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DOI: https://doi.org/10.15688/mpcm.jvolsu.2017.3.7
UDC 517 LBC 22.161
EQUIVALENCE OF RECURRENCE AND LIOUVILLE PROPERTY FOR SYMMETRIC DIRICHLET FORMS1
Naotaka Kajino
Associate Professor, Department of Mathematics, Graduate School of Science,
Kobe University
Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan
Abstract. Given a symmetric Dirichlet form (S, T) on a (non-trivial) a-finite measure space (E, B,m) with associated Markovian semigroup {Tt}te(o,^), we prove that (S, T) is both irreducible and recurrent if and only if there is no non-constant B-measurable function u : E ^ [0, <x>] that is S-excessive, i.e., such that Ttu < u m-a.e. for any t E (0, ro). We also prove that these conditions are equivalent to the equality {u E Te | S(u,u) = 0} = R1, where Te denotes the extended Dirichlet space associated with (S, T). The proof is based on simple analytic arguments and requires no additional assumption on the state space or on the form. In the course of the proof we also present a characterization of the S-excessiveness in terms of Te and S, which is valid for any symmetric positivity preserving form.
Key words: symmetric Dirichlet forms, symmetric positivity preserving forms, extended Dirichlet space, excessive functions, recurrence, Liouville property.
1. Introduction and the statement of the main theorem
Since the classical theorem of Liouville saying that there is no non-constant bounded holomorphic function on C, non-existence of non-constant bounded (super-)harmonic functions on the whole space, so-called Liouville property, has been one of the main concerns of harmonic analysis on various spaces. One of the most well-known facts about Liouville ~ property is that the non-existence of non-constant bounded superharmonic functions on the ^ whole space is equivalent to the recurrence of the corresponding stochastic process. Such 2 an equivalence is known to hold for standard processes on locally compact separable metriz-.1 able spaces by Blumenthal and Getoor [1, Chapter II, (4.22)] and also for more general £ right processes by Getoor [9, Proposition (2.4)]. Getoor [8, Proposition 2.14] provides the @ same kind of equivalence in terms of excessive measures. The purpose of this paper is to
give a completely elementary proof of this equivalence in the framework of an arbitrary symmetric Dirichlet form on a (non-trivial) a-finite measure space. Our proof is purely functional-analytic and free of topological notions on the state space, although we need to assume the symmetry of the Dirichlet form.
In the rest of this section, we describe our setting and state the main theorem. We fix a a-finite measure space (E, B,m) throughout this paper, and below all B-measurable functions are assumed to be [—ro, ro]-valued. Let (S, T) be a symmetric Dirichlet form on L2(E,m) and let {Tt}te(o,^) be its associated Markovian semigroup on L2(E,m). Let L+(E,m) := {f | f : E ^ [0, ro], f is B-measurable} and L°(E,m) := {f | f : E ^ ^ R, f is B-measurable}, where we of course identify any two B-measurable functions which are equal m-a.e. Let 1 denote the constant function 1 : E ^ {1}, and we regard R1 := {c1 | c e R} as a linear subspace of L°(E,m). Also let Lp+(E,m) := Lp(E,m) n R L+(E,m) for p e [1, ro] U {0}. Note that Tt is canonically extended to an operator on L+(E,m) and also to a linear operator from V[Tt] := {/ e L°(E,m) | Tt|/1 < ro m-a.e.} to L°(E,m); see Proposition 1 below.
Definition 1. u e L+(E,m) is called S-excessive if and only if Ttu < u m-a.e. for any t e (0, ro). Similarly, u e Hie(0 <x>) ^Pt] is called S-excessive in the wide sense if and only if Ttu < u m-a.e. for any t e (0, ro).
Remark 1. As stated in [1; 2; 6; 7; 14], when we call a function u excessive, it is usual to assume that u is non-negative, which is why we have added "in the wide sense" in the latter part of Definition 1.
S-excessive functions will play the role of superharmonic functions on the whole state space, and the main theorem of this paper (Theorem 1) asserts that (S, T) is irreducible and recurrent if and only if there is no non-constant S-excessive function.
