (WBMD) obtained from the Walsh s Brownian motion X on R 2 by rendering a star-like compact set K = {(r, θ) : r ψ(θ), θ [, 2π)} into a single point. In contrast with a BMD, a WBMD is highly reducible until approaching to the hole K. We end this introduction by pointing out that we can consider other types of Markov processes in the plane or higher dimensional spaces by adopting censored stable processes [BBC, for instance, in place of the absorbing Brownian motions in Sections 5 and 6. 2 Normalized local time and canonical entrance law Let X = (X t, ζ, P x ) be a Borel right process on a Lusin space E with lifetime ζ and with cadlag paths up to ζ. We fix a point a E which is X-regular for itself, that is, P a (σ a = ) = 1. (2.1) Here σ a = inf{s > : X s = a}. To avoid triviality, we assume that P x (σ a < ) is not identically zero on E := E \ {a}. Denote by X = {Xt, ζ, P x) the part process of X killed upon leaving E. The transition semigroup of X (resp. X ) is denoted by {P t } (resp. {Pt }). Since a is a regular point, there exists a positive continuous additive functional (PCAF) l t of X supported by {a} in the sense that t I {a}(x s )dl s = l t (cf. [BG, Theorem V.3.13). A PCAF supported by {a} is called a local time of X at the point a. Clearly, if l is a local time at a, then so is Cl t for any positive constant C. Given a local time l t at a, its inverse τ t = inf{s : l s > t} gives rise to an excursion point process (p, P a ) in the following manner ([I, CFY2). Let Ω be the space of cadlag paths from [, ) to E, where is an isolated point added to E serving as a cemetery point. Define the operators 3

4 θ t, k t and i t on Ω by (θ t ω)(s) = ω(t + s), (k t ω)(s) = { ω(s) s < t s t,, i t = k σa θ t. We introduce spaces of excursions around {a} by { W = {kσa (ω) : ω Ω, σ a > }, W + = {ω W : σ a (ω) < }, W = {ω W : σ a (ω) =, ζ > }. (2.2) Then W = W + W { }, where denotes the path identically taking value. Define D p(ω) = {s (, ) : τ s (ω) < τ s (ω)}, p s (ω) = i τs ω for s D p(ω). Note that {p s (ω) : s D p(ω) } W and {p s (ω) : s D p(ω), s < l } W +. When T = l <, then τ T < and τ T =. In this case, T D p and p T (ω) = θ τt (ω) W { }. Thus T is the only time of the occurrence of a non-returning excursion. By [FG2, Proposition 2.1, T is exponentially distributed δ [, ) under P a. It follows from [Me, 1-2 (see also the proof of [CFY2, Lemma 3.1) that, under probability measure P a, p is an absorbed Poisson point process with absorbing time T in the sense of Meyer [Me. Let n be the characteristic measure of (p, P a ), which is a σ-finite measure on the excursion space W. Define ν t (B) = n(w W : w(t) B, t < σ a ), B B(E ). Then {ν t } is then an X -entrance law in the sense that ν s P t = ν s+t for s, t >, (2.3) The measures {ν t } is called the entrance law associated with the local time l t. Note that, if {ν t } is an X -entrance law, then so is {Cν t } for any constant C >. By the X -entrance law {ν t } associated with l, the above characteristic measure n is uniquely determined by f 1 (w(t 1 ))f 2 (w(t 2 )) f n (w(t n ))n(dw) W = ν t1 f 1 P t 2 t 1 f 2 P t n 1 t n 2 f n 1 P t n t n 1 f n. (2.4) The σ-finite measure n is sometimes called the excursion law associated with l. Let m be an X-excessive measure, that is, m is σ-finite and mp t m for every t >. The existence of m is a mild assumption; see [FG2, Theorem 4.5. The Revuz measure ν A of a PCAF A t of X relative to an X-excessive measure m is defined by 1 lim t t E m [ t f(x s )da s = ν A (f), f B + (E). By assumption (2.1), the set {a} is not semipolar so that the Dirac measure δ a concentrated on {a} is smooth and admits a unique PCAF l t of X whose Revuz measure relative to an excessive measure m is δ a ([FG1): [ 1 t lim t t E m f(x s )dl s = f(a), f B + (E). (2.5) 4

5 The PCAF l t is called the normalized local time of X at a relative to m. From now on, we assume that there exist a σ-finite measure m on E and a Borel right process X = ( X t, ζ, P x ) on E such that X and X are in weak duality relative to m: for the transition function { P t } of X, P t f(x)g(x)m(dx) = E E f(x)p t g(x)m(dx) for every t > and f, g B + (E). (2.6) Clearly, m has to be X-excessive. Consider the hitting probability of a for X: φ(x) =: P x (σ a < ), x E. Let m be the restriction of m to E = E \ {a}. Then φ m is purely X -excessive in the sense that lim t ( φ m ) Pt =. By a theorem of P. Fitzsimmons ([G), there exists a unique X -entrance law {µ t } such that in other words, φ m = µ t dt; (2.7) µ t (dx)dt = P x (σ a dt)m (dx). (2.8) We call {µ t } the canonical entrance law of X at the point a relative to m. If P x (σ a < and X σa = a) = P x (σ a < ) for x E, (2.9) then φ(x) = P x( X ζ = a, ζ < ) and consequently µ t (dx)dt = P x( ζ dt, X ζ = a)m (dx). (2.1) Thus under assumption (2.9), the canonical entrance law {µ t } of X at a is determined by the dual minimal process X. Theorem 2.1 Let l be the normalized local time of X at a relative to m and {µ t } be the canonical entrance law of X at a relative to m. Then (i) {µ t } is the entrance law associated with l. (ii) For any constant C >, {C 1 µ t } is the entrance law associated with C l. Proof. When X is an m-symmetric diffusion, (i) of Theorem 2.1 was first proved in [FT, Theorem 3.1, Remark 4.2. In [FT, the local time was always assumed to be a normalized one but a similar proof works to obtain also (ii) (by taking CL(m, ψ) 1 in place of L(m, ψ) 1 on the third line of page 452 of [FT). In the present generality, (i) of Theorem 2.1 was first established in [CFY2, Theorem 3.2 using the notion of an exit system (Q, l) due to Maissonouve [Ma defined as follows. Fix any local time l of X at a. Let M(ω) be the closure of the visiting time set {t [, ) : X t (ω) = a}. The complement of M(ω) in [, ) is a disjoint union of open intervals called excursion intervals. 5

