Kasahara, Y. and Kotani, S. Osaka J. Math. 53 (26), 22 249 TAUBERIAN THEOREM FOR HARMONIC MEAN OF STIELTJES TRANSFORMS AND ITS APPLICATIONS TO LINEAR DIFFUSIONS YUJI KASAHARA and SHIN ICHI KOTANI (Received July 4, 24, revised December 8, 24) Abstract When two Radon measures on the half line are given, the harmonic mean of their Stieltjes transforms is again the Stieltjes transform of a Radon measure. We study the relationship between the asymptotic behavior of the resulting measure and those of the original ones. The problem comes from the spectral theory of second order differential operators and the results are applied to linear diffusions neither boundaries of which is regular.. Introduction Let H be the totality of the functions on (, ½) having the following representation: h(s) a [,½) d (), s (a s ) where Ï R[,½) is a nondecreasing, right-continuous function vanishing on ( ½,) such that d () ½. [,½) Let us call the spectral function of h (the reason will be clear in Section 5). For h, h 2 ¾ H define h by (.) h(s) h (s) h 2 (s). Then as is well known we again have h ¾ H (a property of Herglotz functions). The aim of the present article is to study the relationship between the asymptotic behavior of () as and those of i () (i, 2), where and i are the spectral functions of h and h i, respectively. Notice that the equation (.) is familiar in the spectral theory of Sturm Liouville operators and is fundamental in the theory 2 Mathematics Subject Classification. Primary 6J6; Secondary 4E5, 34B24. This work was supported by JSPS KAKENHI Grant Number 2454.
222 Y. KASAHARA AND S. KOTANI of linear diffusions. Indeed, our problem is motivated by a study of the asymptotic behavior of the transition probability in the long term and this problem will be discussed in Section 6. Especially, Example 6. will illustrate the motivation of our problem. To describe our results let us prepare some notation. We define l i h i () (i, 2) and l h() ( ½). By (.) it holds (.2) l l l 2 with the convention that ½ (namely, if l ½ and l 2 ½ then l l, while if l l 2 ½ then l ½). We also define (.3) p l 2 l l 2 l l, q l l l 2 l l 2, when they make sense. Very roughly speaking our result is as follows: Under a certain regularity condition, it holds (.4) () () 2 () () 2 () p 2 ()q 2 2 () l 2 2 d () 2() (l l 2 ½), (l ½, l 2 ½), (l ½, l 2 ½), where, is the dual of. The precise statement will be given in Section 2 and will be proved in Section 4. In Section 3 we prepare some intermediate results we need in the proofs of the main results. Sections 5 and 6 are devoted to applications of the mains results to linear diffusions. Since we shall repeatedly make use of Tauberian theorems for Lebesgue Stieltjes transforms, we listed necessary facts in Appendix for the convenience of the reader. REMARK.. We shall discuss only the case of (.), but the results can easily be extended to the case where h(s) h (s) h 2 (s) h n (s). So our results may have applications to diffusions on some sort of graphs as well as linear diffusions.
TAUBERIAN THEOREM 223 2. Main results We denote by R «() the totality of ultimately positive functions (defined on some interval (, A)) varying regularly at with index «(¾ R): i.e., lim s f (cs) f (s) c«(c ). A regularly varying function with index «is said to be slowly varying. Clearly f ¾ R «() if and only if f (s) s «L(s) with slowly varying L. Theorem 2. (Case I). Suppose that l ½, l 2 ½ and let p, q be as in (.3). If ³ ¾ R «() («), then (i) if and only if (ii) As a special case, if for c, c 2 (c c 2 ), then () ³() ( ) pq () Ï p 2 ()q 2 2 () ³() i () c i ³() ( ), i, 2, ( ). (2.) () (c p 2 c 2 q 2 )³() ( ). Here and throughout, f cg means f g c including the case c. Theorem 2.2 (Case II). Suppose that l l 2 ½ and let ³ ¾ R «() («). If i () c i ³() for c i ¾ (, ½), (i, 2), then (2.2) () c c 2 c c 2 ³() ( ). The assertion remains valid in the extreme case c ½, c 2 ½, with the convention c c 2 (c c 2 ) c. (The restriction «is necessary for l l 2 ½.) Apparently (2.2) may look quite different from (2.), but (2.2) is in fact the extreme case of (2.) as l, l 2 ½ with l l 2 c c 2.