Yet another possible way of formulation of harmonicity of functions (on the whole space E) is to use the extended Dirichlet space Te associated with (S,T); u e Te could be called "superharmonic" if S(u,v) > 0 for any v e Te n L+(E,m), and "harmonic" if S(u,v) = 0 for any v e Te, or equivalently, if S(u,u) = 0. In fact, as a key lemma for the proof of the main theorem, in Proposition 3 below we prove that u e Te is "superharmonic" in this sense if and only if u is S-excessive in the wide sense. Under this formulation of harmonicity, if (S, T) is recurrent, i.e., 1 e Te and S(1,1) = 0, then the non-existence of non-constant harmonic functions amounts to the equality
{u e Te |S(u,u) = 0} = R1. (1.1)
Oshima [10, Theorem 3.1] proved (1.1) (and the completeness of (Te/R1, S) as well) for the Dirichlet form associated with a symmetric Hunt process which is recurrent in the sense of Harris; note that the recurrence in the sense of Harris is stronger than the usual recurrence of the associated Dirichlet form. Fukushima and Takeda [7, Theorem 4.2.4] (see also [2, Theorem 2.1.11]) showed (1.1) for irreducible recurrent symmetric Dirichlet forms (S, T) under the (only) additional assumption that m(E) < ro. In the recent book [2], Chen and Fukushima has extended this result to the case of m(E) = ro when (S, T) is regular, by using the theory of random time changes of Dirichlet spaces. As part of our main theorem, we generalize (1.1) to any irreducible recurrent symmetric Dirichlet form. In fact, this generalization could be obtained (at least when L2(E,m) is separable) also by applying the theory of regular representations of Dirichlet spaces (see [6, Section A.4]) to
reduce the proof to the case where (S, T) is regular. The advantage of our proof is that it is based on totally elementary analytic arguments and is free from any use of time changes or regular representations of Dirichlet spaces.
Here is the statement of our main theorem. See [2, Section 1.1] or [4, Section 1] for basics on Te, and [6, Sections 1.5 and 1.6] or [2, Section 2.1] for details about irreducibility and recurrence of (S,T). We remark that Te C rite(o,^) ^Pt] by Lemma 2-(1) below. We say that (E, B,m) is non-trivial if and only if both m(A) > 0 and m(E \ A) > 0 hold for some A E B, which is equivalent to the condition that L2(E,m) C R1 since (E, B,m) is assumed to be a-finite.
Theorem 1. Consider the following six conditions.
1) (S, T) is both irreducible and recurrent.
2) {u E Te | S(u,u) = 0} = R1.
3) {u E Te n L™(E,m) | S(u,u) = 0} = {c1 | c E [0, w)}.
4) If u E Te is S-excessive in the wide sense then u E R1.
5) If u E L°+(E,m) is S-excessive then u E R1.
6) If u E Te n Lc_¡°(E,m) is S-excessive then u E R1.
The three conditions 1), 2), 3) are equivalent to each other and imply 4), 5), 6). If (E, B,m) is non-trivial, then the six conditions are all equivalent.
The organization of this paper is as follows. In Section 2, we prepare basic results about the extended space Te and S-excessive functions, which are valid as long as (S, T) is a symmetric positivity preserving form. The key results there are Propositions 3 and 4, which are essentially known but seem new in the present general framework. Furthermore Proposition 4 provides a characterization of the notion of S-excessive functions in terms of Te and S. Making use of these two propositions, we show Theorem 1 in Section 3.
2. Preliminaries: the extended (Dirichlet) space and excessive functions
As noted in the previous section, we fix a a-finite measure space (E, B,m) throughout this paper, and all B-measurable functions are assumed to be [-w, w]-valued. Note that by the a-finiteness of (E, B, m) we can take n E L1(E, m) n L™(E, m) such that n > 0 m-a.e. Notation. (0) We follow the convention that N = {1,2, 3,... }, i.e., 0 E N.