6 Denote by G(ω) the collection of left end points of those intervals. Then, by [Ma, there exists a σ-finite measure Q on W characterized by [ [ E x Z s Γ i s = Q (Γ) E x Z s dl s, x E, (2.11) s G for every non-negative predictable process Z s and every non-negative random variable Γ on W. Define ν t (B) = Q (w W : w(t) B, t < σ a ), B B(E ). {ν t, t > } is then an X -entrance law, which we call the entrance law induced by the exit system (Q, l). In [CFY2, Proposition 2.1, the entrance law induced by the exit system (Q, l) was identified with the canonical entrance law {µ t, t > } of X at a relative to m. As a consequence, the entrance law induced by the exit system ( 1 C Q, C l) can be identified with { 1 C µ t, t > } for any positive constant C. On the other hand, the excursion law n associated with the local time l can be identified with Q characterized by (2.11): Q = n. (2.12) Identity (2.12) was established in [CFY2, Theorem 3.2 in the case that l = l but exactly the same proof works for any local time l. In particular, the entrance law associated with a local time l coincides with the entrance law induced by the exit system (Q, l) and so we get the desired conclusion (i) and (ii). As a matter of fact, Theorem 2.1 holds for any Borel right process X on E and for any point a E that is X-regular for itself. Indeed, for each choice of an X-excessive measure m, it is known that m automatically admits an m-dual moderate Markov process as a substitute of X in the above ([FG1, FG2) so that the canonical entrance law {µ t } of X at a relative to m still makes sense by the characterization (2.7). Fitzsimmons and Getoor have identified in [FG2, (3.23) the canonical entrance law with the entrance law induced by the exit system (Q, l), and so (i) and (ii) of Theorem 2.1 follow from the identity (2.12). From (2.11) and (2.12), we have the following characterization of the excursion law n associated with a local time l of X at a: [ [ E x Z s Γ i s = n(γ) E x Z s dl s, x E, (2.13) s G for every non-negative predictable process Z s and every non-negative random variable Γ on W. This will be utilized in Section 3. Example 2.2 Let X = (X t, P x ) be one-dimensional reflecting Brownian motion on E = [, ) and X = (X t, ζ, P x) be the absorbing Brownian motion on E = (, ). It is known that a = is X-regular (i.e. P (σ = ) = 1) and the Lebesgue measure m = dx (resp. m = dx) on [, ) (resp. (, )) is X (resp. X )-excessive. Let l t be the normalized local time of X at relative to m and {µ t } be the canonical entrance law of X at relative to m. 6

7 Since P x (σ dt) = P BM (σ x dt) = 1 (2πt 3 ) 1/2 xe x2 /(2t) dt, we have from (2.8) or (2.1), µ t (dx) = 1 (2πt 3 ) 1/2 xe x2 /(2t) m (dx). (2.14) By virtue of Theorem 2.1, {µ t } is the entrance law of X at associated with l t. Consequently, by Theorem 2.1(ii), { µ t := 2µ t } is the entrance law of X at associated with l t := 1 2 l t. Note that l t has been a standard choice of the local time of X by the following reason. The Dirichlet form of X on L 2 ([, ); dx) is ( 1 2 D, H1 (, )). For u(x) = x, which is in Hb,loc 1 (, ), and for v C1 c ([, ), we have ν u (dx) = dx and 1 2 u v dx = 1 2 v dx = 1 2 v() = 1 2 δ, v. Thus Fukushima s decomposition ([FOT, Theorem 5.5.5) yields the well known Lévy-Skorohod decomposition of the RBM X: X t X = B t + l t, (2.15) where B t is Brownian motion on R with B =. The measures µ t = 2µ t has been adopted as a standard choice of the entrance law of the RBM on [, ) in [FY, IW, RY, Wat. The standard ones l t, µ t can be also considered as normalized and canonical ones relative to 2m, respectively. [I and [B adopted a different normalization of the local time: the Blumenthal-Getoor local time l t at a characterized as in [BG, pp 217 by [ E x e t [ d l t = E x e σ a. In the above example, let Cδ, C >, be the Revuz measure of l t relative to m. Denote by R α (x, y) the resolvent density of the RBM X on [, ). By [FOT, (5.1.14), we have for u 1 (x) = E x [e σ u 1 (y) = CR 1 (, y) = CR 1 (y, ) = Cu 1 (y)r 1 (, ) = 2Cu 1 (y), y >. Hence C = 1 2 and l t = 1 2 l t. The entrance law associated with l is therefore { 2 µ t }. 3 Normalized excursion laws for reflecting diffusions Let s and m be a canonical scale and a canonical measure on (, r ), where r (, ; s is a strictly increasing continuous function with s(+) = and m is a positive Radon measure of full support. We assume that the left boundary is regular, while the right boundary r is non-regular with respect to (s, m). We extend the measure m to [, r ) by setting m({}) =. Define r E s du dv (u, v) = ds ds ds F s = {u : absolutely continuous in ds on (, r ) with E s (u, u) < }. (E s, F s L 2 ([, r ); m)) is a regular symmetric strongly local irreducible Dirichlet form on L 2 ([, r ), m). Since r is assumed to be non-regular, its extended Dirichlet space is given by {u F s : u(r ) = if s(r ) < }, (3.1) 7