224 Y. KASAHARA AND S. KOTANI (i) Corollary 2. (Lopsided case I). Suppose that ¾ R «() and 2 ¾ R (). If «, then (ii) If «, then () () () p 2 () ( ). ( ). Proof. (i) Apply Theorem 2.2 with ³() () and c, c 2 ½. (ii) Apply Theorem 2. (ii) with c, c 2. It remains to discuss the case where l ½, l 2 ½. To this end we need to consider the dual of h: For a given h ¾ H its dual h is defined by h (s) sh(s). As is well known it holds h ¾ H. So let a and correspond to h : i.e., (2.3) h (s) a [,½) We also define # Ï R [, ½) by # () [,] d () s. d () ( ), ( ). Another characterization of # will be given in (3.). Theorem 2.3 (Case III). Suppose that l ½, l 2 ½ and let ³ ¾ R () ( ). Then, (2.4) () ³() ( ) if and only if (2.5) l 2 2 # () 2() ³() ( ). (The restriction is necessary for the assumption l 2 ½.) Corollary 2.2. Let ¾ R «() ( «) and c, c 2. If () c () (¾ R «()), 2 () c 2 2 () (¾ R 2 «()) ( ),
TAUBERIAN THEOREM 225 then () «2 «l 2 2 c {¼(«)¼( «)} 2 c 2 2 () ¾ R 2 «() ( ). The assertion remains valid even if (c ½; c 2 ½) or ( c ½; c 2 ; «) with the convention that ½. As we shall see in Proposition 3.2, the assumption () c () im- Proof. plies that # () «l2 2 2 2 «c {¼(«)¼( «)} 2 () ( ). Therefore, the assertion follows immediately from Theorem 2.3. Note that when «we may use the last part of Proposition 3.2. By the last part of Corollary 2.2 we have (i) Corollary 2.3 (Lopsided case II). Let ¾ R «() and 2 ¾ R (). If «and «2, then, () 2 () ( ). (ii) If «and «2, then (2.6) () «l2 2 2 2 «{¼(«)¼( «)} 2 () ( ). (In (ii) we excluded the case «because the the right-hand side of (2.6) vanishes.) Proof of Corollary 2.3. (i) Let () 2 2 () (¾ R 2 ()). Then ()()¾ R «2 () and, hence, «2 implies ()() ½. Therefore, we can apply Corollary 2.2 with c ½, c 2. For the proof of (ii) put () (), then appeal to Corollary 2.2 with c, c 2. Let us next consider the case «which we excluded in Corollary 2.3. (i) Corollary 2.4 (Lopsided case III: «). Suppose that ¾ R () and 2 ¾ R (). If, then () () ( ).
226 Y. KASAHARA AND S. KOTANI (ii) If, then (2.7) () l 2 () ÇL() 2 ( ), where Note that Ç L(s) l written as ÇL(s) Ï ½ s (u) u 2 du ¾ R (). (cf. (3.2)). So in (ii), if l ½, then (2.7) may also be () p 2 () ( ), because p ll. This means that (ii) of Corollary 2. remains valid in the extreme case «when l ½. Proof of Corollary 2.4. (i) If l ½, we can apply Theorem 2.2 with c 2 ½ because o( 2 ). Next consider the case l ½. By Theorem 2.3, () l 2 # 2 () (). By Proposition 3.2, we have # 2 ¾ R 2 () so that # 2 o( ). Thus we have the assertion. (ii) l ½, as we shall prove in Proposition 3.3, it holds When l ½, just apply Theorem 2. with ³, c, c 2. When # () () ÇL() 2 ¾ R (). On the other hand 2 ¾ R () with implies 2 o( # ). Therefore, we deduce the assertion from Theorem 2.3. In Corollary 2.2 we discussed the case where l ½, l 2 ½ and ¾ R () with 2. For the case 2, we have the following result. Here, notice that the condition () trivially implies ¾ R () and l ( h ()) ½. Corollary 2.5. Suppose () and ³ ¾ R () with 2. If and 2 () c 2 ³() ( ) (2.8) () () c ³() 2 ( ),
for c, c 2 (c c 2 ), then TAUBERIAN THEOREM 227 () c l 2 2 () 2 2 c 2 ³() ¾ R (). Proof. As we shall see in Proposition 3.4 the condition (2.8) is equivalent to # () c 2 () 2 Therefore, the assertion follows from Theorem 2.3. ³() c 2 ³(). () 2 Theorem 2.4 (Case IV). Suppose that () and 2 (). Then, (2.9) () () 2 () () 2 () ( ). Furthermore, if (2.) i () i () c i ³() ( ), i, 2 for ³ ¾ R «() («) and c, c 2 (c c 2 ), then (2.) () () (p 2 c q 2 c 2)³() ( ), where p 2 () () 2 (), q () () 2 (). 3. Intermediate results In this section we prepare a few propositions we need for the proofs of Theorems 2. 2.4 in Section 2. Throughout the paper, we put Çh(s) h(s), Ç hi (s) h i (s) (i, 2). Since Ç h(s) sh (s), we have Çh ¼ (s) h (s)sh ¼ (s) a ½ a ½ d () (s ) 2. d () s ½ s d () (s ) 2
228 Y. KASAHARA AND S. KOTANI Therefore, (3.) Ç h (n) (s) a ½ ( ) n n! d # () (s ) 2 (n ), ½ d # () (s ) n (n 2). Of course we have similar formulas for h, h 2 ¾ H and we may define # i (i, 2) in the obvious manner. Since (.) can be written as (3.2) Ç h(s) Ç h (s) Ç h2 (s), it holds Çh ¼ (s) Ç h ¼ (s) Ç h ¼ 2 (s) and hence, by (3.), we have (3.3) # () # () # 2 (). Thus the proofs of the results in the previous section are reduced to the study of the relationship between the asymptotic behavior of,, 2 and that of #, #, # 2. Next we define and Proposition 3.. #(Ç h) and #(h) inf{k Á h () (k) ½} inf k,, 2, Á d () ½ k (3.4) lim s [,) #(Ç h) inf{k Á Ç h (k) () ½}. If n Ï #(h) (i.e., l ½), then it holds that #(h) Çh (n) (s) h (n) (s) l 2, n n. For the proof of Proposition 3. we prepare (i) Lemma 3.. Suppose #(h) n (i.e., l ½). Then; lim h Ç (s) (k) ( k n ), s ½ (k n ).