(1) For a,b E [-w, w], we write a V b := max{a, b}, a A b := min{a, b}, a+ := a V 0 and a- := —(a A 0). For {a„}„eN C [-w, w] and a E [-w, w], we write an t a (resp. an ^ a) if and only if {a„}„eN is non-decreasing (resp. non-increasing) and an = a. We use the same notation also for (m-equivalence classes of) [-w, w]-valued functions.
(2) As introduced before Definition 1, identifying any two B-measurable functions that are equal m-a.e., we set L+(E,m) := {/ | f : E ^ [0, w], f is B-measurable}, L0(E,m) := := {/ | f : E ^ R, f is B-measurable} and Lp+(E,m) := Lp(E,m) n L+(E,m), p E E [1, w] U {0}. We regard R1 := {c1 | c E R} as a linear subspace of L0(E,m). Let || ■ ||p denote the norm of Lp(E,m) for p E [1, w]. Finally, let (f,g) := JE fgdm for f,g E L+ (E,m) and also for f,g E L°(E,m) with fg E Ll(E,m).
Recall the following definitions regarding bounded linear operators on L2(E,m).
Definition 2. Let T : L2(E,m) ^ L2(E,m) be a bounded linear operator on L2(E,m).
(1) T is called positivity preserving if and only if Tf > 0 m-a.e. for any f E L+(E,m).
(2) T is called Markovian if and only if 0 < Tf < 1 m-a.e. for any f E L2(E,m) with 0 < f < 1 m-a.e.
Clearly, if T is positivity preserving then so is its adjoint T*. Note that if T is Markovian, then it is positivity preserving, ||Tf < ||/||^ for any L2(E,m) n L^(E,m) and ||T*/1| 1 < ||/||i for any f e L1(E,m) n L2(E,m). Moreover, using the a-finiteness of (E, B,m), we easily have the following proposition.
Proposition 1. Let T : L2(E,m) ^ L2(E,m) be a positivity preserving bounded linear operator on L2(E,m).
(1) T(E,m) uniquely extends to a map T : L+(E,m) ^ L+(E,m) such that Tfn t Tf m-a.e. for any f e L+(E,m) and any {/n}ngn C L+(E,m) with fn t f m-a.e. Moreover, let f, g e L+(E,m) and a e [0, ro]. Then T(f + g) = Tf + Tg, T (af) = aT f, (Tf, g) = = (f,T*g), and if f < g m-a.e. then Tf < Tg m-a.e.
(2) Let V[T] := {f e L°(E,m) | T|f| < ro m-a.e.}. Then T : L2(E,m) ^ L2(E,m) is extended to a linear operator T : V[T] ^ L°(E,m) given by Tf := T(f+) — T(f-), f e V[T], so that it has the following properties:
(1) If f,g e V[T] and f < g m-a.e. then Tf < Tg m-a.e.
(ii) If {/njnen C V[T] and f,g e V[T] satisfy limn^^ fn = f m-a.e. and |/n| < |g| m-a.e. for any n e N, then limn^^ Tfn = Tf m-a.e.
Throughout the rest of this paper, we fix a closed symmetric form (S, T) on L2(E,m) together with its associated symmetric strongly continuous contraction semigroup {Tt}te(0,^) and resolvent {Ga}ae(0,^) on L2(E,m); see [6, Chapter 1.3] for basics on closed symmetric forms on Hilbert spaces and their associated semigroups and resolvents.
Let us further recall the following definition. Definition 3. (1) (S, T) is called a positivity preserving form if and only if u+ e T and S(u+,u+) < S(u,u) for any u e T, or equivalently, Tt is positivity preserving for any t e (0, ro).
(2) (S, T) is called a Dirichlet form if and only if u+ A1 e T and S(u+ A 1,u+ A1) < S(u,u) for any u e T, or equivalently, Tt is Markovian for any t e (0, ro).