8 in view of [CF3, Theorem Let X = (X t, ζ, P x ) be the associated m-symmetric diffusion process on [, ) and X = (Xt, ζ, P x) be the subprocess of X killed upon hitting the point. X is then the minimal diffusion for (s, m) and X is the reflecting extension of X at (see [CF3, Example 3.5.7, 7.3 (3)). The point is X-regular for itself. Take any local time l of X at and let n be the excursion law associated with l. By the strong Markov property of n, we have for < x < y < r, ( ) n(σ y < ) = n(σ x <, σ y θ σx < ) = n P X (σ σx y < ); σ x < = P x(σ y < )n(σ x < ) = s(x) s(y) n(σ x < ). So c = s(x)n(σ x < ) does not depend on x (, r ). Consequently, n(σ x < ) = Now, for the entrance law µ t associated with l, q t := n(w(t) (, x), σ x θ t < ) = c s(x), x (, r ). (3.2) x µ t (dξ) P ξ (σ x < ), and hence lim t n(t < σ x < ) = lim t q t = 1 x s(x) s(ξ)µ t(dξ). Accordingly we have from (3.2) c = lim t x s(ξ)µ t (dξ), for any x (, r ). (3.3) On the other hand, we have by (2.1) and (3.3) that for u α (ξ) := E ξ [e ασ, ( x ) x c = lim αe αt s(ξ)µ t (dξ) dt = lim α s(ξ)u α (ξ)m(dξ), x (, r ), (3.4) α α In his study of general boundary conditions for linear diffusions using excursion theory, K. Yano [Y, (2.34) adopts a special normalization n(σ x < ) = 1 s(x), < x < r, for the excursion law n of X, which will now be shown to characterize the normalization of the local time of X at given in the preceding section. Theorem 3.1 The local time l at of X is the normalized one relative to m if and only if the excursion law n associated with l satisfies n(σ x < ) = 1 s(x) for any x (, r ). (3.5) Proof. (i) First, observe that the theorem holds for the reflecting Brownian motion X on [, ). In this case, r = and we may take s = x, m = 2dx so that the canonical entrance law at relative to m is given by (2.14) with m = 2dx. The Laplace transform of dµ t /dm is u α (x) = 2e 2αx. Thus by (3.4), we have x x c = lim α ξu α (ξ)dξ = lim α 2ξe 2αξ dξ = 1, x >, α α and hence (3.5) follows from (3.2) and Theorem

9 (ii) The same is true if we replace the RBM X by its part process X b on I = [, b), < b <, being killed upon hitting b. In fact, the corresponding α-potential u b α is By (3.4), we have for x (, b) u b α(ξ) = E ξ [ e ασ ; σ < σ b = uα (ξ) u α (b)e ξ [ e ασ ; σ b < σ. x c = lim α ξu b α α(ξ)dξ o x = lim α α x = lim α α = 1. xu α (ξ)dξ lim α αu α(b) 2ξe 2αξ dξ lim α αe x x 2αb E ξ [ e ασ ; σ b < σ dξ E ξ [ e ασ ; σ b < σ dξ (iii) For the general reflecting diffusion X = (X t, ζ, P x ) specified in the above, we shall prove the theorem by making use of the characterization (2.13) of the excursion law. To this end, we transform X into RBM on [, b) absorbed at some b > by a scale change followed by a time change that carries a normalized local time over to a normalized one. Let I = [, b) with b = s(r ) and D(u, v) = I u v dx. Consider the transformed process Y = {Y t, ζ, P Y x, x I} of X by the scale s defined by Y t = s(x t ), P Y x = P s 1 (x). In view of [F1, Lemma 4.3, Y is symmetric with respect to the measure m = m s 1 on I, and its Dirichlet form (E Y, F Y ) on L 2 (I; m) is given by E Y = D, F Y = BL(I) L 2 (I; m). On account of (3.1), its extended Dirichlet space is given by F Y e = {u BL(I) : u(b ) = whenever b < }. We now consider Fukushima s decomposition for the strong Markov process Y. In view of [F2, Lemma 4.1, the resolvent kernel R α, α >, of Y admits the reproducing kernel g α (x, y) of (F Y, Eα Y ) as its density relative to m that satisfies sup{g α (x, y) : x J, y I} < for any compact interval J I. Hence the strict decomposition theorem [FOT, Theorem applies to the function u(x) = x, x I, that belongs to Fb,loc Y. Its energy measure µ u equals 2dx on I and D(u, v) = v() for v Cc 1 (I). Therefore we have Y t Y = M t + l t, t ζ, P s 1 (x)-a.e. for any x I, (3.6) where M is a local martingale additive functional (local MAF) of Y in the strict sense having 2dx as the Revuz measure for its predictable quadratic variation M relative to m, and l t is a normalized local time of Y at relative to m. Since 2dx is of full quasi-support on I, M is strictly increasing ([CF3, Theorems and 5.2.1). Let τ t = inf{s > : M s > t} for t < M, and define Ỹt = Y τt. Then by [CF3, Theorem 5.2.2, the time changed process Ỹ = (Ỹt, ζ, P Y x ) having lifetime ζ = A is associated with Dirichlet form (D, Fe Y L 2 (I; 2dx)). Thus Ỹ has the same distribution as the part process of the reflecting Brownian motion X on [, ) killed upon hitting b. Substituting τ t into (3.6), we obtain Ỹ t Ỹ = M τt + l τt, t [, ζ). (3.7) 9

10 Then, up to the time ζ, M τt is a Brownian motion with M = and l τt is a continuous increasing process. On the other hand, Fukushima s decomposition (2.15) for the RBM X gives another decomposition of Ỹt as a sum of a Brownian motion and a continuous increasing process l t up to time ζ. Due to the uniqueness of a decomposition of a continuous semi-martingale, we have l τt = l t, t < ζ. (3.8) As l t is a normalized local time of X at relative to 2dx on [, ), lτt ζ is a normalized local time at of Ỹ relative to 2dx I in view of [CF3, Proposition (iv) Let nỹ be the excursion law of Ỹ at associated with a normalized local time of Ỹ at relative to 2dx on I = [, b), where b = s(r ). As established in Step (ii) of this proof, nỹ has the property nỹ (σ y < ) = 1 y for every y (, s(r )). (3.9) nỹ satisfies (2.13) with l τt ζ in the above and the law PỸx of Ỹ in place of l and P x, respectively. Let n Y be the excursion law of Y associated with a normalized local time of Y at relative to the measure m on (, s(r )). n Y then satisfies (2.13) for l in (3.6) and with the law P Y x of Y in place of P x. Those characterizations of nỹ and n Y combined with time change by τ t lead us to the equality n Y (σ y < ) = nỹ (σ y < ), y (, s(r )), and so we get from (3.9) n Y (σ y < ) = 1 y for every y (, s(r )). (3.1) (v) The scale function s induces a transformation (denoted by s again) from the space W of [, r )-valued excursions to the space W of [, s(r ))-valued excursions. Define a measure n on W by n(a) = n Y (s(a)) for A W. (3.11) We then have n(σ x < ) = 1 s(x), x (, r ). from (3.1). Let n X be the excursion law of X associated with a normalized local time of X at relative to m. We shall show n X = n. (3.12) Once (3.12) is established, then we have the only if part of Theorem 3.1 as well as the if part because, for any C >, 1 C nx is associated with a normalized local time multiplied by C in view of (2.13). For every non-negative random variable Γ on W and Z s = k j=1 f j(x sj )1 (t1,t 2 (s), where f j are non-negative Borel measurable functions and s 1 < s 2 < s k t 1 < t 2 <, we have by (2.13) for (n Y, l), [ E x Z r Γ i r = E Y k s(x) f j s 1 (Y sj )1 (t1,t 2 (r) s(γ) i r r G r G j=1 k = n Y (s(γ)) E Y s(x) f j s 1 (Y sj )1 (t1,t 2 (s)d l s = n(γ) E x [ 1 Z s d l s. j=1