TAUBERIAN THEOREM 229 (ii) (3.5) h (n ) (s) l 2 Ç h (n ) (s) (s ). Proof. (i) By the Leibniz formula we have, for m, Therefore, m k (3.6) Ç h (m) (s) h(s) mc k h (k) (s)ç h (m k) (s) (h(s)ç h(s)) (m). m k mc k h (k) (s)ç h (m k) (s). Now we have the assertion by induction on m, 2,, n. (ii) As we have seen in (3.6), it holds that (3.7) h Ç (n ) (s) h (s)ç (n) h(s) h(s) n and hence by (i) we have k Çh (n) (s) Ç h (n) h(s) (s) h(s) O() h(n ) (s) nc k h (k) (s) Ç h (n k) (s) µ O(). h(s) 2 Since h (n ) (s)½ by the definition of n, we can neglect the O() in the right-hand side and deduce the assertion. Lemma 3.2. Let n. If l ½ and h (n) ()½, then for k, 2,, n it holds (3.8) h (k) (s) o(h (n) (s) kn ) (s ) and (3.9) h (k) (s)h (n k) (s) o(h (n) (s)) (s ). Proof. Since (3.9) follows immediately from (3.8), we shall prove (3.8) only. If h (k) () ½, then the assertion is obvious. So we assume that h (k) ()½. For every, applying Hölder s inequality to d () (s ) k d () (s ) (kn) (s ) (n)kn
23 Y. KASAHARA AND S. KOTANI with p (kn) and q kn, we have d () (s ) k d () s d () (kn) (kn) ½ Therefore, using the condition h (n) () ½, we deduce lim sup s ½ d () º ½ (s ) k d () kn (s ) n d () kn (s ) n d () kn. (s ) n d () (kn). Since the right-hand side converges to as Ê ½ because () d () l ½ by assumption, we obtain (3.8). Proof of Proposition 3.. Let us prove the assertion by induction on n ( n ). The case n n is proved in Lemma 3. (ii). Next suppose that (3.4) holds for n n, n,, m, and let us see that (3.4) remains valid for n m. By Lemma 3. we have Çh (k) (s) O() ( k n ), l 2 h (k) (s) (n k m). So in any case, Çh (k) (s) O(h (k) (s)), k, 2,, m. Therefore, for any k, 2,, m, and, hence by Lemma 3.2, we see or, changing the variable k, we have h (m k) Ç h (k) O(h (m k) h (k) ), h (m k) Ç h (k) o(h (m) ), k,, m, (3.) h (k) Ç h (m k) o(h (m) ), k,, m. Now as in (3.7), we have Çh (m) (s) h(s) h (s)ç (m) h(s) m nc k h (s)ç (k) h (m k) (s). k µ
TAUBERIAN THEOREM 23 So, applying (3.) to the right-hand side we deduce Çh (m) (s) h(s) h(m) (s)ç h(s)o(h (m) (s)). Thus we have Çh (m) (s) l 2 h (m) (s), completing the induction. Proposition 3.2. Let ³ ¾ R «() ( «). Then () ³() if and only if # () «2 2 «{¼(«)¼( «)} 2 ³() ( ). As an extreme case, if ¾ R (), then # () 2 o ³() ( ). Proof. The assertion follows from Tauberian theorem (Theorem 7.) as follows. For the definition of C,«see (7.). () ³() and the last one is also equivalent to iff ³(s) iff ä h(s) C,«ä h (s) s sh(s) C,«³(s) iff ä () (¾ R «()), C,«C, «³() # «() 2 C,«C, «2 «³() by Lemma 7. (apply with «). When «, the above argument does not hold because C, «sense. So let us prove directly. If ³ ¾ R (), then does not make h(s) (s), h ¼ (s) (s) s s, 2 h¼¼ (s) 2 (s) s. 3 Therefore, h ¼¼ (s) h(s) 2 2 s (s), h ¼ (s) 2 h(s) 3 s (s)
232 Y. KASAHARA AND S. KOTANI and so Çh ¼¼ (s) h¼¼ (s) 2 h¼ (s) 2 o h(s) 2 h(s) 3 s (s) o. s³(s) Thus, recalling (3.), we have ½ d # () (s o ) 3 s³(s) and hence # () o( 2 ³()) (see Theorem 7.). Proposition 3.2 does not include the extreme case «(the right-hand side diverges). So, in this case we need a slight modification as follows in order to know the exact order of # (): Proposition 3.3. Let ¾ R (). Then varies slowly as s and ÇL(s) Ï ½ s (u) u 2 du, s (3.) h(s) Ç L(s) (s ). Furthermore, it holds # () () ÇL() 2 ( ). (3.2) Proof. Since ½ (u) u 2 du ½ d (u) u ( l), we see Ç L() l. Therefore, if l ½ then the slowly varying property of Ç L and (3.) are clear. So let us consider the case where Ç L() ( l) ½. Note first that (3.3) lim s cç L ¼ (cs) ÇL ¼ (s) lim s c (cs) (cs) 2 (s) s 2 (cs) lim s c (s). The last equality holds by the assumption ¾ R (). Combining (3.3) with the condition Ç L() ½, we deduce lim s ÇL(cs) (Ç L(cs)) ¼ ÇL(s) lim s (Ç L(s)). ¼