See, e.g., [11, Section 2] for the equivalences stated in Definition 3. In the rest of this section, we assume that (S, T) is a positivity preserving form. The following definition is standard (see [12, Definition 3], [2, Definition 1.1.4] or [4, Definition 1.4]).
Definition 4. We define the extended space Te associated with (S, T) by
-p .= [ue L0(E m\ lim^œ un = u m-a.e. for some {wra}ra€N С 6 { ( ' J with limfcAi^œ S(uk - щ,ик -щ) = 0 ]'
(2.1)
For u e Te, such {«n}neN C T as in (2.1) is called an approximating sequence for u. When (S, T) is a Dirichlet form, Te is called the extended Dirichlet space associated with (S, T).
Obviously T C Te and Te is a linear subspace of L0(E,m). By virtue of [13, Proposition 2], T = Te n L2(E,m), and for u,v e Te with approximating sequences {«n}neN and {^n}neN, respectively, the limit limn^^ S(un, vn) e R exists and is independent of particular choices of {«n}neN and {fn}neN, as discussed in [12, before Definition 3]. By setting S(u, v) := limn^^ S(un, vn), S is extended to a non-negative definite symmetric bilinear form on Te. Then it is easy to see that limn^^ S(u — un,u — un) = 0 for u e Te and any approximating sequence {«n}neN C T for u. Moreover, we have the following proposition due to Schmuland [12], which is easily proved by utilizing a version [2, Theorem A.4.1-(ii)] of the Banach — Saks theorem.
Proposition 2 ([12, Lemma 2]). Let u E L°(E,m) and {«ra}raeN C T satisfy un =
= u m-a.e. and liminfra^^ S(un,un) < w. Then u E Te, S(u,u) < liminfra^^ S(un,un), and liminf^^, S(un,v) < S(u, v) < limsupra^^ S(un,v) for any v E Te.
In particular, we easily see from Proposition 2 that u+ E Te and S(«+,«+) < S(«,«) for any u E Te.
Remark 2. For symmetric Dirichlet forms, the properties of Te stated above are well-known and most of them are proved in the textbooks [2, Section 1.1] and [7, Section 4.1] and also in [4, Section 1]. In fact, we can verify similar results in a quite general setting; see Schmuland [12] for details.
The next proposition (Proposition 3 below) requires the following lemmas.
Lemma 1. Let n E L1(E,m) n L2(E,m) be such that n > 0 m-a.e., and set ЦuЦTe := := S(u, u)l/2 + fE(|«| A 1)n dm for u E Te. Then we have the following assertions:
(1) !u + vHfs < ЦuЦтe + |H|t£ and ЦauЦтe < (|a| V 1)|M|t£ for any u,v E Te and any a E R.
(2) Te is a complete metric space under the metric dTe given by dTe(u,v) := Un - vH^e.
Proof. (1) is immediate and dTe is clearly a metric on Te. For the proof of its completeness, let {«ra}raeN C Te be a Cauchy sequence in (Te,dTe). Noting that T is dense in (Te,dTe), for each n E N take vn E T such that - Un!^ < n-1. Then {i>ra}raeN is also a Cauchy sequence in (Te,dTe). A Borel-Cantelli argument easily yields a subsequence {vnk}ken of {^ra}raeN converging m-a.e. to some u E L0(E,m), which means that u E Te with approximating sequence {vnk}ken and hence that mu - vnk ||Te = 0. The same
argument also implies that every subsequence of {fra}raeN admits a further subsequence converging to u in (Te,dTe), from which Цu-v-nm^ = 0 follows. Thus Цu-
- «Jt = 0.
Lemma 2. (1) Te C P|te(0 V[Tt] and Tt(Te) C Te for any t E (0, w). (2) Let n and || ■ ||Te be as in Lemma 1, and let u E Te. Then S(Ttu,Ttu) < S(u,u), h-Ttu||2 < iS(u,u) and mu^ < (3+ hh^M^ for any t E (0, w), TsTtu = Ts+tu for any s,t E (0, w), and lim^0 Цu - Tturn^^ = 0.