11 Since every non-negative predictable process Z s with respect to the minimal augmented filtration generated by X can be expressed as an increasing limit of linear combination of the above type of Z s, we conclude that [ [ E x Z r Γ i r = n(γ) E x Z s d l s, x [, r ), (3.13) r G holds for every non-negative random variable Γ on W and every non-negative predictable process Z s with respect to the minimal augmented filtration generated by X. Finally we show that l in the decomposition (3.6) is a normalized local time of X at relative to m. We can then get the desired (3.12) from (2.13) and (3.13). Just as we did for the process Y and the function u(x) = x, we can apply the strict decomposition theorem [FOT, Theorem to the process X and the function s F s b.loc. The energy measure dµ s equals 2ds on [, r ). For v C 1 c (I), v s F s and E s (s, v s) = D(x, v) = v() = v(s()). Since such function v s is E s 1 -dense in F s, we have the decomposition s(x t ) s(x ) = M X t + l X t, t [, ζ), P x -a.e. for any x [, r ), (3.14) where M X is a local MAF of X and l X is a normalized local time of X at relative to m. By the uniqueness of the decomposition of continuous semimartingale Y t = s(x t ), we have from (3.6) and (3.14) that l t = l X t for every t. 4 Elastic one-point reflection In this section, we first review a construction in [CFY2 of a one-point extension X possibly with a killing at the boundary point a of a general symmetric Markov process X using the canonical entrance law of X at a. The process X can also be called an elastic one-point reflection of X at a. We then study basic properties of the constructed X. Let E be a locally compact separable metric space and m a positive Radon measure with full support on E. Let E = E be the one point compactification of E. When E is compact, is added as an isolated point. We fix a non-isolated point a E with m({a}) =. We consider an m-symmetric Hunt process X = (X t, ζ, P x) on E = E \ {a} satisfying the following three conditions. The quantities related to X are designated by having the superscript. The resolvent of X is denoted by G α. (A.1) φ(x) =: P x(ζ <, X ζ = a) > for any x E. (A.2) P x(x ζ {a, }) = 1 for any x E, regardless the length of ζ. (A.3) There exists a neighborhood of a such that inf x V G 1φ(x) > for any compact set V U \ {a}. When X is not a diffusion, we assume an additional condition (A.4) on the jumping measure J of X that is given by J (dx, dy) = N(x, dy)µ H (dx) in terms of the Lévy system (N, H) of X (cf. [CF3): (A.4) Either E \ U is compact for any neighborhood U of a in E, or for any neighborhood U 1 of a in E, there exists an open neighborhood U 2 of a in E with U 2 U 1 such that J (U 2 \ {a}, E \ U 1 ) <. 11

12 Let {µ t } be the canonical entrance law of X at a relative to m: Introduce the excursion space by W = {w : cadlag function from (, ζ(w)) to E µ t dt = φ m. (4.1) for some ζ(w) (,, w(+) = a, w(ζ(w) ) {a, }} W + = {w W : ζ(w) <, w(ζ(w) ) = a}, W = W \ W +, and define a measure n on W by f 1 (w(t 1 ))f 2 (w(t 2 )) f n (w(t n ))n(dw) W = µ t1 f 1 P t 2 t 1 f 2 P t n 1 t n 2 f n 1 P t n t n 1 f n. (4.2) For functions u, v on E, denote the integral E u(x)v(x)m(dx) by (u, v). The energy functional L (φ, v) of the X -excessive measure φ m and an X -excessive function v is defined by By [CFY2, Lemma 5.5, we then have L 1 (φ, v) = lim t t (φ P t φ, v). L (φ, 1 φ) = n(w ) <. (4.3) The equality in the above follows from (4.1) and n(ζ > t; W ) = µ t, 1 φ, while the finiteness is due to our assumptions (A.3),(A.4) on X. We add to W a path identically equal to and extend n to W { } by setting n({ }) = κ for a fixed value κ indicating a killing rate at a. We then consider a Poisson point process p defined on a certain probability space (Ω, P) taking values in W { } with the characteristic measure n. Let p + and p be the range restrictions of p to W + and W { }, respectively. We define a subordinator J by J(s) = r s ζ(p + r ) for s >, and J() =. Let T be the first time of occurrence of the point process {p s, s > }. Then J and T are independent and T is exponentially distributed with rate n(w ) + κ. By piecing together the returning excursions p + s at each jumping time of J until the time T and then joining the path p T, we can obtain an E-valued continuous process (Xa t, ζ ω, P) with X a = a and X ζω {a, }. Define [ ζω G α f(a) = E e αt f(xt a )dt, α >, f B(E). According to this way of constructing X a t, we have n(f α ) G α f(a) = α e αt n(t < ζ <, w(ζ ) = a)dt + n(w ) + κ, (4.4) where f α = ζ(w) e αt f(w(t))dt, w W. See 5.1 and 5.2 of [CFY2 for more details. 12

