On Phragme'n - Lindelo"f principle for non-divergence type elliptic equations and mixed boundary conditions

DOI:

give a completely elementary proof of this equivalence in the framework of an arbitrary symmetric Dirichlet form on a (non-trivial) σ-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 ст-finite measure space (E , B, m) throughout this paper, and below all B -measurable functions are assumed to be to, to] -valued. Let (8 , E ) be a symmetric Dirichlet form on L ² (E,m) and let {T _t } t e (o , ^ ) be its associated Markovian semigroup on L²(E,m) . Let L ₊ (E,m) := { / | / : E ^ [0. to], / is B -measurable } and L ⁰ (E,m) := { / | / : E ^ ^ R. / 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 | с E R } as a linear subspace of L ⁰ (E,m) . Also let L + (E,m) : = L ^p (E,m) П П L ₊ (E,m) for p E [1, to] U { 0 } . Note that T _t is canonically extended to an operator on L ₊ (E,m) and also to a linear operator from D[T _t ] := { / E L°(E,m) | T _t |/ 1 <  to m -a.e. } to L ⁰ (E,m) ; see Proposition 1 below.

Definition 1. и E L ₊ (E,m) is called 8 -excessive if and only if T _t и <  и m -a.e. for any t E (0, to) . Similarly, и E Q _te(₀ ^) ^ [T _t ] is called 8 -excessive in the wide sense if and only if Ти <  и m -a.e. for any t E (0, to) .

Remark 1. As stated in [1; 2; 6; 7; 14], when we call a function и excessive , it is usual to assume that и is non-negative , which is why we have added “in the wide sense” in the latter part of Definition 1.

8 -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 ( 8 , E ) is irreducible and recurrent if and only if there is no non-constant 8 -excessive function.

Yet another possible way of formulation of harmonicity of functions (on the whole space E ) is to use the extended Dirichlet space E _e associated with ( 8 , E ) ; и E E _e could be called “superharmonic” if 8 ( u,v ) >  0 for any v E E _e П L ₊ (E,m) , and “harmonic” if 8 ( u,v ) = 0 for any v E E _e , or equivalently, if 8(и, и) = 0 . In fact, as a key lemma for the proof of the main theorem, in Proposition 3 below we prove that и E E _e is “superharmonic” in this sense if and only if и is 8 -excessive in the wide sense. Under this formulation of harmonicity, if ( 8 , E ) is recurrent , i.e., 1 E E _e and 8 (1,1) = 0 , then the non-existence of non-constant harmonic functions amounts to the equality

{ и E E _e | 8 (и,и) = 0 } = R1. (1.1)

Oshima [10, Theorem 3.1] proved (1.1) (and the completeness of (Ee/R1,8) 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 (8, E) under the (only) additional assumption that m(E) < to. In the recent book [2], Chen and Fukushima has extended this result to the case of m(E) = to when (8, E) 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 (8, У) 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 У _е , and [6, Sections 1.5 and 1.6] or [2, Section 2.1] for details about irreducibility and recurrence of ( 8 , У ) . We remark that У _е С П1е(о,^) '^ [T _t ] 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 £ B , which is equivalent to the condition that L²(E,m) С R1 since (E, B, m) is assumed to be σ-finite.

Theorem 1. Consider the following six conditions.

• 1)    (8 , У ) is both irreducible and recurrent.

• 2)    {и £ У _е | 8(и, и) = 0 } = R1 .

• 3)    {и £ У _е П /А (E, m) | 8(и, и) = 0 } = { с1 | с £ [0, то ) } .

• 4)    If и £ У _е is 8-excessive in the wide sense then и £ R1 .

• 5)    If и £ L + (E,m) is 8-excessive then и £ R1 .

• 6)    If и £ У _е П /А (E, m) is 8-excessive then и £ 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 У _е and 8 -excessive functions, which are valid as long as (8, У ) 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 ℰ -excessive functions in terms of У _е and 8 . Making use of these two propositions, we show Theorem 1 in Section 3.

