4215:. The Poisson boundary of the Brownian motion on the associated symmetric space is also the Furstenberg boundary. The full Martin boundary is also well-studied in these cases and can always be described in a geometric manner. For example, for groups of rank one (for example the isometry groups of
4246:
of its Cayley tree. The identification of the full Martin boundary is more involved; in case the random walk has finite range (the step distribution is supported on a finite set) the Martin boundary coincides with the minimal Martin boundary and both coincide with the Gromov boundary.
4242:, under rather weak assumptions on the step distribution which always hold for a simple walk (a more general condition is that the first moment be finite) the Poisson boundary is always equal to the Gromov boundary. For example, the Poisson boundary of a free group is the space of
4198:
The
Poisson and Martin boundaries are trivial for symmetric random walks in nilpotent groups. On the other hand, when the random walk is non-centered, the study of the full Martin boundary, including the minimal functions, is far less conclusive.
3928:
881:. This gives an identification of the extremal positive harmonic functions on the group, and to the space of trajectories of the random walk on the group (both with respect to a given probability measure), with the topological/measured space
3163:
3436:
2566:
409:
3728:
3285:
The Martin kernels are positive harmonic functions and every positive harmonic function can be expressed as an integral of functions on the boundary, that is for every positive harmonic function there is a measure
325:
1912:
3007:
507:
159:
3986:
50:
behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity. Despite being called a boundary it is in general a purely measure-theoretical object and not a
3215:
1649:
4107:
2181:
3280:
609:
1788:
850:
825:
800:
638:
1084:
2401:
1465:
3241:
2918:
543:
4068:
2315:
1365:
1233:
1177:
2725:
2030:
1517:
1739:
578:
3787:
3552:
4134:
901:
879:
238:
3812:
1269:
4179:
3472:
3317:
2806:
2470:
2246:
188:
2641:
2368:
2217:
1967:
1820:
1579:
740:
4042:
3337:
2609:
2266:
1759:
1692:
1599:
1543:
1431:
1398:
700:
208:
3062:
2764:
2341:
3590:
1669:
1336:
1204:
1001:
775:
676:
3619:
2661:
2589:
2086:
1987:
1712:
1104:
954:
258:
3036:
802:
endowed with the class of
Lebesgue measure (note that this identification can be made directly since a path in Wiener space converges almost surely to a point on
4006:
3807:
3751:
3520:
3496:
3067:
2866:
2846:
2826:
2681:
2441:
2421:
2286:
2106:
2066:
1935:
1313:
1289:
1148:
1128:
1021:
974:
934:
92:
3345:
2482:
4431:
Kaimanovich, Vadim A. (1996). "Boundaries of invariant Markov operators: the identification problem". In
Pollicott, Mark; Schmidt, Klaus (eds.).
3217:
has a relatively compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A point
4136:
is the bottom of the spectrum. Examples where this construction can be used to define a compactification are bounded domains in the plane and
333:
3630:
263:
1828:
2926:
4490:
4219:) the full Martin boundary is the same as the minimal Martin boundary (the situation in higher-rank groups is more complicated).
414:
100:
3933:
4388:
Ballmann, Werner; Ledrappier, François (1994). "The
Poisson boundary for rank one manifolds and their cocompact lattices".
3168:
1604:
777:. Thus the Poisson formula identifies this measured space with the Martin boundary constructed above, and ultimately to
4211:(with step distribution absolutely continuous with respect to the Haar measure) the Poisson boundary is equal to the
4073:
2111:
3246:
583:
54:. However, in the case where the random walk is on a topological space the Poisson boundary can be related to the
852:
as the space of trajectories for a Markov process is a special case of the construction of the
Poisson boundary.
743:
17:
51:
4485:
211:
2727:
associated to a random walk. Much of the theory can be developed in this abstract and very general setting.
1764:
855:
Finally, the constructions above can be discretised, i.e. restricted to the random walks on the orbits of a
58:, which is an analytic construction yielding a genuine topological boundary. Both boundaries are related to
830:
805:
780:
618:
1026:
4441:. London Math. Soc. Lecture Note Ser. Vol. 228. Cambridge Univ. Press, Cambridge. pp. 127–176.
4227:
3478:
Martin boundary, whose elements can also be characterised by being minimal. A positive harmonic function
2373:
1436:
3220:
2871:
4480:
520:
4047:
2291:
1341:
1209:
1153:
4185:
on the Martin boundary. With this measure the Martin boundary is isomorphic to the
Poisson boundary.