13 Theorem 4.1 (i) By patching X and (Xt a, P) together, we have an m-symmetric right process X = (X t, ζ, P x ) on E possessing the resolvent (u α, f) G α f(a) = α(u α, φ) + L (φ, 1 φ) + κ, (4.5) G α f(x) = G αf(x) + u α (x)g α f(a), x E. (ii) The part process of X on E equals X. The singleton {a} is regular for itself relative to X. The process X admits no sojourn at a nor jump from E to a. (iii) The sample path {X t, t < ζ} is cadlag on [, ζ), continuous when X t = a and satisfies X ζ {a, } when ζ <. (iv) If X is a diffusion on E, then X is a diffusion on E. (v) For a constant C >, let X C be the symmetric right process on E obtained as above like X but with the canonical entrance law {µ t } and κ being replaced by {C 1 µ t } and κ, respectively. Then, the statements (i), (ii), (iii), (iv) hold for X C but with κ in (4.5) being replaced by C κ. In particular, X C has the same distribution as X if and only if κ = C 1 κ. Proof. (i), (ii), (iii) and (iv) are taken from Theorem 5.15 in [CFY2, 5, which was formulated under a more general setting that X is a Borel standard process on E possessing an m-weakly dual Borel standard process X on E. (v) If the canonical entrance law {µ t } is replaced by {C 1 µ t }, then the characteristic measure n specified by (4.2) is changed into C 1 n and so (v) follows from (4.4). A right process X = (X t, ζ, P x ) on E is called a one-point extension or one-point reflection of X if its part on E equals X, X is m-symmetric and X admits no killing at {a}. By virtue of [CF3, Theorem 7.5.4, a one-point reflection X of X is unique in law. Theorem 4.1 with κ = provides us with a construction of a one-point reflection X of X. Theorem 4.1(iv) in particular implies that the one-point reflection of X becomes automatically a diffusion process on E. We next study the Dirichlet form and the L 2 -infinitesimal generator of the symmetric right process X on E constructed in Theorem 4.1 (i) from X. We recall some concepts introduced in [CF3 in relation to the process X. Let (E, F ) be the Dirichlet form of X on L 2 (E, m) and ((F ) ref, E,ref ) be its reflected Dirichlet space; see [CF3 for related definitions. Define the active reflected Dirichlet space (F ) ref a to be (F ) ref L 2 (E ; m). We then consider an operator L on L 2 (E, m) specified by f D(L) with Lf = g L 2 (E, m) if and only if f (F ) ref a with E,ref (f, v) = (g, v), v F. 13

14 Denote by u α the α-order approach probability of X to a: u α (x) = E x [e αζ ; Xζ = a. The flux N(f)(a) of f D(L) at a is then defined by N(f)(a) = E,ref (f, u α ) + (Lf, u α ), which is independent of α >. We say that an X -q.e. finely continuous function f on E has an X - fine limit value c R at a if ( P x lim t ζ f(x t ) = c ) ζ <, Xζ = a = 1 for q.e. x E. (4.6) Theorem 4.2 X. Let X be the m-symmetric right process on E constructed in Theorem 4.1 from (i) Let (E, F) be the Dirichlet form of X on L 2 (E; m) and (F e, E) be its extended Dirichlet space. Then F is the linear subspace of (F ) ref a spanned by F and u α, and F e is the linear subspace of (F ) ref spanned by Fe and φ. Moreover, for f = f + cφ, f Fe, c R, E(f, f) = E,ref (f, f) + c 2 κ. (4.7) Denote by k the killing measure on E in the Beurling-Deny decomposition of E. Then k({a}) = κ. (4.8) (ii) Suppose the active reflected Dirichlet space (F ) ref a of X satisfies the following condition: If f (F ) ref a admits an X -fine limit value at a, then f F. (4.9) Let A be the L 2 -infinitesimal generator of X. Then f D(A) if and only if f D(L), f admits an X -fine limit value f(a) at a and satisfies the boundary condition In this case, Af = Lf. N(f)(a) + κ f(a) =. (4.1) Proof. (i) Since X is an m-symmetric Borel right process, (E, F) is quasi-regular Dirichlet form on L 2 (E; m) and the transfer method as in the proof of [CF3, Theorem applies. As the single point set {a} is not m-polar, we have F = {f + cu α : f F, c R}, F e = {f + cφ : f F e, c R}. On the other hand, we can verify as in [FT, (2.19 that G α f(a) = (u α, f) α(u α, φ) + E(φ, φ) for f C c (E). (4.11) 14

15 In fact, we have for w = G α f, g F the equation E(w, g) + α(w, g) = (f, g). By Lemma of [CF3, we can find a uniformly bounded sequence g n F such that lim g n(x) = φ(x) m-a.e., lim E(g n φ, g n φ) =. n n Substituting g n in the above equation and letting n, we get E(w, φ) + α(w, φ) = (f, φ). (4.12) As w = G α f F F e, there is some u F e so that w = u + cφ with c = w(a) = G α f(a). By the strong Markov property, we also have w = G αf + cu α. Putting the above expressions for w in the first and second terms on the left hand side of (4.13), we get ce(φ, φ) + cα(u α, φ) = (f, φ) α(g αf, φ). As the righthand side equals (f, u α ), we have (4.11). A comparison of (4.11) with (4.5) yields E(φ, φ) = L (φ, 1 φ) + κ. (4.13) In order to deduce (4.7) and (4.8) from (4.13), we use again the transfer method: without loss of generality, we can assume that (E, F) is a regular Dirichlet form on L 2 (E; m) associated with a Hunt process X on E. We first note that the local part µ c φ of the associated energy measure of φ F e does not charge on the one-point set {a}: µ c φ ({a}) =. (4.14) To see this, take a function g F C c (E) with g = 1 in a neighborhood of a and put φ(x) = φ(x)g(x), x E. Then φ F b and µ c φ ({a}) = µc φ ({a}) = in view of [CF3, Theorem Owing to the energy image density property formulated in [CF3, Theorem 4.3.8, µ c φ ( φ 1 (1)) =, which implies (4.14) because {a} φ 1 (1). In view of Theorem 4.1 (ii), X admits no jump from E to a so that Theorem of [CF3 applies. By combining this theorem with (4.14), we can conclude that F e (F ) ref and, for f F e, f = f + cφ, f Fe, c R, E(f, f) = E,ref (f, f) + c 2 k({a}), E,ref (f, f) = E (f, f ) + c 2 V ({a}), (4.15) where V ({a}) is the supplementary Feller measure at the point a as the boundary of E. According to the definition [CF3, (5.5.7) of V, we have V ({a}) = L (φ, 1 φ). Therefore we are led to (4.7) and (4.8) from (4.13) and (4.15). (ii) This follows from [CF3, Theorem on account of (i) and (4.14). When κ >, the process X constructed in Theorem 4.1 can be obtained from the one-point reflection of X by killing it with rate κ at a by virtue of (4.8) and [CF3, Theorem Thus when κ >, we can call X the elastic reflection of X at a. When X is the absorbing Brownian motion on (, ), then N(f)() = 1 2 f (). If we employ the standard choice µ t = 2µ t of the entrance law, then κ for the constructed process X is changed 15

