TAUBERIAN THEOREM FOR HARMONIC MEAN OF STIELTJES TRANSFORMS AND ITS APPLICATIONS TO LINEAR DIFFUSIONS

Size: px
Start display at page:

Download "TAUBERIAN THEOREM FOR HARMONIC MEAN OF STIELTJES TRANSFORMS AND ITS APPLICATIONS TO LINEAR DIFFUSIONS"

Transcription

1 Kasahara, Y. and Kotani, S. Osaka J. Math. 53 (26), 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.

2 222 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.

3 TAUBERIAN THEOREM 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.

4 224 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 «()) ( ),

5 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 # () «l «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) () «l «{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 () () ( ).

6 226 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 ( ),

7 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 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

8 228 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 ).

9 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

10 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

11 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)

12 232 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

13 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().

14 234 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.

15 TAUBERIAN THEOREM Proofs of Theorems 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.

16 236 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().

17 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.

18 238 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 ) ( ).

19 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 ]). 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 )¼((2 )2) 2.

20 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() ( ),

21 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

22 242 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 )

23 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,

24 244 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 ().

25 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 )

26 246 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 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 () «,

27 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()) ( ).

28 248 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), [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), [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, [9] Y. Yano: On the occupation time on the half line of pinned diffusion processes, Publ. Res. Inst. Math. Sci. 42 (26),

29 TAUBERIAN THEOREM 249 Yuji Kasahara Institute of Mathematics University of Tsukuba Tsukuba, Ibaraki Japan Shin ichi Kotani Department of Mathematical Sciences Kwansei Gakuin University Sanda, Hyogo Japan Current address: Osaka University

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family

More information

The Azéma-Yor Embedding in Non-Singular Diffusions

The Azéma-Yor Embedding in Non-Singular Diffusions Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let

More information

On non negative solutions of some quasilinear elliptic inequalities

On non negative solutions of some quasilinear elliptic inequalities On non negative solutions of some quasilinear elliptic inequalities Lorenzo D Ambrosio and Enzo Mitidieri September 28 2006 Abstract Let f : R R be a continuous function. We prove that under some additional

More information

EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS

EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS Qiao, H. Osaka J. Math. 51 (14), 47 66 EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS HUIJIE QIAO (Received May 6, 11, revised May 1, 1) Abstract In this paper we show

More information

ONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS

ONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS ONLINE APPENDIX TO: NONPARAMETRIC IDENTIFICATION OF THE MIXED HAZARD MODEL USING MARTINGALE-BASED MOMENTS JOHANNES RUF AND JAMES LEWIS WOLTER Appendix B. The Proofs of Theorem. and Proposition.3 The proof

More information

SOBOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOUBLING MORREY SPACES

SOBOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOUBLING MORREY SPACES Mizuta, Y., Shimomura, T. and Sobukawa, T. Osaka J. Math. 46 (2009), 255 27 SOOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOULING MORREY SPACES YOSHIHIRO MIZUTA, TETSU SHIMOMURA and TAKUYA

More information

Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing

Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes

More information

Sufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems

Sufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions

More information

THE CENTER OF MASS OF THE ISE AND THE WIENER INDEX OF TREES

THE CENTER OF MASS OF THE ISE AND THE WIENER INDEX OF TREES Elect. Comm. in Probab. 9 (24), 78 87 ELECTRONIC COMMUNICATIONS in PROBABILITY THE CENTER OF MASS OF THE ISE AND THE WIENER INDEX OF TREES SVANTE JANSON Departement of Mathematics, Uppsala University,

More information

Conway s RATS Sequences in Base 3

Conway s RATS Sequences in Base 3 3 47 6 3 Journal of Integer Sequences, Vol. 5 (0), Article.9. Conway s RATS Sequences in Base 3 Johann Thiel Department of Mathematical Sciences United States Military Academy West Point, NY 0996 USA johann.thiel@usma.edu

More information

PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS

PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 207 222 PERTURBATION THEORY FOR NONLINEAR DIRICHLET PROBLEMS Fumi-Yuki Maeda and Takayori Ono Hiroshima Institute of Technology, Miyake,