Clearly, if T is positivity preserving then so is its adjoint T * . Note that if T is Markovian, then it is positivity preserving, ^ Tf ||_ro< ||/||_ro for any L ² (E,m) П L ^x (E,m) and || T * /||i <  ||f ||i for any / G L1(E,m) П L ² (E,m) . Moreover, using the ^-finiteness of (E, B,m) , we easily have the following proposition.

Proposition 1. Let T : L ² (E,m) ^ L ² (E,m) be a positivity preserving bounded linear operator on L²(E,m).

• (1)    T | _L ^ (_E , _m) uniquely extends to a map T : L ₊ (E,m) ^ L ₊ (E,m) such that Tf _n f T/ m-a.e. for any / G L ₊ (E,m) and any { / _n } _nGN C L ₊ (E,m) with f _n f / m-a.e. Moreover, let /,g G L + (E,m) and a G [0, to] . Then T(/ + g) = T/ + Tg, T(a/ ) = aT /, (T^g) =

= ( /, T * g ) , and if / <  g m-a.e. then T/ <  Tg m-a.e.

• (2)    Let T>[T ] := { / G L ⁰ (E,m) | T | / 1 <  to m-a.e. } . Then T : L ² (E,m) ^ L ² (E,m) is extended to a linear operator T : P [T] ^ L ⁰ (E,m) given by T/ := T(/ ⁺ ) — T(/ ^- ) , / G E[T ] , so that it has the following properties:

• (i)    If /,g G E[T ] and / <  g m-a.e. then T/ <  Tg m-a.e.

• (ii)    If { / n } nGN C P [T] and /,g G D[T ] satisfy lim n^ / n = / m-a.e. and | / n | < | g | m-a.e. for any n G N , then lim _n^^ T/ _n = T/ m-a.e.

Throughout the rest of this paper, we fix a closed symmetric form (8, E ) on L ² (E,m) together with its associated symmetric strongly continuous contraction semigroup {T _t } _tG(₀ , _ro) and resolvent { G _a } _aG(₀,^) on L ² (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) ( 8 , E ) is called a positivity preserving form if and only if u ⁺ G E and 8 (u ⁺ ,u ⁺ ) <  8(и, и) for any и G E , or equivalently, T _t is positivity preserving for any t G (0, to) .

• (2)    ( 8 , E ) is called a Dirichlet form if and only if и ⁺ Л 1 G E and 8(и ⁺ Л 1,u ⁺ Л 1) <  8(и, и) for any и G E , or equivalently, T _t is Markovian for any t G (0, to) .

See, e.g., [11, Section 2] for the equivalences stated in Definition 3.

In the rest of this section, we assume that ( 8 , E ) 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 E _e associated with ( 8 , E ) by

E e := {и G L ⁰ (E,m)

lim _n^^ и _n = и m -a.e. for some { и _n } _nGN C E with lim i i ,_x 8(и _к — u i , и _к — U i ) = 0

}•

(2.1)

For и G E _e , such { u _n } _neN C E as in (2.1) is called an approximating sequence for u. When ( 8 , E ) is a Dirichlet form, E _e is called the extended Dirichlet space associated with ( 8 , E ) .

Obviously E C E _e and E _e is a linear subspace of L ⁰ (E,m) . By virtue of [13, Proposition 2], E = E _e П L ² (E,m) , and for u,v G E _e with approximating sequences { u _n } _nGN and { v _n } _nGN , respectively, the limit lim _n^^ 8(u _n ,v_n) G R exists and is independent of particular choices of { u _n } _nGN and { v _n } _nGN , as discussed in [12, before Definition 3]. By setting 8 ( u,v ) := lim _n^^ 8 (u _n ,v _n ) , 8 is extended to a non-negative definite symmetric bilinear form on E _e . Then it is easy to see that lim _n^^ 8(и — u _n , и — u _n ) = 0 for и G E _e and any approximating sequence { u _n } _nGN C E for и . 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 и E L°(E,m) and {u _t } _{t g N} С E satisfy lim _T^^ и _п = = и m-a.e. and liminf _п ^ _ю 8 (и _п ,и _п ) <  то . Then и E E _e , 8 (u,u) <  liminf _Т ^^ 8 (и _п ,и_п'), and liminf _Т^^ 8(и _п ,и) <  8(и, и) <  limsup _T^^ 8(и^и) for any и E E _e .