16 into 1 2 κ by Theorem 4.1 (iv) so that its boundary condition (4.9) is reduced to f () = κf(), the elastic boundary condition with killing rate κ. For κ =, the constructed process X r is the RBM on [, ). Let X r be the RBM on [, ), l t = 1 2 l t be its standard local time at and T be an independent exponential holding time with rate κ. In [IM the diffusion X on [, ) with the stated elastic boundary condition was constructed by killing X r upon the first time that l t exceeds T, in other words, at time J(T ) for the inverse J of l. Since the entrance law of the characteristic measure n of J equals { µ t }, this procedure matches what we have done above in constructing X from X by starting with { µ t } instead of l. 5 Walsh s Brownian motion as a one-point reflection Consider the case where E = R 2, a = the origin and E = R 2 \ {}. The pinched plane E can be represented in terms of the polar coordinate: E = {x = (r, θ) : r (, ), θ [, 2π)}. Let λ(dr) = dr be the Lebesgue measure on (, ) and η be an arbitrary probability measure on [, 2π). We equip E = (, ) [, 2π) with the product measure m = λ η. For simplicity, we assume that S = Supp[η = [, 2π). (Otherwise it suffices to set E = (, ) S). Let X = (Xt, ζ, P x) be the diffusion process on E that behaves, when started from x = (r, θ) E, as the one-dimensional absorbing Brownian motion on the ray R θ connecting (r, θ) with the origin. Denote by {p t } the transition semigroup of the absorbing Brownian motion on (, ) that is symmetric with respect to λ(dr) = dr. For a function f on E, write f θ (r) = f(r, θ), r >, θ [, 2π). Then the transition semigroup {Pt } of X satisfies (Pt f)(r, θ) = (p t f θ )(r) and we have by Fubini s theorem that for every f B + (E ) f(x)pt g(x)m(dx) = η(dθ) f θ (r)(p t g θ )(r)dr E = η(dθ) (p t f θ )(r)g θ (r)dr = Pt f(x)g(x)m(dx). E Thus X is m-symmetric. Clearly X satisfies conditions (A.1), (A.2), (A.3). We extend m on E = R 2 \ {} to E = R 2 by setting m() =. Therefore there exists a unique one-point reflection X = (X t, ζ, P x ) of X at. On account of (2.1) and (2.14), the X -canonical entrance law {µ t } to produce X from X is given by µ t (dx) = 1 (2πt 3 ) 1/2 re r2 /(2t) dr η(dθ). (5.1) By virtue of Theorem 4.1(v) with κ = κ =, we can take any positive constant multiple of µ t as an entrance law. We call X Walsh s Brownian motion (WBM in abbreviation). The entrance law (5.1) and the construction of X from X given in 4 (with κ = ) fits well the intuitive description of the motion that Walsh gave in [Wal, Epilogue, which says it moves like a one-dimensional Brownian motion along the ray but, upon hitting the origin, it reflects in a random direction θ with a given distribution η on [, 2π). Describing the motion in terms of Itô s excursion theory, he argued, would give the reader a feeling of watching a butterfly being assassinated with an elephant gun. Now the notion of the one-point reflection provides us with a neat apparatus to capture the butterfly alive. 16

17 Walsh s Brownian motion has previously been rigorously constructed or characterized by several authors using different approaches ([R, BC, S). In particular Barlow-Pitman-Yor [BPY gave a rigorous construction of its semigroup. Let {p r t } be the semigroup of the reflecting Brownian motion on [, ) that is symmetric with respect to λ(dr) = dr. Define for f C (R 2 ) P t f() = p r t f(), P t f(r, θ) = p r t f(r) + p t (f θ f)(r), r >, θ [, 2π). (5.2) It was then shown in [BPY that {P t } is a Feller semigroup on C (R 2 ) and that the associated Hunt process Y on R 2 admits X as its part on E, namely, X is obtained from Y by killing upon hitting. In the same way as we have verified for Pt in the above, we can use Fubini s theorem again to check that the semigroup P t defined by (5.2) is symmetric with respect to m = λ η. Therefore Y is a one-point reflection of X at so that it automatically becomes a diffusion on R 2 and coincides in law with our X. Let (E, F ) be the Dirichlet form of X on L 2 (R 2 \{}, m). Using {P t }, we immediately obtain C c (R 2 \ {}) F, E (f, g) = 1 2 D(f θ, g θ )η(dθ), (5.3) where D(f, g) = f g dx. We introduce the following spaces of functions on the interval (, ): BL((, )) = {f : absolutely continuous on (, ), D(f, f) < }, ((, )) = {f BL((, )) : f(+) = }, H 1 e H 1 ((, )) = BL((, )) L 2 ((, )), H 1((, )) = H1 e ((, )) L2 ((, )). Notice that, in view of [CF3, p 352, the Dirichlet space (resp. the extended Dirichlet space) of X Rθ for each θ [, 2π) is given by H 1((, )) (resp. H1 e ((, ))) which is a Hilbert space with inner product 1 2 D 2 (resp. 1 2 D). Here D α(f, g) = D(f, g) + α fgdx, α >. The Dirichlet form (E, F ) is transient as it is associated with the transient semigroup {Pt } of X. Denote by (Fe, E ) its extended Dirichlet space. Define the space (G, a) by G = {f : f θ H 1 e((, )), for η-a.e. θ, a(f, g) = 1 2 D(f θ, f θ )η(dθ) < }. (5.4) D(f θ, g θ )η(dθ), f, g G. (5.5) We denote by G the -order resolvent of X : G f(x) = P t f(x)dt for any bounded Borel function f on R 2 \ {}. The next theorem gives explicit description the Dirichlet space F, extended Dirichlet space F e and reflected Dirichlet space F,ref of X. Theorem 5.1 (i) It holds that Moreover, C c (R 2 \ {}) is E -dense in F e. (F e, E ) = (G, a). (5.6) 17