Proof. Let n, || ■ Цтe and dTe be as in Lemma 1. First we prove (2) for u E T. The fourth assertion is clear. Ttu E T and S(Ttu,Ttu) < S(u,u) for t E (0, w) by [6, Lemma 1.3.3-(i)], and limt|0 Цu - Tturn^^ =0 by [6, Lemma 1.3.3-(iii)]. Let t E (0, w). Noting that (f - Ttf,Ttf) = ||Tt/2/1|2 -mf ||2 > 0 for f E L2(E,m), we have ||u - T^ = (u -
- Ttu,u) - (u - Ttu,Ttu) < (u - Ttu,u) < tS(u,u) by [6, Lemma 1.3.4-(i)]. Applying these estimates to ||u - T^T < S(u,u)l/2 + S(Ttu,Ttu)l/2 + ||n- Ttu^ easily yields mu^ < (3+ N^Nk.
Now since T is dense in a complete metric space (Te, dTJ, it follows from the previous paragraph that Tt|T is uniquely extended to a continuous map T^r from (Te,dTe) to itself, and then clearly T.f is linear and the assertions of (2) are true with T.f in place of Tt.
Let t E (0, w) and u E Te n L+(E, m). It remains to show T^ru = Ttu, as v+, v~ E Te for v E Te. Since v+ A u E Te n L2(E, m) = T and S(v+ A u,v+ A u)l/2 < S(v, v)l/2 + + S(u,u)l/2 for any v E T by the positivity preserving property of (S, T), an application of the Banach-Saks theorem [2, Theorem A.4.1-(ii)] assures the existence of an approximating sequence {u>ra}raeN for u such that 0 < wn < u m-a.e. A Borel — Cantelli argument yields
a subsequence {wnfc}fceN such that limk^^Ttwnk = Tfu m-a.e., and Tfu = Ttu follows by letting k ^ ro in Tt(inij>kwnj) < TtWnk < Ttu m-a.e.
The following proposition (Proposition 3), which seems new in spite of its easiness, plays an essential role in the proof of 1) ^ 2) of Theorem 1. Proposition 3-(2) is an extension of a result of Chen and Kuwae [3, Lemma 3.1] for functions in T to those in Te, and Proposition 3-(3) extends a basic fact for functions in T to those in Te. Proposition 3. (1) Let u e Te and v e T. Then
1 f*
lim — (u — Ttu, v) = S (u, v) and (u — Ttu, v) = S (u,Tsv )ds, te (0, ro). (2.2) 4° t Jo
(2) Let u e Te. Then u is S-excessive in the wide sense if and only if S(u, v) > 0 for any v e T n L+(E, m), or equivalently, for any v e Te n L+(E,m).
(3) Let u e Te. Then Ttu = u for any t e (0, ro) if and only if S(u, u) = 0.
Proof. (1) Let u e Te, v e T and set p(t) := (u — Ttu, v) for t e [0, ro), where T°u := u. Then t-1^^ < S(u,u)1/2S(v, v)1/2 for t e (0, ro) and lim4° t-1q(t) = S(u, v) if u e T by [6, Lemma 1.3.4-(i)], and the same are true for u e Te as well by Lemma 2. Using Lemma 2, we easily see also that ty'(t) = S(u,Ttv) for t e [0, ro) and that cp' is continuous on [0, ro), proving (2.2).
(2) The third assertion of Proposition 2 together with the positivity preserving property of (S, T) easily implies that S(u, v) > 0 for any v e TnL+(£,m) if and only if the same is true for any v e Te n L+(E,m). The rest of the assertion is immediate from (2.2).
(3) This is an immediate consequence of (2).
The next proposition (Proposition 4), which characterizes the notion of S-excessive functions in terms of Te and S, is of independent interest. The proof is based on a result [11, Corollary 2.4] of Ouhabaz which provides a characterization of invariance of closed convex sets for semigroups on Hilbert spaces. A similar argument in a more general framework can be found in Shigekawa [14].