More information

Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems

Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems Corina Constantinescu and Enrique Thomann Abstract The classical result of Cramer-Lundberg states that if the rate

More information

ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS

ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)

More information

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)

More information

CONTINUOUS STATE BRANCHING PROCESSES

CONTINUOUS STATE BRANCHING PROCESSES CONTINUOUS STATE BRANCHING PROCESSES BY JOHN LAMPERTI 1 Communicated by Henry McKean, January 20, 1967 1. Introduction. A class of Markov processes having properties resembling those of ordinary branching

More information

hal , version 1-22 Nov 2009

hal , version 1-22 Nov 2009 Author manuscript, published in "Kinet. Relat. Models 1, 3 8) 355-368" PROPAGATION OF GEVREY REGULARITY FOR SOLUTIONS OF LANDAU EQUATIONS HUA CHEN, WEI-XI LI AND CHAO-JIANG XU Abstract. By using the energy-type

More information

COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL. Ross G. Pinsky

COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL. Ross G. Pinsky COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL Ross G. Pinsky Department of Mathematics Technion-Israel Institute of Technology Haifa, 32000 Israel

More information

MODULUS OF CONTINUITY OF THE DIRICHLET SOLUTIONS

MODULUS OF CONTINUITY OF THE DIRICHLET SOLUTIONS MODULUS OF CONTINUITY OF THE DIRICHLET SOLUTIONS HIROAKI AIKAWA Abstract. Let D be a bounded domain in R n with n 2. For a function f on D we denote by H D f the Dirichlet solution, for the Laplacian,

More information

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3) M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation

More information

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS (2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University

More information

Prime numbers and Gaussian random walks

Prime numbers and Gaussian random walks Prime numbers and Gaussian random walks K. Bruce Erickson Department of Mathematics University of Washington Seattle, WA 9895-4350 March 24, 205 Introduction Consider a symmetric aperiodic random walk

More information

LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION

LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in

More information

The Hilbert Transform and Fine Continuity

The Hilbert Transform and Fine Continuity Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval

More information

ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics.

ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS. Citation Osaka Journal of Mathematics. ANALYTIC SMOOTHING EFFECT FOR NONLI TitleSCHRÖDINGER EQUATION IN TWO SPACE DIMENSIONS Author(s) Hoshino, Gaku; Ozawa, Tohru Citation Osaka Journal of Mathematics. 51(3) Issue 014-07 Date Text Version publisher

More information

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator

More information

Absolutely convergent Fourier series and classical function classes FERENC MÓRICZ

Absolutely convergent Fourier series and classical function classes FERENC MÓRICZ Absolutely convergent Fourier series and classical function classes FERENC MÓRICZ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary, e-mail: moricz@math.u-szeged.hu Abstract.

More information

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary

More information

The Subexponential Product Convolution of Two Weibull-type Distributions

The Subexponential Product Convolution of Two Weibull-type Distributions The Subexponential Product Convolution of Two Weibull-type Distributions Yan Liu School of Mathematics and Statistics Wuhan University Wuhan, Hubei 4372, P.R. China E-mail: yanliu@whu.edu.cn Qihe Tang

More information

An Abel-Tauber theorem for Fourier sine transforms

An Abel-Tauber theorem for Fourier sine transforms J. Math. Sci. Univ. Tokyo 2 (1995), 33 39. An Abel-Tauber theorem for Fourier sine transforms By Akihiko Inoue Abstract. We prove an Abel-Tauber theorem for Fourier sine transforms. It can be considered

More information

The Skorokhod reflection problem for functions with discontinuities (contractive case)

The Skorokhod reflection problem for functions with discontinuities (contractive case) The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection

More information

Functions of several variables of finite variation and their differentiability

Functions of several variables of finite variation and their differentiability ANNALES POLONICI MATHEMATICI LX.1 (1994) Functions of several variables of finite variation and their differentiability by Dariusz Idczak ( Lódź) Abstract. Some differentiability properties of functions

More information

Random Bernstein-Markov factors

Random Bernstein-Markov factors Random Bernstein-Markov factors Igor Pritsker and Koushik Ramachandran October 20, 208 Abstract For a polynomial P n of degree n, Bernstein s inequality states that P n n P n for all L p norms on the unit

