14
1 ( (Stochastics) ”. ”. ( ) ”. 1. (x (1) ,x (2) , ··· ,x (d) ), . ( ), x 0 =(x (1) 0 ,x (2) 0 , ··· ,x (d) 0 ). x 1 =(x (1) 1 ,x (2) 1 , ··· ,x (d) 1 ). A 0 =(a (0) ij ), i (a (0) ij ) j A 0 1 1997 ”( 21 100 1998, 816–823) ”( 34:6, 2005, 661–672). 1

CH28 - Beijing Normal Universitymath0.bnu.edu.cn/~chenmf/files/SciPopul/3.pdf4 G & u @ A > λ1 x v Q ] " > B V A} $, E E = {i,j,k,···} t (O H Z) @ A i i r = t ^ a j ` n N

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  • B�SH�?ZPF1lnm('���-K��$���h�A,J>�AJ (Stochastics) �QX�.mjf*o"Um{}W�WC$.�v “`n”. =je “�A�” > “(_�” �℄b��F+7��/J�Im�+X

  • 2W/�a > $��a. 1W{℄r x1, C1� ∑di=1 x(i)1 a(0)ij m��� j ?℄3�$��� l1K�C,M= (1KM=LzPhj�S0�,Jt�+KT?), H℄g x0ZGdl��℄:�℄ &℄r x1. �|Yb� x0 � x1 S0� A0 ?Zzk����x0 = x1A0._|!h�r

    x0 = xnAn−1 · · ·A0.kW) An ≡ A(�2;�$��℄n�)), $tf{ x0 = xnAn, n ≥ 0. k A $�$�rxn = x0A

    −n, n ≥ 0. (1)5�12>�q�℄g�4nH6��7$$ n �℄r xn. 1��[� I:��. |M�TY.m&�� � 1968 N�

    M67fFX{�L�>�u��� 1974 7��5J�Ldl℄r=�2. DbV-��Lk�='���_B� “�” zDb�

  • 3� a(n)ij e 1/6 �`nL?VA (1 ± 0.01)aij . He 1/3 2`nq 1% �2y-���4�x0 = (44.344, 20) r== n ,1�`nk�=�

    n 1 2 3`n 0 0.09 0.651=��p �,1�`n� 0.74, �G�� T = 8 ).)����t$eA"JL�M>�Z�M>��8.bp�>�d{8.>��1P�SYd{8.�SH POE. J8.p��Sd{ NPOE. 4�℄�U�SKA��'�OE�p)Z 144 me�,r℄Y��Bz15e�:}e�D��℄~�m20j�iy 144 me�1m,f�D���F�l�A7�0j$z 143! Y�X>�1m,z8���`}��{M>℄gYj\h1�/}}�℄g~) �M>A7>�{BCX�M>�Z�143!

    109 × 365× 24× 60× 60 ≈ 10231( ).1ml�w0�S��℄m#�e2`nG��$��$6 NP �S,r8.>� [20],[26]. D�~f�A�nlM>X��Cq�e1.f�h=M>A"J�mfg (f^g).`���8�l�mw0�S����XMSm2A�uD�}1GzR8D2A�dz�>�__9%l�G&u (}FiuVAm2�H/j). �5 ='r℄O�A>� (k 34J��). 1P>��X"��vv�Q (+_�3�BeYd{ 2>,Monte Carlo), ℄mD2A�\�r=e℄)�`n/℄D2Aw!g�i�Fe�m2`nu�wjj�1�3�Utl�GD2A�&u:D�e��zW��1P>�J_z4�v�kt)�?;Æ�S�v1P>�$e�/,r℄Y 30421 wG�0j1��GB>�m�s� 30380 wG)\I:39�Z�mwt__�b��GBMD�F��+_ [17]. 5R?m�MSmwtI9{mwt$�A>�$� NP �00{ P �0���

    xF�4B_��QEU��A>�H�dV�℄�/rrf� [28],[29].

  • 4

    �G&u �A>�� λ1BBN ����xxv�Q�℄"℄�>�B�v�Q�VA}$,E E = {i, j, k, · · · } t� (OH�Z) �A�i� i r=���Z t ^a� j �`nN{ pij(t). 1W��z pij(t) ≥ 0 F∑j∈E pij(t) = 1. =℄"��,N{ qij = p′ij(t)|t=0. 1m�6 Q = (qij) _zk�℄��K�J6k(��{℄ (∑j qij = 0). x&℄�KK> Q qv}_A/T 1{ 0, H Q1 = 0 = 0 · 1, H�6 Q z4CP7A λ0 = 0. e_X� −Q ��℄mP7A λ1 ��{��_$��X�����$�D l�M=�6}P�6 −Q �:$�OH~1z λ1 > λ0(= 0). &0�39zv����Bd;W�M0�1�m{_z

    pij(t) → πj � t → ∞. (∗)m~℄F��P=0z8/�{���� λ1. H |pij(t)− πj | ∼ e−λ1t, t → ∞. 6}�A>�`_� (πi) V{ “E.” }� i w��,AH�,A�2>�℄b�℄w�$�����1W>��z4�YX�V�}Bz!�v�Q (pij(t)) -�|LE�=0�5R?λ1 ��)q��A>�����$_ ℄ (J4C) P7A λ1 �1GAbq+�qv�z�|qS�s+��q)$P [21]. 17RF (∗) L�D'_z℄JxkHi