Proposition 4. Let u e L+(E,m). Then u is S-excessive if and only if v Au e Te and S (v Au,v Au) < S (v, v) for any v e Te.
1. The notion of S-excessive functions is determined solely by the pair (Te, S) of the extended space Te and the form S : Te x Te ^ R.
2. Let u e L+(E,m) be S-excessive and v e Te. Suppose u < v m-a.e. Then u e Te and S(«, m) < S(w, w).
Remark 3. Chen and Kuwae [3, Lemma 3.3] gave a probabilistic proof of Corollary 2 for the Dirichlet forms associated with symmetric right Markov processes.
Proof of Proposition 4. Let Ku := {f e L2(E,m) | f < u m-a.e.}, which is clearly a closed convex subset of L2(E,m). We claim that
u is S-excessive if and only if Tt(Ku) C Ku for any t e (0, ro). (2.3)
Indeed, let t e (0, ro). If Ttu < u m-a.e. then Ttf < Ttu < u m-a.e. for any f e Ku and hence Tt(Ku) C Ku. Conversely if Tt(Ku) C Ku, then choosing n e L2(E,m) so that n > 0 m-a.e., we have (nn) Au t « m-a.e., (nn)Aw e Ku and hence Ttu = limn^^ Tt((nn) Au) < m m-a.e.
On the other hand, since the projection of f E L2(E,m) on Ku is given by f A u, [11, Corollary 2.4] tells us that Tt(Ku) C Ku for any t E (0, w) if and only if
v A u ET and S(v A u,v A u) <S(v,v) for any v ET. (2.4)
Finally, Te n L2(E,m) = T and Proposition 2 easily imply that (2.4) is equivalent to the same condition with Te in place of T, completing the proof.
3. Proof of Theorem 1
We are now ready for the proof of Theorem 1. We assume throughout this section that our closed symmetric form (S,T) is a Dirichlet form. The proof consists of three steps. The first one is Proposition 5 below, which establishes 1) ^ 2) of Theorem 1 and whose proof makes full use of Proposition 3-(3). Recall the following notions concerning the irreducibility of (S,T); see [6, Section 1.6] or [2, Section 2.1] for details. Definition 5. (1) A set A E B is called S-invariant if and only if 1ATt(f 1E\a) = 0 m-a.e. for any f E L2(E, m) and any t E (0, w).
(2) (S, T) is called irreducible if and only if either m(A) = 0 or m(E \ A) = 0 holds for any S-invariant A E B.
Lemma 3. Let u E L+(E,m) be S-excessive. Then {u = 0} is S-invariant.
Proof. In fact, the following proof is valid as long as (S, T) is a symmetric positivity preserving form. Let B := {u = 0}, f E L2(E,m) and set fn := U| A (nu) for n E N, so that fn t ^|1e\b m-a.e. Then 0 < 1BTtfn < 1BTt(nu) < n1Bu = 0 m-a.e., and letting n ^ w leads to |1BTt(f 1E\B)| < 1BTtdf|1E\B) = 0 m-a.e. Thus B = {u = 0} is S-invariant.
Proposition 5. Suppose that (S, T) is irreducible. If u E Te and S(u,u) = 0 then u E R1.
Proof. We follow [2, Proof of Theorem 2.1.11, (i) ^ (ii)]. Let u E Te satisfy S(u,u) = 0. We may assume that m({u > 0}) > 0. Let A E [0, w) and uA := u - u A A. Since (S, T) is assumed to be a Dirichlet form, uA E Te n L+ (E,m) and S(uA,uA) = 0 (see Proposition 4), and therefore TtuA = uA for any t E (0, w) by Proposition 3-(3). Then {«A = 0} is S-invariant by Lemma 3, and the irreducibility of (S, T) implies that either m({uA = 0}) = 0 or m({uA > 0}) = 0 holds. Now setting k := sup{A E [0, w) | m({uA = 0}) = 0}, we easily see that k e (0, w) and that u = k m-a.e.
