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by: Addison Beer


Addison Beer
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This 8 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 426 at University of Washington taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/192099/math-426-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15
ON FELLER S BOUNDARY PROBLEM AND DARNING COUNTABLE HOLES FOR MARKOV PROCESSES Masatoshi FUKUSHIMA a joint work with Zhen Qing CHEN Let E be a locally compact separable metric space and m be a o nite Eorel measure on E We consider a pair of Borel standard processes X Xt7 Pat7 X Xt7 Pay on E which are in weak duality with respect to m and satisfy the following X1 Every semi polar set is m polar for X Let F a1a2 e be a nite or countable closed subset of E and E0 E F X2 0 X3 reg Ex e WF XUF oi gt 07 Ex e WFgXaF oi gt O for qe w E E0 X4 X and X admit no jumps from EOAto F A Let X0 XE7 P2 and X0 XE7 P2 be the part processes of X and X7 respectively killed upon leaving E0 X0 and X0 are in weak duality with respect to mg the restriction ofm to E0 We view X and X as most general duality preserving extensions of X 0 and X 0 respectively from E0 to E E0 U E Our objectives are to characterize those extensions at the resolvent level as well as at the generator level using only the quantities intrinsic to X0 and X07 thus solving in the duality setting the boundary problem of Markov processes going back to WFeller i Let U27 W U3 be the Feller measures on F determined by X0X0 and let rim2 be the restrictions to F of the jumping and killing measures of X Let A be the matrix with entries given by 127 Uij Jij for 2 7amp9 and Zajzj J27 Vi m 2 By using a general result in 37 we get the following If F is nite7 then Gav age 7 Habit 7 Ua 1 av for 11 e BbE0 1 where GmGg are resolvent of X7 X07 and Hm39 av are vectors with entries reg ggtvm0 respectively If E if in nte7 then we have the expression of Ga as the limit of ii Consider the symmetric case X Then X0 X0 De ne the function space on F by He f FltffgtltooU1ltIfIIfIgtltoo 2 with Ff f being de ned by 5Fffgt 2 NM 7 fajgtgt2U j J27 Z aaz m S 00 3 721259 2 21 Let fofo be the Dirichlet space of X0 on L2E0m0 and f0ref ref be its re ected Dirichlet space Let us introduce a linear operator E on L2E0m0 speci ed by f D fg L2E0m0 ltgtf f0fff Ereffv7gv Vvefo We also introduce the ux Nfa off 6 D at a by Nltfgtltaigt frefltf7u 3gtgt was which turns out to be independent of 04 gt 0 Let A be the L2Em in nitesimal generator of X Using a general result in 2 we see that E is an extension of A and furthermore under cetain conditions on X0 and F we have the next characterization f E DA if and only if f E D f admits an XD ne limit function yf E f 4 Nfa2 Z foai YfgtajgtgtJijVfaigt 07 2 21 5 7217744 iii Let K be a disjoint union of compact sets which are locally nite E0 E K and E E0 U 041720427 be the topological extension of E0 obtained by regarding each set X as a one point a2 Given a pair of standard processes X 5 on E which are in weak duality with respect to a a nite measure m on E we consider their part processes X0 X0 on E0 which are in weak duality in m0 m0 is extended to E by setting 0 Under certain conditions on X0 TO we can repeat the procedure of darning each hole Ki as has been performed in 45 to construct duality preserving extensions X X of X0 X0 to E They admit no jumps from F to F but they may admit killings on F Their resolvents and generators can be characterized as in i ii 1 Z Q Chen and M Fukushima One point extensions of Markov processes by darning Preprint 2006 2 Z Q Chen M Fukushima and J Ying Traces of symmetric Markov processes and their characterizations Ann Probab 342006 10521102 3 Z Q Chen M Fukushima and J Ying Entrance lawexit system and Levy system of time changed processes To appear in Ill J at 4 M Fukushima and H Tanaka Poisson point processes attached to symmetric diHuions Ann Inst Henri Poincare Probab Statist 41 2005 419459 5 Z Q Chen M Fukushima and J Ying Extending Markov processes in weak duality by Poisson point processes of excursions To appear in Proceedings of the Abel Symposium 2005 Stochastic Analysis and ApplicationsiA Symposium in Honor of Kiyosi lto Springer http2abelsymposiumno2005preprints 6 W Feller On boundaries and lateral conditions for the Kolmogorov differential equations Ann Math 651957 527570 7 M Fukushima On boundary conditions for multidimensional Brownian motions with symmetric resolvent densities J Math Soc Japan 211969 5893 8 M L Silverstein Symmetric Markov Processes Lecture Notes in Math 426 SpringerVerlag 1974 On Arithmetic Of Hyperelliptic Curves Jing Yu Abstract In this expose Pell7s equation is put in a geometric perspective and a version of Artinls primitive roots conjecture is formulated for hyperel liptic jacobiansi Also explained are some recent results which throw new lights having to do with AnkenyArtinChowla s conjecture class number relations and CohenLenstra heuristicsi Introduction It is well known that there are close connections between the arithmetic behavior of algebraic number elds and that of the algebraic function elds in one variable over nite elds This connection has been a constant source for exciting developments of number theory in 20th century In this article we shall further explore