More information

Introduction to Random Diffusions

Introduction to Random Diffusions Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales

More information

Citation Osaka Journal of Mathematics. 41(4)

Citation Osaka Journal of Mathematics. 41(4) TitleA non quasi-invariance of the Brown Authors Sadasue, Gaku Citation Osaka Journal of Mathematics. 414 Issue 4-1 Date Text Version publisher URL http://hdl.handle.net/1194/1174 DOI Rights Osaka University

More information

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also

More information

Integral mean value theorems and the Ganelius inequality

Integral mean value theorems and the Ganelius inequality Proceedings of the Royal Society of Edinburgh, 97.4, 145-150, 1984 Integral mean value theorems and the Ganelius inequality Ian Knowles Department of Mathematics, University of Alabama in Birmingham, Birmingham,

More information

ON CONTINUITY OF MEASURABLE COCYCLES

ON CONTINUITY OF MEASURABLE COCYCLES Journal of Applied Analysis Vol. 6, No. 2 (2000), pp. 295 302 ON CONTINUITY OF MEASURABLE COCYCLES G. GUZIK Received January 18, 2000 and, in revised form, July 27, 2000 Abstract. It is proved that every

More information

arxiv: v2 [math.mg] 14 Apr 2014

arxiv: v2 [math.mg] 14 Apr 2014 An intermediate quasi-isometric invariant between subexponential asymptotic dimension growth and Yu s Property A arxiv:404.358v2 [math.mg] 4 Apr 204 Izhar Oppenheim Department of Mathematics The Ohio State

More information

Disconjugate operators and related differential equations

Disconjugate operators and related differential equations Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic

More information

arxiv: v1 [math.ap] 28 Mar 2014

arxiv: v1 [math.ap] 28 Mar 2014 GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard

More information

Yunhi Cho and Young-One Kim

Yunhi Cho and Young-One Kim Bull. Korean Math. Soc. 41 (2004), No. 1, pp. 27 43 ANALYTIC PROPERTIES OF THE LIMITS OF THE EVEN AND ODD HYPERPOWER SEQUENCES Yunhi Cho Young-One Kim Dedicated to the memory of the late professor Eulyong

More information

NUMBER THEORY, BALLS IN BOXES, AND THE ASYMPTOTIC UNIQUENESS OF MAXIMAL DISCRETE ORDER STATISTICS

NUMBER THEORY, BALLS IN BOXES, AND THE ASYMPTOTIC UNIQUENESS OF MAXIMAL DISCRETE ORDER STATISTICS NUMBER THEORY, BALLS IN BOXES, AND THE ASYMPTOTIC UNIQUENESS OF MAXIMAL DISCRETE ORDER STATISTICS Jayadev S. Athreya Department of Mathematics, Iowa State University, Ames, Iowa 50011, U.S.A. Lukasz M.

More information

ON PARABOLIC HARNACK INEQUALITY

ON PARABOLIC HARNACK INEQUALITY ON PARABOLIC HARNACK INEQUALITY JIAXIN HU Abstract. We show that the parabolic Harnack inequality is equivalent to the near-diagonal lower bound of the Dirichlet heat kernel on any ball in a metric measure-energy

More information

RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI.

RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI. RIMS-1898 A Note on an Anabelian Open Basis for a Smooth Variety By Yuichiro HOSHI January 2019 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan A Note on an Anabelian Open Basis

More information

Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian

Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian Variational eigenvalues of degenerate eigenvalue problems for the weighted p-laplacian An Lê Mathematics Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720 e-mail: anle@msri.org Klaus

More information

An essay on the general theory of stochastic processes

An essay on the general theory of stochastic processes Probability Surveys Vol. 3 (26) 345 412 ISSN: 1549-5787 DOI: 1.1214/1549578614 An essay on the general theory of stochastic processes Ashkan Nikeghbali ETHZ Departement Mathematik, Rämistrasse 11, HG G16

More information

A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

More information

THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r)

THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION. Juha Kinnunen. 1 f(y) dy, B(x, r) B(x,r) Appeared in Israel J. Math. 00 (997), 7 24 THE HARDY LITTLEWOOD MAXIMAL FUNCTION OF A SOBOLEV FUNCTION Juha Kinnunen Abstract. We prove that the Hardy Littlewood maximal operator is bounded in the Sobolev