In particular, we easily see from Proposition 2 that u ⁺ E E _e and 8 (u ⁺ ,u ⁺ ) <  8(и, и) for any и E E _e .

Remark 2. For symmetric Dirichlet forms, the properties of E _e 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 L 1 (E,m) П L ² (E,m) be such that n > 0 m-a.e., and set ||u | _{E e} : = ^:= ^{8 ( и,и ) 1/2} + J_E ^{( |} и | Л 1) n dm for и E E _e . Then we have the following assertions:

• (1)    1и + v | _{E e} < | и | _{Е е} + | v | _{E e} and | аи | _{Е е} <  ( | a | V 1) | и | _{Е е} for any и, и E E e and any ct E R .

• (2)    E _e is a complete metric space under the metric d ^ _e given by d ^ _e (и,и) := ||и — v | _{E e} .

Proof. (1) is immediate and d ^ _e is clearly a metric on E _e . For the proof of its completeness, let {u _t } _{t g N} С E _e be a Cauchy sequence in (E _e ,d ^ _e ) . Noting that E is dense in ( E _e , d ^ _e ) , for each n E N take и _Т E E such that | и _п — u _n | _{E e} <  n ^-1 . Then {v _t } _{t g N} is also a Cauchy sequence in ( E _e ,d _{E e} ) . A Borel-Cantelli argument easily yields a subsequence {и _{Т к} } ^ _{g N} of {u _t } _{t g N} converging m -a.e. to some и E L°(E,m) , which means that и E E _e with approximating sequence {и _{п к} } _fcGN and hence that lim _G ,_x ||и — и _{п к} ||_Ee = 0 . The same argument also implies that every subsequence of { v _T } _raGN admits a further subsequence converging to и in ( E _e ,d ^ _e ) , from which lim _T^^ ||u — v _T | _{E e} = 0 follows. Thus lim _T^^ ||u — ^— " т _E = ⁰

Lemma 2. (1) E _e С Q _fG(₀,^) ^ [T _t ] and T _t (E _e ) С E _e for any t E (0, то ) .

• (2)    Let n and || • ||_Ee be as in Lemma 1 , and let и E E _e . Then 8(T _t u,T _t и) <  8 (u,u) , lu — T t u | 2 <  t8 (u,u) and НЗД! ^ <  (3+ | n | 2 y t) | u | _E e for any t E (0, то ) , T _s T t U = T _s+t U for any s,t E (0, то ) , and lim _t^₀ ||u — T _t u | _{E e} = 0 .

Proof. Let η, H • ll^e and d ^ _e be as in Lemma 1. First we prove (2) for и E E . The fourth assertion is clear. T _t u E E and 8(T _t u,T _t u) <  8 (u,u) for t E (0, то ) by [6, Lemma 1.3.3-(i)], and lim _t^₀ Ци — T _t u^ ^ _e = 0 by [6, Lemma 1.3.3-(iii)]. Let t E (0, то ) . Noting that ( / — T t f,T t / ) = | T t/2 f ||2 — || T t / ||2 >  0 for / E L ² (E,m) , we have Ци — T t u | 2 = ( и — — T _t u,u) — (и — T _t u,T _t u) < (и — T _t u,u) <  t8 (и, и) by [6, Lemma 1.3.4-(i)]. Applying these estimates to Ци — T _t u | _{E e} <  8 (и, u ) ^{1 / 2} + 8(T _t u, T _t u) ^1/2 + | n | ₂ | u — T _t u | ₂ easily yields » T < U « Z , <  (3+ | n « 2 V t) | u | , . .

Now since E is dense in a complete metric space ( E _e , d ^ _e ) , it follows from the previous paragraph that T _t | _E is uniquely extended to a continuous map T _t ^e from (E _e ,d ^ _e ) to itself, and then clearly T _t ^e is linear and the assertions of (2) are true with T _t ^e in place of T _t .