18 (ii) It holds that F = { f : f θ H 1 ((, )) for η-a.e. θ, E α(f, g) = 1 2 Moreover, C c (R 2 \ {}) is E 1 -dense in F. } D 1 (f θ, f θ )η(dθ) <. (5.7) D 2α (f θ, g θ )η(dθ) for f, g F, α >. (5.8) (iii) Let ((F ) ref, E,ref ) be the reflected Dirichlet space of (E, F ) and (F ) ref a = (F ) ref L 2 (R 2 \ {}) be its active reflected Dirichlet space. Then (F ) ref = { f : f θ BL((, )) for η-a.e. θ, { (F ) ref a = f : f θ H 1 ((, )) for η-a.e. θ, E,ref (f, g) = 1 2 } D(f θ, f θ )η(dθ) <. (5.9) } D 1 (f θ, f θ )η(dθ) <. (5.1) D(f θ, g θ )η(dθ) for f, g (F ) ref. (5.11) Proof. When the support of the measure η is a finite set, this theorem has been proved in [CF2, 7.6(3). But here η is a general probability measure on [, 2π). (i). We first establish (5.6). By virtue of [CF3, Theorem , it suffices to prove that (a) G is a real Hilbert space with inner product a. (b) For any f C c (R 2 \ {}), G f G and a(g f, v) = R 2 \{}) fvdm for any v G. (5.12) To prove (a), suppose {u n }, n 1, is an a-cauchy sequence in G. Taking a subsequence if needed, we may and do assume that We then have ( n=1 a(u n+1 u n, u n+1 u n ) 2 n. ( ) ) 2 1/2 r (u n+1 u n )(r, θ) dr η(dθ) < This in particular implies that there is a set A [, 2π) with η(a) = so that, for every θ [, 2π) \ A, u n (, θ) He 1 ((, )) for any n 1 and ( n=1 ( ) ) 2 1/2 r (u n+1 u n )(r, θ) dr <. 18

19 It follows that for each θ [, 2π) \ A, u n (r, θ) converges locally uniformly (this follows from Cauchy-Schwarz inequality) on [, ) and in D-norm to some function v θ He 1 ((, )). Moreover, by Fatou s Lemma, we have D(v θ, v θ )η(dθ) < and lim D(u n,θ v θ, u n,θ v θ )η(dθ) =, n in other words, v(r, θ) = v θ (r) G and u n is a-convergent to v. To show (b), we consider the reproducing kernel {g (r, s), (He 1 ((, )), 1 2D) (cf. [F2): < r, s < } of the space g (, s) H 1 e((, )) and 1 2 D(g (, s), v) = v(s) for s (, ), v H 1 e((, )). g (r, s) is positive and continuous on (, ) (, ). For any f C c (R 2 \ {}), (G f) θ (r) = g (r, s)f θ (s)ds so that (G f) θ (+) = and 1 2 D((G f) θ, (G f) θ ) = g (s, s )f θ (s)f θ (s )dsds. Hence G f G and a(g f, G f) = R 2 \{} f(x)(g f)(x)m(dx). The equation in (5.12) can be obtained in the same way. Next suppose u G satisfies a(u, v) = for any v Cc (R 2 \ {}). Taking v(r, θ) = ϕ(r)ψ(θ) with ϕ Cc (, ) and ψ C (S), where S is the unit circle, we have D(u θ, ϕ)ψ(θ)η(dθ) =. Since this holds for any ψ C (S), there exists A S with η(a) = such that, for every θ S\A, u θ He ((, )) and D(u θ, ϕ) = for a countable family of ϕ Cc (, ) uniformly dense in Cc (, ). It follows that u θ =, θ S \ A and hence u = m-a.e. This proves that Cc (R 2 \ {}) is a-dense in G. (ii). (5.7) and (5.8) follow from (i) as F = Fe L 2 (R 2 \ {}, m) and Eα(u, v) = E (u, v) + α R 2 \{} uvdm. The last statement can be shown as the proof of the corresponding one in (i) by using (5.8) in place of (5.5). (iii) follows immediately from (ii), [CF3, Theorem 6.2.5, and the fact that (E, F ) is a local Dirichlet form. Define the approaching probability to by X and its α-order version by φ(x) = P x(ζ <, Xζ = ) [ u α (x) = E x e αζ ; X ζ =, x R 2 \ {}, α >. Let (E, F) be the Dirichlet form of the Walsh s Brownian motion X on L 2 (R 2 ; m) and (F e, E) be its extended Dirichlet space. By virtue of Theorem 4.2 with κ =, F e (resp. F) is the linear subspace of F,ref (resp. Fa,ref ) spanned by Fe and φ (resp. F and u α ). Since φ equals identically 1 on R 2 \ {}, we can deduce from Theorem 5.1 the following theorem. 19

20 Theorem 5.2 It holds that { F e = f : for η-a.e. θ, f θ BL((, )) and f θ (+) = c } for some c independent of θ, D(f θ, f θ )η(dθ) <, (5.13) F = E(f, g) = 1 2 D(f θ, g θ )η(dθ), f, g F e. (5.14) { f : for η-a.e. θ, f θ H 1 ((, )) and f θ (+) = c } for some c independent of θ, D 1 (f θ, f θ )η(dθ) <, (5.15) Moreover E α (f, g) = 1 2 D 2α (f θ, g θ )η(dθ), f, g F. (5.16) F = {f = f + cu α : f F, c R}, (5.17) and, (E, F) is a strongly local, irreducible recurrent and regular Dirichlet form on L 2 (R 2 ; m). Property (5.15) is a consequence of (5.13) and F = F e L 2 (R 2 ; m). The function u α (r, θ) = u α (r) is independent of θ and u α (r), r (, ), is a decreasing function belonging to the space H 1 ((, )) so that u α C (R 2 ). Therefore the regularity of the Dirichlet form (E, F) follows from (5.17) and Theorem 5.1(ii). Remark 5.3 Let A be a non-empty Borel subset of [, 2π). Then Γ = (, ) A E is X -finely open, and m(γ) = if and only if η(a) =. Hence Γ is m-polar with respect to X if and only if η(a) = in view of [CF3, Theorem A Moreover, it is easy to see from the description of X, a set Γ E is m-polar if and only if there is a Borel set A [, 2π) with η(a) = so that Γ (, ) A. Take any function f in the space (F ) ref a of (5.1). We can then see from the above observation that f has an X -fine limit value c R at in the sense of (4.6) if and only if f θ (+) = c for η-a.e. θ. In particular, the space (F ) ref a satisfies the condition (4.9) imposed in Theorem 4.2 (ii) on account of the description (5.7) of the space F. Let A be the L 2 -infinitesimal generator of X. Theorem 4.2(ii) therefore applies and we can deduce from Theorem 5.1 the following characterization of A: Theorem 5.4 f = f(r, θ) D(A) if and only if the following three conditions are satisfied: (i) f belongs to the space F specified by (5.15). (ii) f θ (iii) lim ϵ is absolutely continuous for η-a.e. θ and f θ L2 (R 2 ; m), f θ (ϵ)η(dθ) =. 2