TAUBERIAN THEOREM 233 Thus Ç L varies slowly. Next note that, by Tauberian theorem (Corollary 7.), ¾ R () implies (3.4) h ¼ (s) (s) s 2 ( Ç L ¼ (s)). Since l ½, (3.4) implies (3.). Next, combining (3.4) and (3.) we deduce namely, by (3.), Çh ¼ (s) (h(s)) ¼ h ¼ (s)h(s) 2 (s) s 2 ÇL(s) 2 (3.5) ½ d # () (s 2Ç (s) ¾ ) 2 s L(s) R (), 2 which proves, by Tauberian theorem (Theorem 7.), # () C, () ÇL() 2 () ÇL() 2 ( ). EXAMPLE 3.. If (), then by Proposition 3.3 we have # () () ÇL() 2 (log()) 2 ( ). For, ÇL(s) ½ s (u) u 2 du log s (s ). In Propositions 3.2 and 3.3 we studied the case where # ¾ R () with 2 and, respectively. The following proposition is concerned with the case 2. Proposition 3.4., Let ³ ¾ R «() («) and A. If () then, as () () A³() iff ä iff ä () A 2 iff ä # () A 2 d () A ««³() ««³() ««2 2 ³().
234 Y. KASAHARA AND S. KOTANI For the proof of Proposition 3.4, we prepare Lemma 3.3. Suppose () and n. Then, (3.6) (h (s)) (n) (sh(s)) (n) (s ) provided that at least one of the two sides diverges to infinity as s. Proof. 2 Since (h ) h, the assertion can be reduced to Proposition 3. as follows. Clearly it holds sh(s). Therefore, h (s) ( (sh(s))) l ½ and we can apply Proposition 3. to h in place of h and (3.4) can be written as (h ) (n) (s) l 2 Ç h (n) (s) 2 (sh(s)) (n) (s ), which proves (3.6). Here we used Ç h (s) h (s) sh(s). Proof of Proposition 3.4. For the proofs of the first and the last relationship see Lemma 7.. So we shall prove the second one. Since (sh(s)) ¼ h(s)sh ¼ (s) a [,½) a [,½) d () (s ) 2, d () s [,½) s d () (s ) 2 it holds ½ (sh(s)) (n) ( ) n n! d (), n (s 2. ) n On the other hand we have (h (s)) (n) ( ) n n! These two combined with (3.6) imply ½ ½ d () (s ) n 2 d (), n (s. ) n ½ d () (s ) n for n 2. Now appeal to the Tauberian theorem to deduce the assertion.
TAUBERIAN THEOREM 235 4. Proofs of Theorems 2. 2.4 Proof of Theorem 2.. Let (4.) h pq (s) p 2 h (s)q 2 h 2 (s), so that h pq (s) [,½) d pq () s. By the Tauberian theorem (Corollary 7.2), it suffices to show that (4.2) h (n) (s) h (n) pq (s) (s ) for some n «. To begin with let us see that #(h pq ) #(h) and (4.2) holds for n n Ï #(h). Since (4.3) Ç h (k) (s) Ç h (k) (s) Ç h (k) 2 (s), we see that h Ç (k) () ½ holds if and only if h Ç (k) () ½ Ç or h (k) 2 (s) ½. So by Lemma 3., #(h)n if and only if min{#(h ),#(h 2 )} n. Similarly, by the definition of #(h), we see that #(h pq ) min{#(h ), #(h 2 )} n. For the proof of (4.2) first consider the case where #(Ç h ) #(Ç h2 ) n. By Proposition 3., for n n, it holds h (n) (s) l Ç 2 h (n) (s) l (Ç 2 h (s) (n) Ç h (n) 2 (s)) l2 l 2 h (s) (n) l2 2 h (n) 2 (s) h (n) pq (s) and hence (4.2) is proved for all n n. When #(Ç h ) #(Ç h2 ), we need a slight modification. Consider the case where #(Ç h ) n and #(Ç h2 ) n. In this case, for n such that n n #(Ç h2 ), the argument above does not hold, but h Ç (n) 2 (s) and h (n) 2 (s) are bounded and hence negligible when compared to h Ç (n) (s) and h (n) (s). Therefore we have the same conclusion. Proof of Theorem 2.2. CASE ( «) C,«c i ³(s)s. Therefore, There are two cases. By Tauberian theorem (Theorem 7.) we have h i (s) h(s) h (s)h 2 (s) h (s)h 2 (s) c c 2 ³(s) C c,«, c 2 s which proves the assertion by Theorem 7.. When c 2 ½, it means o( 2 ) so that h o(h 2 ) and hence hh h 2 ((h h 2 )). Thus we have h h, which proves c ³.