Let t E (0, то) and и E Ee П L+(E,m). It remains to show Tteu = Ttu, as и+,и- E Ee for и E Ee. Since и+ Ли E Ee П L2(E,m) = E and 8 (и+ Л и,и+ Л и)1/2 < 8 (и,и)1/2 + + 8(и,и)1/2 for any и E E by the positivity preserving property of (8, E), an application of the Banach-Saks theorem [2, Theorem A.4.1-(ii)] assures the existence of an approximating sequence {wn}nGN for и such that 0 < wn < и m-a.e. A Borel — Cantelli argument yields a subsequence {wnk }fceN such that limfc^^ Ttw„fe = Tteu m-a.e., and 7/ u = Ttu follows by letting к ^ то in Tt(infj>k wnj) < 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 У to those in У _е , and Proposition 3-(3) extends a basic fact for functions in У to those in У _е .

Proposition 3. (1) Let u G Уе and v G У. Then lim-(u — Tu,v) = 8(u,v)   and   (u - Ttu,v^ = / 8(u,Tsv^ds, t G (0, то). (2.2)

A0 t '                 ' ^x '                   '              ' ' J 0

• (2)    Let u G У _е . Then u is 8-excessive in the wide sense if and only if 8 (u, v) >  0 for any v G У П L + (E,m), or equivalently, for any v G У _е П L + (E,m).

• (3)    Let u G У _е . Then T _t u = u for any t G (0, то ) if and only if 8(u, u) = 0 .

Proof. (1) Let u G У _е , v G У and set

(u — T _t u,v) for t G [0, то ) , where T 0 u := u . Then t ^-1 | ^ (t )| <  8 (u,u) ^1/2 8 (v,v) ^1/2 for t G (0, то ) and lim _t^₀ t ^-1 ^ (t) = 8 (u,v) if u G У by [6, Lemma 1.3.4-(i)], and the same are true for u G У _е as well by Lemma 2. Using Lemma 2, we easily see also that ^ ‘ (t) = 8(u,T _t v) for t G [0, то ) and that ф ‘ is continuous on [0 , то ) , proving (2.2).

• (2)    The third assertion of Proposition 2 together with the positivity preserving property of ( 8 , У ) easily implies that 8 (u,v) >  0 for any v G У П L + (E,m) if and only if the same is true for any v GУ _е П 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 ℰ -excessive functions in terms of У _е and 8 , 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 G L + (E,m). Then u is 8-excessive if and only if v ^u G У _е and 8 (v A u, v A u) <8 (v,v) for any v G У _е .

• 1.    The notion of 8-excessive functions is determined solely by the pair (У_е, 8) of the extended space У_еand the form 8 : У_е x У_е ^ R.

• 2.    Let u G L+(E,m) be 8-excessive and v G У_е. Suppose u< v m-a.e. Then u G У_е and 8(u,u) < 8(v,v).

3.    Proof of Theorem 1

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 := {/ G L2(E,m) | / < u m-a.e.}, which is clearly a closed convex subset of L2(E,m). We claim that u is 8-excessive if and only if Tt(Ku) C Ku for any t G (0, то).        (2.3)

Indeed, let t G (0, то ) . If T _t u < u m -a.e. then T _t f <  T _t u < u m -a.e. for any / G K _u and hence T _t (K _u ) C K _u . Conversely if T _t (K _u ) C K _u , then choosing n G L ² (E,m) so that n > 0 m -a.e., we have (n n ) A u t u m -a.e., (n n ) A u G K _u and hence T _t u = lim _n^^ T _t ((n n )Au) <  u m -a.e.

On the other hand, since the projection of / G L2(E,m) on Ku is given by / Л и, [11, Corollary 2.4] tells us that Tt(Ku) C Ku for any t G (0, to) if and only if v Ли GT and 8 (v Ли, и Ли) <8 (и, и)    for any и GT.         (2.4)

Finally, E _e П L ² (E,m) = T and Proposition 2 easily imply that (2.4) is equivalent to the same condition with T _e in place of T , completing the proof.