More information

ARTICLE IN PRESS. J. Math. Anal. Appl. ( ) Note. On pairwise sensitivity. Benoît Cadre, Pierre Jacob

ARTICLE IN PRESS. J. Math. Anal. Appl. ( ) Note. On pairwise sensitivity. Benoît Cadre, Pierre Jacob S0022-27X0500087-9/SCO AID:9973 Vol. [DTD5] P.1 1-8 YJMAA:m1 v 1.35 Prn:15/02/2005; 16:33 yjmaa9973 by:jk p. 1 J. Math. Anal. Appl. www.elsevier.com/locate/jmaa Note On pairwise sensitivity Benoît Cadre,

More information

Some SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen

Some SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text

More information

Continued Fraction Digit Averages and Maclaurin s Inequalities

Continued Fraction Digit Averages and Maclaurin s Inequalities Continued Fraction Digit Averages and Maclaurin s Inequalities Steven J. Miller, Williams College sjm1@williams.edu, Steven.Miller.MC.96@aya.yale.edu Joint with Francesco Cellarosi, Doug Hensley and Jake

More information

SMSTC (2007/08) Probability.

SMSTC (2007/08) Probability. SMSTC (27/8) Probability www.smstc.ac.uk Contents 12 Markov chains in continuous time 12 1 12.1 Markov property and the Kolmogorov equations.................... 12 2 12.1.1 Finite state space.................................

More information

JASSON VINDAS AND RICARDO ESTRADA

JASSON VINDAS AND RICARDO ESTRADA A QUICK DISTRIBUTIONAL WAY TO THE PRIME NUMBER THEOREM JASSON VINDAS AND RICARDO ESTRADA Abstract. We use distribution theory (generalized functions) to show the prime number theorem. No tauberian results

More information

FACTORS OF GIBBS MEASURES FOR FULL SHIFTS

FACTORS OF GIBBS MEASURES FOR FULL SHIFTS FACTORS OF GIBBS MEASURES FOR FULL SHIFTS M. POLLICOTT AND T. KEMPTON Abstract. We study the images of Gibbs measures under one block factor maps on full shifts, and the changes in the variations of the

More information

A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1.

A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE. 1. A LOWER BOUND ON BLOWUP RATES FOR THE 3D INCOMPRESSIBLE EULER EQUATION AND A SINGLE EXPONENTIAL BEALE-KATO-MAJDA ESTIMATE THOMAS CHEN AND NATAŠA PAVLOVIĆ Abstract. We prove a Beale-Kato-Majda criterion

More information

Common fixed points of two generalized asymptotically quasi-nonexpansive mappings

Common fixed points of two generalized asymptotically quasi-nonexpansive mappings An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Common fixed points of two generalized asymptotically quasi-nonexpansive mappings Safeer Hussain Khan Isa Yildirim Received: 5.VIII.2013

More information

From Calculus II: An infinite series is an expression of the form

From Calculus II: An infinite series is an expression of the form MATH 3333 INTERMEDIATE ANALYSIS BLECHER NOTES 75 8. Infinite series of numbers From Calculus II: An infinite series is an expression of the form = a m + a m+ + a m+2 + ( ) Let us call this expression (*).

More information

56 4 Integration against rough paths

56 4 Integration against rough paths 56 4 Integration against rough paths comes to the definition of a rough integral we typically take W = LV, W ; although other choices can be useful see e.g. remark 4.11. In the context of rough differential

More information

arxiv: v2 [math.pr] 30 Sep 2009

arxiv: v2 [math.pr] 30 Sep 2009 Upper and Lower Bounds in Exponential Tauberian Theorems Jochen Voss 3th September 9 arxiv:98.64v [math.pr] 3 Sep 9 Abstract In this text we study, for positive random variables, the relation between the

More information

Functional central limit theorem for super α-stable processes

Functional central limit theorem for super α-stable processes 874 Science in China Ser. A Mathematics 24 Vol. 47 No. 6 874 881 Functional central it theorem for super α-stable processes HONG Wenming Department of Mathematics, Beijing Normal University, Beijing 1875,