For the rest of the proof of Theorem 1, let us recall basic notions concerning recurrence and transience of Dirichlet forms. See [6, Sections 1.5 and 1.6] or [2, Section 2.1] for details. For t E (0, w), we define St : L2(E,m) ^ L2(E,m) by Stf := ¡0 Tsf ds, where the integral is the Riemann integral in L2(E,m). Then t-lSt is a Markovian symmetric bounded linear operator on L2(E,m), and therefore it is canonically extended to an operator on L+(E,m) by Proposition 1. Furthermore, for any s,t E (0, w) we easily see that Ss+t = Ss + TsSt = Ss + StTs as operators on L+(E,m) or on L2(E,m).
Let f E L+(E,m). Then 0 < Ssf < Stf m-a.e. and 0 < Gpf < GJ m-a.e. for 0 < s < t, 0 < a < p. Therefore there exists a unique Gf E L+(E,m) satisfying SNf t Gf m-a.e. It is immediate that Gfn t Gf m-a.e. for any {/„}raeN C L+(E,m) with fn t f TO-a.e. Since, on L2(E,m), {Ga}ae(0,^) is the Laplace transform of {Tt}te(0,^), we see that Stn f t Gf m-a.e. and Ga„ f t Gf m-a.e. for any {tn}neN, {ara}ra€N C (0, w) with
tn t œ, an | 0. Moreover, since St+Nf = Stf + TtSNf > TtSNf m-a.e. for t G (0, œ) and N G N, by letting N ^ œ we have TtGf < Gf m-a.e., that is, Gf is 8-excessive. We call this operator G : L+(E,m) ^ L+(E,m) the 0-resolvent associated with (8, T).
Definition 6 (Transience and Recurrence). (1) (8, T) is called transient if and only if Gf < œ m-a.e. for some f G L+(E,m) with f > 0 m-a.e.
(2) (8, T) is called recurrent if and only if m({0 < Gf < œ}) = 0 for any f G L+(E,m).
By [6, Lemma 1.5.1], (8,T) is transient if and only if Gf < œ m-a.e. for any f G L+(E,m). On the other hand, by [6, Theorem 1.6.3], (8,T) is recurrent if and only if 1 G Te and 8(1,1) = 0.
The following proposition is the second step of the proof of Theorem 1.
Proposition 6. Assume that (8, T) is recurrent. If u G L°+(E,m) is 8-excessive then u G Te and 8(u, u) = 0.
Proof. Let n G N. Then uAn < ni m-a.e., ni G Te and 8(n1,n1) = 0 by the recurrence of (8, T), and u An is 8-excessive since so are u and 1. Thus u An G Te and 8 (u An, u An) = 0 by Corollary 2. Lemma 1-(2) implies that limn^œ \\v — u A n||^-e = 0 for some v G Te with | ■ \\^e as defined there, and then we easily have u = v G Te and 8(u,u) = 0.
As the third step, now we finish the proof of Theorem 1.
Proof of Theorem 1. 1) ^ 2) follows by Proposition 5, and so does 1) ^ 5) by Propositions 5 and 6. 2) ^ 3), 4) ^ 6) and 5) ^ 6) are trivial.
1) ^ 4): Let u G Te be 8-excessive in the wide sense, n G N and un := u A n. Then un G Te, un is also 8-excessive in the wide sense, n1 — un G Te n L+(E,m) and hence 8(un,un) = 8(un,un — n1) < 0 by Proposition 3-(2). As in the proof of Proposition 6, letting n ^ œ we get 8(u,u) = 0 by Lemma 1-(2), and hence u G R1 by Proposition 5.
3) ^ 1): (8, T) is recurrent since 1 G Te and 8(1,1) = 0. Let A G B be 8-invariant. Then 1A = 1A1 G Te nL™(E,m) and 0 < 8(1A, 1A) < 8(1,1) = 0 by [6, Theorem 1.6.1]. Now 3) implies 1A G R1, and hence either m(A) = 0 or m(E \ A) = 0.