We are now ready for the proof of Theorem 1. We assume throughout this section that our closed symmetric form (8, 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 (8, T ) ; see [6, Section 1.6] or [2, Section 2.1] for details.

Definition 5. (1) A set A G В is called 8 -invariant if and only if 1 ^ T _t (/ 1 е \ л ) = 0 m -a.e. for any / G L²(E, m) and any t G (0, to) .

• (2)    (8, T ) is called irreducible if and only if either m(A) = 0 or m(E \ A) = 0 holds for any 8 -invariant A G В .

Lemma 3. Let и G L ₊ (E,m) be 8-excessive. Then {u = 0 } is 8-invariant.

Proof. In fact, the following proof is valid as long as (8, T ) is a symmetric positivity preserving form. Let В := {и = 0 } , / G L²(E,m) and set / _n := | / | Л (uu) for и G N , so that / _n t I / | 1 e \ b m -a.e. Then 0 <  1 _B T _t / _n <  1 _B T _t (nu) <  u 1 _B и = 0 m -a.e., and letting и ^ to leads to | 1 _B T _t (/ 1 e \ _B ) | <  1 _B T _t (I/ | 1 e \_B ) = 0 m -a.e. Thus В = { и = 0 } is ℰ -invariant.

Proposition 5. Suppose that ( 8 , T ) is irreducible. If и G T _e and 8(и,и) = 0 then и G R1 .

Proof. We follow [2, Proof of Theorem 2.1.11, (i) ^ (ii)]. Let и G Te satisfy 8(и, и) = 0. We may assume that m({u > 0}) > 0. Let Л G [0, to) and ил := и — и Л Л. Since (8, T) is assumed to be a Dirichlet form, ил G Te П L+(E,m) and 8(ил,ил) = 0 (see Proposition 4), and therefore Ttuл = ил for any t G (0, to) by Proposition 3-(3). Then {ил = 0} is 8-invariant by Lemma 3, and the irreducibility of (8, T) implies that either m({uл = 0}) = 0 or m({uл > 0}) = 0 holds. Now setting к := вир{Л G [0, to) | m({uл = 0}) = 0}, we easily see that к G (0, to) and that и = к 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 G (0, to), we define St : L2(E,m) ^ L2(E,m) by St/ := J^Tg/ds, where the integral is the Riemann integral in L2(E,m). Then t-1St 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 G (0, to) we easily see that Ss+t = Ss + TsSt = Ss + StTs as operators on L+(E,m) or on L2(E,m).

Let / G L+(E,m). Then 0 < Ss/ < St/ m-a.e. and 0 < Ge/ < Ga/ m-a.e. for 0 < s < t, 0 < a < в. Therefore there exists a unique G/ G L+(E,m) satisfying Sn/ t G/ m-a.e. It is immediate that G/n t G/ m-a.e. for any {/„}„eN C L+(E, m) with /n t / m-a.e. Since, on L2(E,m), {Ga}ae(o,^) is the Laplace transform of {Tt}te(o,^), we see that St„/ t G/ m-a.e. and Ga„/ t G/ m-a.e. for any {t„}„GN, {a„}„GN C (0, to) with tn t то, an ^ 0. Moreover, since St+N/ = St/ + TtSN/ > TtSN/ m-a.e. for t E (0, to) and N E N, by letting N ^ to we have TtG/ < G/ m-a.e., that is, G/ is 8-excessive. We call this operator G : L+(E,m) ^ L+(E,m) the 0-resolvent associated with (8, E).

Definition 6 (Transience and Recurrence) . (1) (8, E ) is called transient if and only if G/ <  to m -a.e. for some / E L ₊ (E,m) with / >  0 m -a.e.

• (2)    (8, E ) is called recurrent if and only if m({0 < G/ <  to}) = 0 for any / E L + (E,m) .