    πj =∑

    i

    πipij(t), t ≥ 0, j ∈ E.x|r=Y�8>wZrU�A,J�y)j “,TtQ�A$0�m��J"”, m{&0�Q60'} “�A$0”.��d�8~R�v�Qm:w|A9xx��v�QH��mbf���_15uq~1}�A>�z��b�_�v�Q�`nqL���.�{,q�,JL���f,�Dfp�YFtg+:n}℄m`nqH�T=v�Q.z�f�0a���v�Q�QH�etw2l!e� ��uq��d�zU`;�B>�A$��v��Ptw}GX�Jktw2�U/^�#5_��KQL?� Springer r�US+Q℄[ [22], [16], U/^� John-Wiley r�U℄+Q℄[ [23], ���9UB�v"Jr�zr�zr�U℄+Q℄[ [5]. 6}15℄[IPYX�,J5q�F.p�U�f�5X�Jk~�� Math. Reviews ����� Math. Abstracts ��5q���℄[ [5]�6��D�ze>_�U��℄m�J*� (School). T. M. Liggett (1994) �|$X{Ld���+�+[q?℄� C. Maes � S. B. Shlosman (1993) $�|$X{7e~�9�L�p+

  • 5�}|

    ��RrUImJ_z4�z℄�oLYb�15uq&7* S. P. Meyn � R. L.Tweedie (1993), K. Hamza � F. C. Klebaner (1995), G. Kersting � F. C. Klebaner (1995) �QH�='�Y� D. A. Dawson, S. Feng� X. G. ZhengQ4�^�=t�xL>d��tpmH�.zU���~'�`��4�℄m2Jd�m℄��M=��� l

  • 6

    λ1 �}P,�80D{�d���P7A�SNBX�=Q�ÆD.�jf�r6}dh� Q, X,r λ1 ��℄A�|zBv�D�p5|M�����,bQh℄�I�H��uqm{1H��uqmo8�zp+���$ [2],[3] � [25]. 1p+$.+U)�8�0VNq ℄P7A�S�=L ℄+4�EU�� 20000P �%�P7A|M�I�J�y(S���D�,A M. Kac e “��\r}�=_w�” {S�[q��,���vI�P7=rP7A�|w��} l|O`Ay=���Xv�pmI�T�=�}, d, �� D � Ricci Un�v K. ℄}M= K = 0, f}R� K > 0, /U*Z K < 0. ���v�v1pmI�T|M=t6�>d ℄P7A��v���DKtv|MF)6jf℄5����v|M:R��YD�KK℄5[|M� ℄ms��\D8 40 Gx A. Lichnerowicz(1958) B����

    λ1 ≥d

    d− 1K, K > 0, (2)v�SmI�T d � K. 1�℄mw~s�m{6}f}*Z���R�F��℄���� 28 ?&||MLx P. Bérard, G. Besson � S. Gallot(1985) GL�_~{λ1 ≥

    dK

    d− 1

    { ∫ π/20

    cosd−1 u du∫ D/20

    cosd−1 u du

    }2/d, K ≥ d− 1. (3)� K = 0 �S0��4��� �����℄.QX~'(} P. Li � S. T. Yau(1980). D��r

    λ1 ≥π2

    2D2, K ≥ 0.G0�v�SD�� Cheeger D� (1970), &0�vQ0|MH��SPH�&7.z℄�X�qv�&℄|MxNTO — U!Q (1984) _~{

    λ1 ≥π2

    D2, K ≥ 0, (4)�℄Un���mw��}ZUnM=z Li–Yau(1980) |M

    λ1 ≥exp

    (− 1−

    √1− 4D2K(d− 1)

    )

    D2(d− 1) , K ≤ 0.x1m=�Y$d�r7ZUnM=XuD�81�I�t__����&�~℄F='zO�Y (1991):λ1 ≥

    π2

    D2+K, K ≤ 0; (5)U!Q (1989) �VH (1991),

    λ1 ≥

    π2

    D2e−α/2, d ≥ 5;

    π2

    2D2e−α

    ′/2, 2 ≤ d ≤ 4.(6)|w α = D√−K(d− 1), α′ = D√−K((d− 1) ∨ 2). U� (1989) �1℄|Md{QfaIV�

  • 7`�et�℄Xxxd�$e�rI�t�} ℄P7A�Qd)�o8��61WQ����_�b�4��p5�{�mq l1m(S�-A�{UQ�zi*qv�PLd��)8>{ (&�f~℄Fxx). �vI�JT�GM,u��H��&7=$vQLd�B='A7�`nH�Y�{v}QI�P7A����tS�[10] �v`nH�=$U℄m℄�w���Da)xx1;Z�F8LZFV:DKK1mw� xxSm℄�N�

    F =� [0,D] tOH���G{9��, f �ZU.C(r) = coshd−1

    (r

    2

    √K

    d− 1

    ), r ∈ [0,D],&0�/U"� d− 1 ~��zU1SmN�7$6rk�℄�w�

    λ1 ≥ supf∈F

    infr∈(0,D)

    4f(r)∫ r0C(s)−1 ds

    ∫ Ds

    C(u)f(u) du.b:1G�}�, f ∈ F Vt℄vm:6}℄�T�, f , DXDI�g.$e��℄mD2��v�JkV4C� f ≡ 1, B�r�|M�I�tY�J4C��1mhjh|w��℄P8Lw��S��`�}1Pv}�v|M�w

    G~z��6.tv|M z��#�D�D2w

    $_7?Df�{Ct|w�4MDCfF_~v}k�q��,�

    f(r) = sin(βr), sin(αr), sin(βr), coshd−1(αr) sin(βr),|w β = π2D

    , α =1

    2

    √|K|d− 1, 7$�rk�7m|M�λ1 ≥

    π2

    D2+max

    { π4d

    , 1− 2π

    }K, K ≥ 0. (7)

    λ1 ≥dK

    d− 1{1− cosd [αD]

    }−1, K ≥ 0. (8)

    λ1 ≥π2

    D2+(π2− 1)K, K ≤ 0. (9)

    λ1 ≥π2

    D2

    √1− 2D

    2K

    π4cosh1−d [αD] , K ≤ 0. (10)�G�BX��oPmw|M.m#_z (7) w} (4), (9) w} (5), (10) w} (6). m K > 0� αD ≤ π/2, (8) w} (2). 4$e{V�15|MI�JT&YrU�7� �[8&M�q�BrBv�t)w�J`nA"J`n�I��L����hh

  • 8 � �t ~ � }1mS���|wTQhI"�1. -g�/�℄��hF�`nq�TdÆC)s��℄Æ��m{TdÆC�+K�Ji5�F�y�`nqzbq+�T?eG}`nq� Kolmogorov wC�dD�v�:J�8,Ju�.z��Dbk1G��+u�?℄� C∗ �,�4zTdÆCr=x R. L. Hudson � K. R. Parthasarathy(1983) Bn~�Td�ABL�1℄℄Sdx&0r�U℄[�Td`n�Gdz℄=D2�5

    YX�'U�x L. Accardi �bBY5��Edr�U��?8��}1H/�J0i�e (D�G�h�Iuq), d��zf��2.

  • 9{H=Æ π r=�i��& t �`nLE�|=�LE π � t �d{ 5N��, �ÆCtd{ 5�C. x15Nq�rk�sq�a) 4 J π ��Jz℄�b) �P}4 J�=0{ e−λ1t.uGj℄F lz'hB*Z�� {−1, +1} 5f E = {−1, +1}S(S z'). �{t)SYsq__fK��uGj℄F l' ($,) hB*Z E = {−1, +1}Zd . ℄�Ctk�� l4�t�8"}℄"tI℄{�dFz9�ZD?L�8m4��=℄m[{U���B�:��+�S���[{U$*{0H “G” 8{ “zG”. 1℄{0�inp U~. )8$0ww$_�0�℄0��0�� 100◦C �D0.�)8�)8Q�XA�&>�b�H�)8�$���v x = (xu : u ∈ Zd) 7=Æ E L�"� u ∈ Zd d{�I>r

  • 10CQ9:��℄m=1_�Y��G,JÆC�`nq�mQX�Y='H/?℄�1DX�7AA “J. of Statistical Physics” � “Communications in Mathematical Physics” SP�F7$�rsDv2`nq�FU�k��Y�℄mo

  • 11

    o�P_`n�PLp (p ≥ 1) �P

    a.s. �P ��PF�?Z���ktfBÆ�1G��PYd{_LE�Pkd> ξn �LE��P}��A8T ξ �LE$|���PY��X}o�PU�&0�X}6}}℄zvOH�, f ,∫

    fdPn →∫

    fdP, n → ∞,|w Pn � P LE� ξn � ξ �LE�15�P�?Zz℄QX\?���P�o�PD_9}P rSmLE P1 � P2 ��B�D�1{jf��_:x)j (11) >d�y� P̃ e-pr W (P1, P2) �tv|M�6}F8qv72tv|M�{U (P?6}�P� W (Pn, P ) → 0). {|��I#y� P̃ , $__�I$>��x|Y$e�ry�H��`n�B?Z�|�N��y���{���$eprk�℄�)j�p)℄m$V*Z (E,E ) t�`nV0 P1 �P2, B� P1 � P2 �℄my� P̃ , ��hB*Z (E2,E 2) t�℄m`nV0xkk�� }`Y:

    P̃ (A× E) = P1(A),P̃ (E ×A) = P2(A), A ∈ E .6}p)� (E,E ) t�^a`n�, p(t, x,A), � t � x �)� p(t, x, ·) �℄`nV0�1W6}�)� t, x � y, ��$)j p(t, x, ·) � p(t, y, ·) �y�~:)jF�� Wasserstein

  • 12�B�$�1W�)j|z8wvwm{ p(t, x, ·) I(.�|z#=Æ�����p�� l�i�P2>d�y� (Hv�y�). 4�e�6M=2�℄��U�,��G�dh�6} Q = (qij), qij =Æ�i_J i Z� j (6= i) �=0���$e℄��=fi → j (6= i) =n qij .$�B�y���Xz!hB*Z E2 t��6�fv�66���uDU�vt)=Æ$�Ljf�* 1 (�k�). � i1 6= i2 �V

    (i1, i2) → (j1, i2) =n qi1j1 (j1 6= i1)→ (i1, j2) =n qi2j2 (j2 6= i2)T$V

    (i, i) → (j, j) =n qij (j 6= i).|J� ℄>=Æ#ly��i� 2 mLTD- 1 LTe=n qi1j1 i1 Z� j1 6= i1. �hj?8�15�BSLT�r="Db (i1 6= i2) �M=�kr=")b$`SLT℄AZ�U19�t)��>�6}&��Jd1 p>�℄W�m:pXD6����1my�Æ{ “ \"f7” y�m{SLT)�?Gojo�j (/K), :)��℄&���zUyb��;�j�* 2 (6�ik�). | E = {0, 1, 2, · · · }.(i1, i2) → (i1 + k, i2 + k)

    → (i1 + k, i2)→ (i1, i2 + k)

    =n qi1,i1+k ∧ qi2,i2+k=n (qi1,i1+k − qi2,i2+k)+=n (qi2,i2+k − qi1,i1+k)+,|w�) qii = 0, qij = 0 m j < 0.|y���`� ℄>F�hj��SLT�$�#l�r=�)b��B�kbG5RR�℄��� “�Fj” �Æa)���>~Li��$�#lD�?Z��BD8�zU ℄>?&&S>$e-6r7m:�JL��fpX&S>D6�etSmz!Lb�8w$ed"�$�`nq (�A�i) ���℄P�H�`n���1mhjt̀ nqso�}I��6}p6k6+KtxSX� 0,qi,i−1 = ai > 0. 6k(�}1SX��E��61PP#M=��4z`℄Py��* 3 (Æ;k�). k i2 = i1 + 1, V

    (i1, i2) → (i1 − 1, i2 + 1),→ (i1 + 1, i2),→ (i1, i2 − 1),

    =n ai1 ∧ bi2=n bi1=n ai2k i2 ≥ i1 + 2, V(i1, i2) → (i1 − 1, i2 + 1), =n ai1 ∧ bi2

    → (i1 + 1, i2 − 1), =n bi1 ∧ ai2 .x6d�$6r i1 > i2 ��=n�

  • 13|z!�m&℄>�e_�m{FXSSmLTu�$�!���Hj�_: ℄�7S>\�D9_FXSSLTu)B�5��Hj�1����`�ZD�o"��r(hV���F8^�1Py�\�m�� [7]. 6}=L���|wD$�+)�D�2�℄��Bz15,�9�mO,� ℄P7A�v|M8Lw�YX�B�?_`n�B$`3�� ety�H�$`3� 1937 �_:F��;A�='��q�v}�A^�Ld��Q��tF�d�&SJ"Q�mQX�,Ju��`n�B�y�H��

  • 14

    [21]. Sinclair, A. (1993), Algorithms for Random Generation and Counting: A Markov Chain Ap-proach, Birkhäuser.

    [22]. tw2�U/^ (1992), The Birth and Death Processes and Markov Chains, Springer-Verlag,Science Press.

    [23]. U/^ (1990), The Construction Theory of Denumerable Markov Processes, John Wiley& Sons.[24]. P�` (1989), PLdv;$U�inq , &��F�Jr�z.[25]. Qfa�AC[ (1988), xLI�, "Jr�z.[26]. Blum, L., Cucker, F., Shub, M. and Smale, S. (1998), Complexity and Real Computation,

    preprint.

    [27]. ���ReQ�F�4 (1997), p}�A Ising �