236 Y. KASAHARA AND S. KOTANI Case 2 («) In this case the above argument is insufficient because h h does not necessarily imply. However, h ¼ h ¼ implies. So let us prove h ¼ h ¼. Observe that () (c c 2 ) 2 () implies h ¼ (s) (c c 2 )h ¼ 2 (s), which also implies h (s) (c c 2 )h 2 (s) (by de l Hospital) and hence h(s) {c (c c 2 )}h 2 (s) and h(s) {c 2 (c c 2 )}h (s). Since (.) implies we see h ¼ (s) h¼ (s) h(s) 2 h h¼ 2 (s) (s) 2 h 2 (s), 2 h ¼ h ¼ 2 h2 h¼ h2 c 2 2 c h 2 h ¼ 2 h 2 2 c c 2 c 2 c 2 c c 2 c. c c 2 Thus we have h ¼ h ¼ 2 c (c c 2 ) and by Tauberian theorem as before we can deduce () {c (c c 2 )} 2 (). In the extreme case c ½, it holds that h 2 o(h ) and h ¼ 2 o(h ¼ ). The rest of the proof is the same. Proof of Theorem 2.3. Since l ½ and l 2 ½, it holds l l 2 ½ and hence #(h),#(h 2 ). Therefore, we have from Lemma 3. that h (n) (s)l Ç 2 h (n) (s) and h (s) (n) 2 l Ç 2 h (n) 2 (s) for all sufficiently large n. So This implies, by (3.), h (n) (s) l 2 Ç h (n) (s) l 2 Ç h (n) (s)l2 Ç h (n) 2 (s) l2 Ç h (n) (s)h(n) 2 (s). ½ d () ½ (s l 2 ) n which proves () l 2 # () 2(). d #() (s ½ ) n d 2 () (s ) n, Proof of Theorem 2.4. Since () lim s sh(s), i() lim s sh i(s), we have from (.) that () () 2 (), which implies (2.9). By Proposition 3.4 we see that (2.) is equivalent to i () i () 2 ««c i ³().
TAUBERIAN THEOREM 237 Since (.) implies h (s)h (s)h 2 (s), it holds () () 2 (), and therefore () () 2 () c () 2 c 2 2 () 2 ««³(). Appealing to Proposition 3.4 again, this is equivalent to () () () 2 c c 2 ³() () 2 2 () 2 ( ). 5. An application to positive recurrent linear diffusions In this section we generalize a result of [5], where the transition density of positive recurrent diffusions is discussed. Let X (X t ) t be a diffusion on I [, ½) with local generator (5.) L 2 d 2 dx 2 b(x) d dx, x, b(x) being assumed to be an element of L loc ([, ½), dx). We put reflecting boundary condition at the left boundary. Define Then Feller s canonical form of (5.) is x W (x) exp b(u) du, x. where L d d dm(x) ds(x), x m(x) 2 W (u) du, s(x) x du W (u). Note that the scale-changed process Y t Ï s(x t ) corresponds to L d d d Ém(s) ds, where Ém( ) Ï m(s ( )). It is well known that the transition density p(t, x, y) with respect to dm(x) exists and p(t,, ) has the following spectral representation: (5.2) p(t,, ) [,½) e t d (), t.
238 Y. KASAHARA AND S. KOTANI The spectral function can be characterized by the following formula: If we define G s (x, y) Ï ½ e st p(t, x, y) dt, s, then (5.2) implies h(s) Ï G s (, ) [,½) d () s, s. How to calculate G s (x, y) (and hence h(s)) from L will be explained in Section 6. We remark that it is known that l (Ï h()) s(½). (This fact will easily be seen from (6.2).) Therefore, it holds that (5.3) l ½ du W (u). The authors recently obtained the following result: We denote by R «(½) the totality of functions varying regularly at ½ with index «. Theorem ([6, Theorem 4.2]). Let. If (5.4) W ( ) ¾ R (½) then, as t ½, (5.5) p(t,, ) 2 2 ¼(2) Ô tw ( Ô t) ¾ R 2(½). If we recall the canonical representation of slowly varying functions (see e.g., [, p. 2]), we easily see that a sufficient condition for (5.4) is (5.6) xb(x) (x ½), or, equivalently, b(x) x o x (x ½). We remark that, by (5.2) and Tauberian theorem for Laplace transforms (see e.g. [, p. 37]), (5.5) is equivalent to (5.7) () 2 (2) ¼(2) 2 Ô W ( Ô ) ( ).