the subject by examining some arithmetic questions about hyperelliptic function elds viewed as analogues of quadratic number 1 2 Jing Yu elds Let C be a hyperelliptic curve over an arbitrary base eld 16 ie a double cover of the projective line libkl Its function eld K is a separable quadratic extension of the rational function eld kti In analogy with the classical situation7 K is called an imaginary quadratic function eld if the point at in nity 00 on libk does not split into two points on C Otherwise K is said to be a real quadratic function eld If the characteristic of k is not 2 we may always write K Maw with D 6 Mt a square free polynomial In that case K is real if and only if degD is even7 positive and the leading coefficient of D is a square in kxi Contents ll Fundamental units of real quadratic function elds 2 A horizontal class number one problem Artin7s conjecture for hyper elliptic Jacobiansi 3 A geometric analogue of AnkenyArtin Chowla s conjecture 4 Class number relations 5 A vertical class number one problem CohenLenstra heuristics 1Fundamental units of real quadratic function elds Given a real quadratic function eld Kki Abel l asked the following question whether there exist functions in K whose divisors supported only On Arithmetic Of Hyperelliptic Curves 3 at in nity In other words7 if 004F and 00 are the two points on C above 007 he is looking for functions With divisors of the form nlt00 7007 n E Z A function in K has this property if and only if it is a nonconstant unit inside the the integral closure B of Mt in Ki If such functions do exist7 then it follows that EX 2 kx X Z If such function does not exist7 then Bgtlt kxi A nonconstant function u such that ukx generates BXkgtlt is called a fundamental unit of K i If u is a fundamental unit With divu n004r 7 007 then is the least positive integer such that the n004r 7 00 N 0 We call the regulator of K or C In the case of characteristic 2 and K Mt m one may Write u 7 SE With 7 78 6 Mt Then 7 2 7 32D 5 E kxi Thus finding units amounts to solving Pell7s equations over polynomial ringsi Geometrically one considers the Jacobian variety of the curve C Then nonconstant units exist in B if and only if the divisor class of 007OO is a torsion point inside the Jacobian JacCi The regulator of K is precisely the torsion in question To find out Whether such units exists for a given real quadratic function field7 Abel makes an appeal to the wellknown continued fractions method for solving Pellls equationi Suppose the characteristic of k is f 2 Expand Dt binomially in the formal series field klt as Dt a0 i Where a0 6 Mt and 041 la1 E kHltHti Then write 11 as a1 i with a1 6 Mt and lag E 02 4 Jing Yu kHltHt Proceed in this way one arrives at the continued fraction expansion VD la07a17 ul a0 a 7 1 a2 i As Abel observed nonconstant unit exists in Mt if and only if this continued fraction expansion is quasiperiodic in the sense that there are positive integers no and 1 such that anol 5am for sorneE E kxi If indeed quasiperiodicity happens one can solve the Pell7s equation nontrivially just as in the classical story of real quadratic number elds In the even characteristic case hyperelliptic curves are given by Artin Schreier curves y2 y Rt with Rt 6 Mt Abel s observation is still correcti One just has to write down the continued fraction expansion of the Artin Schreier root instead of the square root and Pellls equation replaced by a norm equation In general one expects that nonconstant units do not always exist in real quadratic function elds The above procedure provides us only with a pseudoalgorithm to answer the question of existence of nontrivial units for real quadratic function eldsi However if k is nite the continued fractions in question are always periodic so that nonconstant units can be found all the timer The most interesting case is certainly the case k Q and D E ZMi On Arithmetic Of Hyperelliptic Curves 5 To decide Whether there are nonconstant units in QtE one reduces the equation y2 Dt modulo various odd primes pi For all but nitely many p namely those dividing the discriminant of the polynomial Dt7 one obtains curve C over the nite eld lei Given such a prime p of good reductioni If the divisor class of 004r 700 is of order n in Jac C then unless p l n7 the divisor class of 004r 7 00 is also of order n in Jac Cpl This fact follows from the theory of reduction of abelian varieties The assertion that 004r 7 00 is of order n in Jac C can be checked easily via continued fraction expansion of m modulo pi Therefore in this case7 one also has at hand an effective algorithm to decide Whether there are nonconstant unitsi Algorithm Given D E and consider the curve C y2 Dt Com pute rst the regulator m of let7 for the least prime p of good reduc tion Then choose another prime p1 where the curve has good reduction and p1 If the regulator m1 of le1 t7 does not coincide with mpi for some i 2 0 then the divisor class of 004r 7 00 must be of in nite order Otherwise one writes down a few terms of the continued fraction expansion of V5 10 Let no be the least integer such that degt one gt 0 and degt 0le lt 0 where 0410 is the conjugate of one over If D is quasiperiodic with period length l then m1 should also be the regulator of


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