More information

HOPF S DECOMPOSITION AND RECURRENT SEMIGROUPS. Josef Teichmann

HOPF S DECOMPOSITION AND RECURRENT SEMIGROUPS. Josef Teichmann HOPF S DECOMPOSITION AND RECURRENT SEMIGROUPS Josef Teichmann Abstract. Some results of ergodic theory are generalized in the setting of Banach lattices, namely Hopf s maximal ergodic inequality and the

More information

L p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by

L p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may

More information

ON AN INEQUALITY OF KOLMOGOROV AND STEIN

ON AN INEQUALITY OF KOLMOGOROV AND STEIN BULL. AUSTRAL. MATH. SOC. VOL. 61 (2000) [153-159] 26B35, 26D10 ON AN INEQUALITY OF KOLMOGOROV AND STEIN HA HUY BANG AND HOANG MAI LE A.N. Kolmogorov showed that, if /,/',..., /'"' are bounded continuous

More information

arxiv: v1 [math.ap] 18 May 2017

arxiv: v1 [math.ap] 18 May 2017 Littlewood-Paley-Stein functions for Schrödinger operators arxiv:175.6794v1 [math.ap] 18 May 217 El Maati Ouhabaz Dedicated to the memory of Abdelghani Bellouquid (2/2/1966 8/31/215) Abstract We study

More information

Reflected Brownian Motion

Reflected Brownian Motion Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide

More information

Appendix A. Sequences and series. A.1 Sequences. Definition A.1 A sequence is a function N R.

Appendix A. Sequences and series. A.1 Sequences. Definition A.1 A sequence is a function N R. Appendix A Sequences and series This course has for prerequisite a course (or two) of calculus. The purpose of this appendix is to review basic definitions and facts concerning sequences and series, which

More information

MATH 426, TOPOLOGY. p 1.

MATH 426, TOPOLOGY. p 1. MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p

More information

Positivity of Turán determinants for orthogonal polynomials

Positivity of Turán determinants for orthogonal polynomials Positivity of Turán determinants for orthogonal polynomials Ryszard Szwarc Abstract The orthogonal polynomials p n satisfy Turán s inequality if p 2 n (x) p n 1 (x)p n+1 (x) 0 for n 1 and for all x in

More information

Relevant sections from AMATH 351 Course Notes (Wainwright): 1.3 Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls): 1.1.

Relevant sections from AMATH 351 Course Notes (Wainwright): 1.3 Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls): 1.1. Lecture 8 Qualitative Behaviour of Solutions to ODEs Relevant sections from AMATH 351 Course Notes (Wainwright): 1.3 Relevant sections from AMATH 351 Course Notes (Poulin and Ingalls): 1.1.1 The last few

More information

On a Class of Multidimensional Optimal Transportation Problems

On a Class of Multidimensional Optimal Transportation Problems Journal of Convex Analysis Volume 10 (2003), No. 2, 517 529 On a Class of Multidimensional Optimal Transportation Problems G. Carlier Université Bordeaux 1, MAB, UMR CNRS 5466, France and Université Bordeaux

More information

arxiv: v1 [math.ag] 30 Apr 2018

arxiv: v1 [math.ag] 30 Apr 2018 QUASI-LOG CANONICAL PAIRS ARE DU BOIS OSAMU FUJINO AND HAIDONG LIU arxiv:1804.11138v1 [math.ag] 30 Apr 2018 Abstract. We prove that every quasi-log canonical pair has only Du Bois singularities. Note that

More information

Jordan Journal of Mathematics and Statistics (JJMS) 8(3), 2015, pp THE NORM OF CERTAIN MATRIX OPERATORS ON NEW DIFFERENCE SEQUENCE SPACES

Jordan Journal of Mathematics and Statistics (JJMS) 8(3), 2015, pp THE NORM OF CERTAIN MATRIX OPERATORS ON NEW DIFFERENCE SEQUENCE SPACES Jordan Journal of Mathematics and Statistics (JJMS) 8(3), 2015, pp 223-237 THE NORM OF CERTAIN MATRIX OPERATORS ON NEW DIFFERENCE SEQUENCE SPACES H. ROOPAEI (1) AND D. FOROUTANNIA (2) Abstract. The purpose