TAUBERIAN THEOREM 239 Now the aim of the present section is to study the case where (5.4) holds for. In this case we see Çm Ï ½ ½ dm(x) 2 W (x) dx ½, which is, probabilistically, equivalent to that the process is positively recurrent. Since, as is well known, (5.8) () Çm, we see that Çm ½ implies () and therefore, p(t,, ) Çm (t ½) (cf. [2, pp. 35 37]). So let us evaluate p(t,, ) Çm as t ½. Since (5.8) implies p(t,, ) Çm (,½) e t d (), our problem will be reduced to the study of () () ( ). To this end let us consider the dual process of (5.): L Ï d 2 b(x) d, x 2 dx 2. dx Note the following argument remains valid under the condition 2 rather than. The functions W, s, m, Ém, h, corresponding to L will be denoted by W, s, m, Ém, h and, respectively. Since they correspond to b in place of b, we have W (x) exp x b(u) du so that W W, and hence, W ¾ R ( )(½) R (½) where Ï 2. If 2, it holds. So we can apply Theorem to b(x) to deduce (5.9) () C Ô W ( Ô ) ¾ R 2() ( ), where C 2 2 ¼( 2) 2 2 2 (2 )¼((2 )2) 2.
24 Y. KASAHARA AND S. KOTANI We next consider the relationship between and defined in (2.3). To this end we recall Krein s correspondence (see e.g. [8]): The correspondence between h and Ém is one-to-one and Ém (x) Ï Ém (x) corresponds to h (s) (sh(s)). Furthermore, cém(cx) corresponds to (c)h(s) (and, hence, cém (cx) to (c)h (s)). Since s (x) (2)m(x) and m (x) 2s(x), we have Ém (x) 2Ém (2x) 2Ém (2x). This proves that h (s) (2)h (s), which implies So by (5.9) we have () 2 (). Proposition 5.. Suppose that (5.6) holds with 2. Then, (5.) () C Ô W ( Ô ) ¾ R 2() ( ), where C 2C 2 (2) (2 )¼((2 )2) 2. REMARK 5.. When 2 we can confirm (5.) directly as follows: Since ¾ R 2 (), we have h ¾ R (2) (), h (s) (sh(s)) ¾ R (2) () and ¾ R (2) (). Therefore, () Combining this with (5.7) we obtain h () C, (2) C, (2) h() C, (2) C,2 (). () 22 ¼(2)¼((2)) C, (2) C,2 Ô W ( Ô ) C Ô W ( Ô ). Now let us return to the case instead of 2. In this case «Ï ( 2) 2 and we can apply Proposition 3.4 to (5.) to obtain () () () 2 «() ¾ R «() R 2(). «Thus we have the following result, which extends Example 3.8 of [5], where only the case 2 is discussed. Theorem 5.. Suppose that (5.6) holds with. Then, (5.) () Çm Çm D 2 Ô W Ô ¾ R 2() ( ),
TAUBERIAN THEOREM 24 where D 2 (2) ¼((2 )2) 2. Notice that, by the reason we explained before, (5.) is equivalent to p(t,, ) Çm Çm 2 D ¼ 6. An application to bilateral diffusions 2 Ô tw ( Ô t) ¾ R 2 (½) (t ½). The aim of this section is study how our results in Section 2 work when we wish to apply Theorem A in the previous section to bilateral diffusions. Here, bilateral means that neither boundary of the state space is regular. To begin with let us quickly review necessary facts on linear diffusions. Let X (X t ) t be a regular, conservative diffusion on an interval in R. For simplicity, we change the scale if necessary so that the local generator is of the form L d d d Ém(x) dx, l x l where l, l ½ and Ém(x) is a nondecreasing right-continuous function defined on I ( l, l ). (We need not to assume that Ém is strictly increasing so that generalized diffusions such as birth death processes are included.) It is well known that the transition density p(t, x, y) with respect to d Ém(x) can be computed as follows (see e.g. [4]): For each ¾, we can define ³ (x) and (x) as the unique solutions of Lu u, x ¾ I with the initial conditions that (u(), u ¼ ( )) (, ) and (u(), u ¼ ( )) (, ), respectively; or, precisely, the solutions of the following integral equations: (6.) ³ (x) (x) x x x (x y)³ (y) d Ém(y), (x y) (y) d Ém(y) Ê x with the convention that Ê x if x. Then, l s(x) (6.2) h (s) Ï lim xl ³ s(x) dx, s ³ s(x) 2 and h (s) Ï lim x l s(x) ³ s(x) l dx ³ s( x) 2, s