By [6, Lemma 1.5.1], (8, E ) is transient if and only if G/ <  to m -a.e. for any / E L + (E,m) . On the other hand, by [6, Theorem 1.6.3], (8, E ) is recurrent if and only if 1 E E e and 8 (1,1) = 0 .

The following proposition is the second step of the proof of Theorem 1.

Proposition 6. Assume that (8, E ) is recurrent. If и E L ^ (E,m) is 8-excessive then и E E _e and 8(и, и) = 0 .

Proof. Let n E N . Then и Л n <  n 1 m -a.e., n 1 E E _e and 8 (n1,n1) = 0 by the recurrence of (8, E ) , and иЛn is 8 -excessive since so are и and 1 . Thus иЛn E E _e and 8(иЛn,иЛn) = 0 by Corollary 2. Lemma 1-(2) implies that lim _n^^ ||v — и Л n|| ^-_e = 0 for some v E E _e with

|| • ||j-_e as defined there, and then we easily have и = v E E _e and 8(и,и) = 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 и E E _e be 8 -excessive in the wide sense, n E N and и _n := и Л n . Then и _n E E _e , и _n is also 8 -excessive in the wide sense, n 1 — и _n E E _e П L ₊ (E,m) and hence 8(и _n ,и _n ) = 8(и _п ,и _п — n1) <  0 by Proposition 3-(2). As in the proof of Proposition 6, letting n ^ to we get 8(и, и) = 0 by Lemma 1-(2), and hence и E R1 by Proposition 5.

• 3)    ^ 1) : (8, E ) is recurrent since 1 E E _e and 8 (1,1) = 0 . Let A E В be 8 -invariant. Then 1 л = 1 ^ 1 E E _e П L ^ (E,m) and 0 < 8 (1 _л , 1 л ) < 8 (1,1) = 0 by [6, Theorem 1.6.1]. Now 3) implies 1 л E R1 , and hence either m(A) = 0 or m(E \ A) = 0 .

• 6)    ^ 3) when (E, В , m) is non-trivial : Choose g E ^^(E,m) so that g >  0 m -a.e., and set E _c := {Gg = to} . Then 1 _{E c} E E _e П L ^ (E,m) and 8 (1 _{E c} , 1 _{E c} ) = 0 by [6, Corollary 1.6.2], and 6) together with Proposition 3-(3) implies 1 _{E c} E R1 , i.e., either m(E _c ) = 0 or m(E \ E _c ) = 0 . In view of 6) and Proposition 3-(3), it suffices to show m(E \ E _c ) = 0 .

Suppose m(E _c ) = 0 , so that ( 8 , E ) is transient, and set n := g/(1 V Gg) . Then 0 <  n <  <   g m -a.e. and ( q , G q ) < {g/ (1 VGg), Gg} <  || g 1 1 <  to . Let / E L + (E, m) ПЕ?(E, m) and set / _п := / Л(п п ) for n E N . Then / _n E L ^ (E,m) , G/ _n <  nG q <  to m -a.e., {/ _n ,G/_n} <  to and / n t / m -a.e. Since 8(G a / n ,G a / n ) <  (/ n ,G a / n ) <  (/ n ,G/ n ) <  to for a E (0, to) ,

Proposition 2 implies G/ _n E E _e . Since G/ _n is 8 -excessive, so is n Л G/ _n E E _e П L ^ (E,m) and 6) yields n Л G/ _n E R1 . Letting n ^ to and noting G/ <  to m -a.e. by the transience of ( 8 , E ) , we get G/ E R1 . Let a E (0, to) . Then G _a / E L^(E,m) П L ² (E,m) and hence GG _a / E R1 . Letting n ^ to in G _a / = G ₁/_n / — ( a — 1/n)G ₁/_n G _a / implies that G _a / = G/ — a GG _a / E R1 . Since a G _a / ^ / in L ² (E,m) as a ^ to , we conclude that L 1 (E,m) ПL‘ ² (E,m) C R1 , contradicting the assumption that (E, В , m) is non-trivial. Thus m(E \ E _c ) = 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

¹ JSPS Research Fellow PD (20 · 6088): The author was supported by the Japan Society for the Promotion of Science.