6) ^ 3) when (E, B,m) is non-trivial: Choose g G L1(E,m) so that g > 0 m-a.e., and set Ec := {Gg = œ}. Then 1Ec G Te n L™(E,m) and 8(1Ec, 1Ec) = 0 by [6, Corollary 1.6.2], and 6) together with Proposition 3-(3) implies 1Ec G R1, i.e., either m(Ec) = 0 or m(E \ Ec) = 0. In view of 6) and Proposition 3-(3), it suffices to show m(E \ Ec) = 0.
Suppose m(Ec) = 0, so that (8, T) is transient, and set n := g/(1 VGg). Then 0 < n < < g m-a.e. and (n,Gn) < (g/(1 VGg),Gg) < \\^\i < œ. Let f G L+(E,m)nL2(E,m) and set fn := fA(nn) for n G N. Then fn G L+(E,m), Gfn < nGn < œ m-a.e., (fn, Gfn) < œ and fn t f m-a.e. Since 8(Gœfn,Gœfn) < (fn,Gœfn) < (fn,Gfn) < œ for a G (0, œ), Proposition 2 implies Gfn G Te. Since Gfn is 8-excessive, so is n A Gfn G Te n L°^(E,m) and 6) yields n A Gfn G R1. Letting n ^ œ and noting Gf < œ m-a.e. by the transience of (8, T), we get Gf G R1. Let a G (0, œ). Then Gaf G L+(E,m) nL2(E,m) and hence GGaf G R1. Letting n ^ œ in Gaf = G1/nf — (a — 1/n)G1/^Gaf implies that Gaf = Gf — aGGaf G R1. Since aGaf ^ / in L2(E,m) as a ^ œ, we conclude that L+(E,m) nL2(E,m) C R1, contradicting the assumption that (E, B,m) is non-trivial. Thus m(E \Ec) = 0 follows.
Acknowledgements
The author would like to express his deepest gratitude toward Professor Masatoshi Fukushima for fruitful discussions and for having suggested this problem to him in [5]. The author would like to thank Professor Masanori Hino for detailed valuable comments on the proofs in an earlier version of the manuscript; in particular, the proofs of Propositions 3 and 6 have been much simplified by following his suggestion of the use of Lemma 1 and Corollary 2. The author would like to thank also Professor Masayoshi Takeda and Professor Jun Kigami for valuable comments.
REMARK
l JSPS Research Fellow PD (20-6088): The author was supported by the Japan Society for the Promotion of Science.
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ЭКВИВАЛЕНТНОСТЬ РЕКУРРЕНТНОСТИ И ЛИУВИЛЛЕВА СВОЙСТВА ДЛЯ СИММЕТРИЧНЫХ ФОРМ ДИРИХЛЕ
Наотака Кадзино
Доцент, Факультет математики, Высшая естественнонаучная школа,
Kobe University
Rokkodai-cho 1-1, Nada-ku, Kobe 657-8501, Japan
Аннотация. Рассмотрим симметричную форму Дирихле (Е, Т) на а-конечном (нетривиальном) метрическом пространстве (Е, Б,т) с ассоциированной марковской полугруппой В работе доказано, что (Е, Т) несократимая и рекуррентная тогда и только тогда, когда не существует непостоянной Б-измеримой и Е-эксцессивной функции и : Е ^ [0, то], то есть такой, что Ttu < и ra-a.e. для всех t Е (0, то). Так же доказано, что эти условия эквивалентны равенству {и Е Те | Е(и, и) = 0} = R1, где Те означает расширенное пространство Дирихле, ассоциированное с (Е, Т). Доказательство чисто аналитическое и не требует дополнительных ограничений на фазовое пространство и форму. В процессе доказательства так же представлена характеристика Е-эксцессивности в терминах Те и Е, которая справедлива для любой симметричной формы, сохраняющей положительность.
Ключевые слова: симметричные формы Дирихле; симметричные формы, сохраняющие положительность; расширенное пространство Дирихле, эксцес-сивные функции, рекуррентность, лиувиллево свойство.