242 Y. KASAHARA AND S. KOTANI are called the characteristic functions of Ém. It holds that h, h ¾ H, and hence we have the following representation: (6.3) h (s) a [,½) For every s, we put d () s, s. u (sá x) ³ s(x) h (s) s(x)á u 2 (sá x) ³ s(x) h (s) s(x). These two are nonnegative solutions of Lu(x) su(x), x ¾ I, u() such that u is nonincreasing and u 2 nondecreasing. The Wronskian is (6.4) W [u (sá ), u 2 (sá )] (Ï u u ¼ 2 u ¼ u 2) h (s) h (s). So the Green function is given by (6.5) G s (x, y) h(s)u 2 (sá x)u (sá y) (x y), h(s)u (sá x)u 2 (sá y) (x y), where h(s) W [u (sá ), u 2 (sá )]; namely, by (6.4), (6.6) Note that it holds h(s) h (s) h (s). (6.7) G s (, ) h(s), which follows immediately from (6.5) because u i (sá ). The transition density p(t, x, y) (with respect to d Ém(x)) can be obtained from G s (x, y) via the following formula: ½ Especially, by (6.7) we have e st p(t, x, y) dt G s (x, y) (s ) d Ém(x) d Ém(y)-a.e. (6.8) ½ e st p(t,, ) dt h(s) (s )
TAUBERIAN THEOREM 243 provided that ¾ Supp{d Ém(x)}. then (6.8) implies Notice that, if is the spectral function of h(s), (6.9) p(t,, ) ½ e s d (), s. In this way the asymptotic behavior of the transition density p(t,, ) will be reduced to those of two diffusions; one is on ( l, ] and the other on [, l ). Let X (X t ) t be a diffusion with generator (5.) on the whole line R with b( ) ¾ Lloc (R). Notice that the results in the above are applicable first to the suitably scaled process Y t s(x t ) and hence to X. So, for example, there exists the transition density p(t, x, y) with respect to dm(x) Ï 2 exp Ê x b(u) du dx and (6.9) remains valid if we choose the scale function s(x) so that s(). Define W (x) exp x W (x) x exp b(u) du, x, b( u) du, x as in Section 5. The reason why b( x) appears in the definition of W (x) is simply because the diffusion ( X t ) t corresponds to L d 2 b( x) d 2 dx 2 dx in place of (5.). By (6.2) we easily see that l (Ï h ()) s (½), where s (x) is defined in a similar way as in the previous section, and hence it holds that l ½ dx W (x) ( ½). So l Ï h() is obtained by l l l l l, (see (.2)). Also as in the previous section we have () Çm, Çm 2 ½ W (u) du and therefore, () () () () () Çm,
244 Y. KASAHARA AND S. KOTANI (see (2.9)), where Çm Çm Çm 2 ½ (W (u) W (u)) du. Theorem 6. (Balanced case). Suppose that and lim x½ xb(x) ( ) (6.) lim A½ (i) A A b(u) du log If ( 2) or ( 2Á l ½), then r r ( r ). () r () ( ). (ii) If ( 2) or ( 2Á l ½), then () p 2 q 2 r r () ( ), where p ll and q ll. (iii) If, then () Çm Çm Çm 2 Çm Çm 2 r r () Çm. Proof. (i) It holds W ¾ R (½) and ¾ R 2() as before (see (5.6) and (5.7)). By the balancing condition (6.), it holds which implies, by (5.7), Therefore, by Theorem 2.2 we deduce (ii) W (x) r, W (x) r () () c Ï r r. () c c () r () Similarly, by Theorem 2., we see () p 2 ()q 2 () p 2 q 2 ( ). r r ().