More information

GENERALIZED COVARIATION FOR BANACH SPACE VALUED PROCESSES, ITÔ FORMULA AND APPLICATIONS

GENERALIZED COVARIATION FOR BANACH SPACE VALUED PROCESSES, ITÔ FORMULA AND APPLICATIONS Di Girolami, C. and Russo, F. Osaka J. Math. 51 (214), 729 783 GENERALIZED COVARIATION FOR BANACH SPACE VALUED PROCESSES, ITÔ FORMULA AND APPLICATIONS CRISTINA DI GIROLAMI and FRANCESCO RUSSO (Received

More information

Constrained Leja points and the numerical solution of the constrained energy problem

Constrained Leja points and the numerical solution of the constrained energy problem Journal of Computational and Applied Mathematics 131 (2001) 427 444 www.elsevier.nl/locate/cam Constrained Leja points and the numerical solution of the constrained energy problem Dan I. Coroian, Peter

More information

n»i \ v f(x + h)-f(x-h)

n»i \ v f(x + h)-f(x-h) PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 72, Number 2, November 1978 SYMMETRIC AND ORDINARY DIFFERENTIATION C. L. BELNA, M. J. EVANS AND P. D. HUMKE Abstract. In 1927, A. Khintchine proved

More information

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.

A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n. Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone

More information

ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES

ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 ITERATED FUNCTION SYSTEMS WITH CONTINUOUS PLACE DEPENDENT PROBABILITIES by Joanna Jaroszewska Abstract. We study the asymptotic behaviour

More information

COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE

COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE Chao, X. Osaka J. Math. 50 (203), 75 723 COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE XIAOLI CHAO (Received August 8, 20, revised December 7, 20) Abstract In this paper, by modifying Cheng Yau

More information

On Kusuoka Representation of Law Invariant Risk Measures

On Kusuoka Representation of Law Invariant Risk Measures MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of

More information

CHAPTER III THE PROOF OF INEQUALITIES

CHAPTER III THE PROOF OF INEQUALITIES CHAPTER III THE PROOF OF INEQUALITIES In this Chapter, the main purpose is to prove four theorems about Hardy-Littlewood- Pólya Inequality and then gives some examples of their application. We will begin

More information

The Codimension of the Zeros of a Stable Process in Random Scenery

The Codimension of the Zeros of a Stable Process in Random Scenery The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar

More information

Pathwise uniqueness for stochastic differential equations driven by pure jump processes

Pathwise uniqueness for stochastic differential equations driven by pure jump processes Pathwise uniqueness for stochastic differential equations driven by pure jump processes arxiv:73.995v [math.pr] 9 Mar 7 Jiayu Zheng and Jie Xiong Abstract Based on the weak existence and weak uniqueness,

More information

Subdifferential representation of convex functions: refinements and applications

Subdifferential representation of convex functions: refinements and applications Subdifferential representation of convex functions: refinements and applications Joël Benoist & Aris Daniilidis Abstract Every lower semicontinuous convex function can be represented through its subdifferential

More information

CHAOTIC BEHAVIOR IN A FORECAST MODEL

CHAOTIC BEHAVIOR IN A FORECAST MODEL CHAOTIC BEHAVIOR IN A FORECAST MODEL MICHAEL BOYLE AND MARK TOMFORDE Abstract. We examine a certain interval map, called the weather map, that has been used by previous authors as a toy model for weather

More information

ON A MAXIMAL OPERATOR IN REARRANGEMENT INVARIANT BANACH FUNCTION SPACES ON METRIC SPACES

ON A MAXIMAL OPERATOR IN REARRANGEMENT INVARIANT BANACH FUNCTION SPACES ON METRIC SPACES Vasile Alecsandri University of Bacău Faculty of Sciences Scientific Studies and Research Series Mathematics and Informatics Vol. 27207), No., 49-60 ON A MAXIMAL OPRATOR IN RARRANGMNT INVARIANT BANACH

More information

On Reflecting Brownian Motion with Drift

On Reflecting Brownian Motion with Drift Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability

More information

A Note on the Central Limit Theorem for a Class of Linear Systems 1

A Note on the Central Limit Theorem for a Class of Linear Systems 1 A Note on the Central Limit Theorem for a Class of Linear Systems 1 Contents Yukio Nagahata Department of Mathematics, Graduate School of Engineering Science Osaka University, Toyonaka 560-8531, Japan.