21 In this case, Af = 1 2 f θ. When the support of the measure η is a finite set, this theorem has been proved in [CF3, 7.6 (3). A similar method of the proof works in interpreting the zero flux condition (4.1) with κ = as the one stated in Theorem 5.4(iii). With the Dirichlet form characterization, Theorem 5.2, of Walsh s Brownian motion X in hands, we can apply results from [C1, CK to give an analytic characterization of harmonic and subharmonic functions of X. The following definition is taken from [CK, Definition 2.2, tailored to Walsh s Brownian motion. Definition 5.5 (Sub/Super-harmonicity) Let D be an open set in E. We say that a nearly Borel function u defined on E is subharmonic (resp. superharmonic) in D if u is q.e. finely continuous in D, and for any relatively compact open subset U with Ū D and for q.e. x U, E x [ u (X τu ) < and u(x) E x [u(x τu ) (resp. u(x) E x [u(x τu )). A nearly Borel measurable function u on E is said to be harmonic in D in the weak sense if u is both superharmonic and subharmonic in D. Theorem 5.6 Let D be an open subset of E. Suppose that a nearly Borel u L loc (D; m). The following are equivalent. (i) u is subharmonic in D; (ii) u (F D ) loc and E(u, v) for every non-negative v F C c (D); (iii) For η-a.e. θ [, 2π), r u(r, θ) is convex on each connected open interval of {r (, ) : (r, θ) D}, and, if D, u r (+, θ) exists, and u (+, θ)η(dθ). (5.18) r Proof. The equivalence of (i) and (ii) is a special case of a more general result established in [CK, Theorem 2.7, which extends [C1, Theorem 2.11 from harmonic functions to subharmonic functions. It remains to show that (ii) is equivalent to (iii). Suppose (ii) holds. For every (r, θ ) D \ {}, take v(r, θ) = ϕ(r)ψ(θ) with non-negative ϕ Cc (, ) and ψ C (S) so that the support of v is contained in D \ {}. Note that the linear span of such functions is E1 -dense in F D. Then E(u, v) = D(u(, θ), ϕ)ψ(θ)η(dθ). This implies that for η-a.e. θ [, 2π), 2 u(r,θ) in the distributional sense on {r (, ) : r 2 (r, θ) D} and so r u(r, θ) is convex there. Suppose now D. since for η-a.e. θ [, 2π), r u(r, θ) is convex on {r (, ) : (r, θ) D}, u u r (+, θ) as the decreasing limit of r (ε, θ) as ε + exists. For every ε >, let v ε (r, θ) = ϕ ε (r), where ϕ ε Cc [, ε) with ϕ ε () = 1, ϕ ε(r) 1 21

22 and ϕ ε(r)dr = 1. Clearly supp[v ε D when ε is sufficiently small. So by the monotone convergence theorem, we have u 2π u (+, θ)η(dθ) = lim r ε + r (r, θ)ϕ ε(r)drη(dθ) = lim E(u, v ε). ε + This establishes (iii). By reversing the above argument, we conclude from Theorem 5.2 that (iii) implies (ii). Taking D = E in Theorem 5.6 recovers the main results (Theorems 3.1 and 4.1) of [FiK, which are proved by using a different method. 6 One-point reflections with darning Let (E, m) be as in 3 and X = (X t, ζ, P x ) be an m-symmetric Hunt process on E with continuous sample paths whose Dirichlet form (E, F) on L 2 (E; m) is regular and irreducible. We assume that X admits no killing inside E. We further assume the following condition on the transition function {P t, t > } of X: P t (x, ) is absolutely continuous with respect to m for every t > and x E. (6.1) Consider a compact subset K of E and put E = E \ K. We extend the topological space E to E = E {a } by adding an extra point a to E whose topology is prescribed as follows: a subset U of E containing the point a is an open neighborhood of a if there is an open set U 1 E containing K such that U 1 E = U \ {a }. In other words, E is obtained from E by identifying the points of K into a single point a. The restriction of the measure m to E will be denoted by m, which is then extended to E by setting m ({a }) =. Let X = (Xt, ζ, P x) be the part process of X on E, which is know to be an m -symmetric diffusion process on E. Denote by {G α, α > } the resolvent of X. In parallel to (A.1), (A.3) in 3, we make the following two assumptions on X : (A o.1) φ K (x) =: P x(ζ <, X ζ K) > for any x E. (A o.3) There exists a neighborhood U of K such that inf x V G 1 φk (x) > for any compact set V U \ K. Since X admits no killing inside E, it holds that φ K (x) = P x (σ K < ) x E. (6.2) Theorem 6.1 Under the above assumptions on X and X, there exists a unique one-point reflection X = (Xt, ζ, P x) on E = E {a } of X. Moreover, X is a diffusion process on E. Proof. As for the existence of X, it suffices to verify that X satisfies conditions (A.1), (A.2), (A.3) in 3 but with a in place of a. Then X on E can be constructed just as in 3 for κ = and it is a diffusion process on E. (A.1), (A.3) with a in place of a follow from the present assumptions (A o.1), (A o.3), respectively. Let us verify that X also enjoys the property 22