TAUBERIAN THEOREM 245 (iii) The assertion can be shown in a similar way by using Theorems 2.4 and 5.. Theorem 6.2 (Lopsided case). Suppose that where and ¾ R. Then; (i) In the following three cases it holds limx½ xb(x), lim x ½ xb(x), (6.) () () ¾ R 2() ( ). () 2, (2) 2, 4, (3) 2. (ii) If 2 or (2, l ½), then () p 2 (). (iii) If ( 2, 4) or (2, l ½), then (6.2) () l 2 # () ¾ R 2 ( 2)() ( ). Proof. First we remark that ¾ R ( )2(). (i) In each of the cases () (3) we can apply Corollary 2. (i), Corollary 2.3 (i), and Corollary 2.4 (i), respectively. (ii) Apply Corollary 2. (ii) and Corollary 2.4 (ii), respectively. (iii) Since ¾ R 2 ( 2)() and ¾ R 2(), the assertion follows from Theorem 2.3. As Ê we mentioned before the transition density p(t, x, y) with respect to dm(x) x 2 exp b(u) du dx exists and especially p(t,, ) satisfies (6.9). So by Karamata Tauberian theorem for Laplace transforms the results for the spectral function can be translated into those for p(t,, ). Thus we have EXAMPLE 6.. (6.3) b (x) Let b(x) b (x)(x) where x (x ), (x ), x (x )
246 Y. KASAHARA AND S. KOTANI and ( ) ¾ L (R, dx). Since W (x) const x (x ½), we see that l ½ if and only if 2, and also Çm ½ if and only if. () If ( 2; ) or 2, then, (2) If 2, then p(t,, ) const t 2 (t ½). p(t,, ) const max{t 2( 2), t 2 } (t ½). (3) If 2, then p(t,, ) const t (log t) 2 (t ½). (4) If, then p(t,, ) Çm const t 2 (t ½). Since () and (2) are immediate from Theorem 6.2 (and Tauberian theorem for Laplace transforms), let us see (3) and (4) only. (3) follows from Theorem 6.2 (iii): Since () const and () const 2 (see (5.7)), we see from (6.2) and Example 3. that () const # () const (log()) 2 ( ). Thus we have (3). Similarly, we can deduce (4) by Theorems 6. (iii) and Theorem 5.. 7. Appendix In this section we briefly sum up some results on Tauberian theorems for Stieltjes transforms. For a nondecreasing, right-continuous function Ï ( ½, ½) [, ½) such that (x) on ( ½, ), we define the generalized Lebesgue Stieltjes transform by H n (Á s) [,½) d () (s ) n n [,½) () d (s ) n2 (n ) provided that the integral converges. The generalized Lebesgue Stieltjes transform determines the measure d () uniquely. For an inversion formula see [9, Appendix]. The most important case is the following: Let «n. Then () «,
TAUBERIAN THEOREM 247 if and only if where (7.) C n,«½ H n (Á s) C n,«s «n, d «() n ¼(n «)¼(«). ¼(n ) The well-known Karamata s extension of Hardy Littlewood Tauberian theorem is Theorem 7.. if and only Let «n, A, and ³ ¾ R «(). Then, () A³() ( ) H n (Á s) AC n,«³(s) s n (s ). For the proofs we refer to [, p. 4] and [7, Appendix]. The assertion holds even if A with the convention that f Ag means f g A. Also,, s may be replaced by, s ½. Let h ¾ H. Since h (n) (s) ( ) n n! H n (Á s) we have, Corollary 7.. Let «n. Then ¾ R «() if and only if h (n) ¾ R «n (), and then, Corollary 7.2. h (n) ¾ R «n ()), then ( ) n h (n) (s) ¼(n «)¼(«)s n (s), s. Let «n and A. If ¾ R «() (or, equivalently, if h (n) 2 (s) A h(n) (s) (s ) iff ä 2 () A () ( ). (i) Lemma 7.. Let ³ ¾ R () ( ) and B. If, then (7.2) () () B³() ( ) if and only if d () B ³() ( ). (ii) If, then (7.2) implies d () o(³()) ( ).
248 Y. KASAHARA AND S. KOTANI Proof. (i) By Tauberian theorem for Laplace transforms (7.2) holds if and only if e s d () B¼( )³ (s ½). s (,½) Then, by the monotone density theorem (see e.g. [, p. 39]) this is equivalent to ½, which is also equivalent to e s d () B¼( ) s ³ s d () B¼( ) ³() ¼( B 2) ³(). (ii) Without loss of generality, we may assume that (). Since d () () () d and the second term of the right-hand side is asymptotically equal to (), we deduce the assertion. References [] N.H. Bingham, C.M. Goldie and J.L. Teugels: Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge Univ. Press, Cambridge, 987. [2] A.N. Borodin and P. Salminen: Handbook of Brownian Motion Facts and Formulae, second edition, Probability and its Applications, Birkhäuser, Basel, 22. [3] K. Itô: Essentials of Stochastic Processes, Translations of Mathematical Monographs 23, Amer. Math. Soc., Providence, RI, 26. [4] K. Itô and H.P. McKean, Jr.: Diffusion Processes and Their Sample Paths, Springer, Berlin, 974. [5] Y. Kasahara: Asymptotic behavior of the transition density of an ergodic linear diffusion, Publ. Res. Inst. Math. Sci. 48 (22), 565 578. [6] Y. Kasahara and S. Kotani: Diffusions with Bessel-like drifts, preprint. [7] Y. Kasahara and S. Watanabe: Asymptotic behavior of spectral measures of Krein s and Kotani s strings, Kyoto J. Math. 5 (2), 623 644. [8] S. Kotani and S. Watanabe: Kreĭn s spectral theory of strings and generalized diffusion processes; in Functional Analysis in Markov Processes (Katata/Kyoto, 98), Lecture Notes in Math. 923, Springer, Berlin, 982, 235 259. [9] Y. Yano: On the occupation time on the half line of pinned diffusion processes, Publ. Res. Inst. Math. Sci. 42 (26), 787 82.
TAUBERIAN THEOREM 249 Yuji Kasahara Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki 35-875 Japan e-mail: kasahara@math.tsukuba.ac.jp Shin ichi Kotani Department of Mathematical Sciences Kwansei Gakuin University Sanda, Hyogo 669 337 Japan Current address: Osaka University e-mail: skotani@outlook.com