More information

ASYMPTOTIC DIOPHANTINE APPROXIMATION: THE MULTIPLICATIVE CASE

ASYMPTOTIC DIOPHANTINE APPROXIMATION: THE MULTIPLICATIVE CASE ASYMPTOTIC DIOPHANTINE APPROXIMATION: THE MULTIPLICATIVE CASE MARTIN WIDMER ABSTRACT Let α and β be irrational real numbers and 0 < ε < 1/30 We prove a precise estimate for the number of positive integers

More information

EXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS

EXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS (February 25, 2004) EXTINCTION TIMES FOR A GENERAL BIRTH, DEATH AND CATASTROPHE PROCESS BEN CAIRNS, University of Queensland PHIL POLLETT, University of Queensland Abstract The birth, death and catastrophe

More information

Independence of some multiple Poisson stochastic integrals with variable-sign kernels

Independence of some multiple Poisson stochastic integrals with variable-sign kernels Independence of some multiple Poisson stochastic integrals with variable-sign kernels Nicolas Privault Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological

More information

P -adic root separation for quadratic and cubic polynomials

P -adic root separation for quadratic and cubic polynomials P -adic root separation for quadratic and cubic polynomials Tomislav Pejković Abstract We study p-adic root separation for quadratic and cubic polynomials with integer coefficients. The quadratic and reducible

More information

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 2: Countability and Cantor Sets

MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION. Chapter 2: Countability and Cantor Sets MATH31011/MATH41011/MATH61011: FOURIER ANALYSIS AND LEBESGUE INTEGRATION Chapter 2: Countability and Cantor Sets Countable and Uncountable Sets The concept of countability will be important in this course

More information

Spatial Ergodicity of the Harris Flows

Spatial Ergodicity of the Harris Flows Communications on Stochastic Analysis Volume 11 Number 2 Article 6 6-217 Spatial Ergodicity of the Harris Flows E.V. Glinyanaya Institute of Mathematics NAS of Ukraine, glinkate@gmail.com Follow this and

More information

MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem.

MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem. MATH 409 Advanced Calculus I Lecture 7: Monotone sequences. The Bolzano-Weierstrass theorem. Limit of a sequence Definition. Sequence {x n } of real numbers is said to converge to a real number a if for

More information

Notes on uniform convergence

Notes on uniform convergence Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean

More information

Citation Osaka Journal of Mathematics. 40(3)

Citation Osaka Journal of Mathematics. 40(3) Title An elementary proof of Small's form PSL(,C and an analogue for Legend Author(s Kokubu, Masatoshi; Umehara, Masaaki Citation Osaka Journal of Mathematics. 40(3 Issue 003-09 Date Text Version publisher

More information

Oscillation theorems for the Dirac operator with spectral parameter in the boundary condition

Oscillation theorems for the Dirac operator with spectral parameter in the boundary condition Electronic Journal of Qualitative Theory of Differential Equations 216, No. 115, 1 1; doi: 1.14232/ejqtde.216.1.115 http://www.math.u-szeged.hu/ejqtde/ Oscillation theorems for the Dirac operator with

More information

212a1214Daniell s integration theory.

212a1214Daniell s integration theory. 212a1214 Daniell s integration theory. October 30, 2014 Daniell s idea was to take the axiomatic properties of the integral as the starting point and develop integration for broader and broader classes

More information

Counting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary

Counting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary Counting geodesic arcs in a fixed conjugacy class on negatively curved surfaces with boundary Mark Pollicott Abstract We show how to derive an asymptotic estimates for the number of closed arcs γ on a

More information

3 Measurable Functions

3 Measurable Functions 3 Measurable Functions Notation A pair (X, F) where F is a σ-field of subsets of X is a measurable space. If µ is a measure on F then (X, F, µ) is a measure space. If µ(x) < then (X, F, µ) is a probability

More information