4044:. In this case there is again a whole family of Martin compactifications associated to the operators
2698:
1992:
1470:
3923:{\displaystyle {\mathcal {K}}_{o,r}(x,y)={\frac {{\mathcal {G}}_{r}(x,y)}{{\mathcal {G}}_{r}(o,y)}}}
3624:
There is actually a whole family of Martin compactifications. Define the Green generating series as
1717:
548:
3760:
3525:
4112:
884:
862:
221:
2033:
1238:
4151:
3444:
3289:
2769:
2446:
2222:
164:
2614:
2346:
2186:
1940:
1793:
1552:
709:
4027:
3322:
2594:
2251:
1744:
1677:
1584:
1522:
1403:
1370:
685:
193:
39:
3041:
2743:
2320:
4463:
4454:
Kaimanovich, Vadim A. (2000). "The
Poisson formula for groups with hyperbolic properties".
4446:
4414:
4397:
4212:
3557:
1654:
1321:
1316:
1182:
979:
753:
654:
3595:
3158:{\displaystyle {\mathcal {K}}_{o}(x,y)={\frac {{\mathcal {G}}(x,y)}{{\mathcal {G}}(o,y)}}}
2646:
2574:
2071:
1972:
1697:
1089:
939:
243:
8:
4021:
4017:
3015:
615:
in the cone of nonnegative harmonic functions. This analytical interpretation of the set
47:
4020:
the Martin boundary is constructed, when it exists, in the same way as above, using the
4243:
3991:
3792:
3736:
3505:
3481:
2851:
2831:
2811:
2666:
2426:
2406:
2271:
2091:
2051:
1920:
1298:
1274:
1133:
1113:
1006:
959:
919:
77:
215:
1917:
It is possible to give an implicit definition of the
Poisson boundary as the maximal
59:
4239:
4223:
4216:
4459:
4442:
4410:
4393:
4137:
703:
856:
679:
63:
3431:{\displaystyle f(x)=\int {\mathcal {K}}_{o}(x,\gamma )\,d\nu _{o,f}(\gamma ).}
4474:
612:
2591:-harmonic bounded functions and essentially bounded measurable functions on
2561:{\displaystyle f(x)=\int _{\Gamma }{\hat {f}}(\gamma )\,d\nu _{x}(\gamma ).}
911:
4405:
Furstenberg, Harry (1963). "A Poisson formula for semi-simple Lie groups".
3754:
2219:
is a discrete-time martingale and so it converges almost surely. Denote by
747:
4293:
4143:
1292:
43:
31:
2643:
is trivial, that is reduced to a point, if and only if the only bounded
404:{\displaystyle f(z)=\int _{\partial \mathbb {D} }K(z,\xi )\,d\mu (\xi )}
3723:{\displaystyle {\mathcal {G}}_{r}(x,y)=\sum _{n\geq 1}p_{n}(x,y)r^{n}.}
2037:
4208:
95:
2695:
on a measured space, a notion which generalises the Markov operator
1694:
is the initial distribution of a random walk with step distribution
74:
The
Poisson formula states that given a positive harmonic function
1023:(a discrete-time Markov process whose transition probabilities are
706:
on the disc with the
Poincaré Riemannian metric), then the process
4365:
4353:
4317:
4305:
4281:
4269:
1150:, which will be the initial state for the random walk. The space
320:{\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} :|z|=1\}}
4341:
3012:
If the walk is transient then this series is convergent for all
1907:{\displaystyle \int _{G}\nu (g^{-1}A)\mu (g)=\nu _{\theta }(A).}
651:
This fact can also be interpreted in a probabilistic manner. If
1295:
of measures; this is the distribution of the random walk after
746:, and as such converges almost everywhere to a function on the
3002:{\displaystyle {\mathcal {G}}(x,y)=\sum _{n\geq 1}p_{n}(x,y).}
4011:
502:{\displaystyle K(z,\xi )={\frac {1-|z|^{2}}{|\xi -z|^{2}}}}
154:{\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}
4230:, is also equal to the Furstenberg boundary of the group.
2288:
along a trajectory (this is defined almost everywhere on
912:
The Poisson boundary of a random walk on a discrete group
3981:{\displaystyle y\mapsto {\mathcal {K}}_{o,r}(\cdot ,y)}
2370:
be the measure obtained by the construction above with
4329:
4257:
4144:
The relationship between Martin and Poisson boundaries
2735:
27:
Mathematical measure space associated to a random walk
4422:
Guivarc'h, Yves; Ji, Lizhen; Taylor, John C. (1998).
4421:
4347:
4299:
4181:
corresponding to the constant function is called the
4154:
4115:
4076:
4050:
4030:
3994:
3936:
3815:
3795:
3763:
3739:
3633:
3598:
3560:
3528:
3508:
3484:
3447:
3348:
3325:
3292:
3249:
3223:
3210:{\displaystyle y\mapsto {\mathcal {K}}_{o}(\cdot ,y)}
3171:
3070:
3044:
3018:
2929:
2874:
2854:
2834:
2814:
2772:
2746:
2701:
2669:
2649:
2617:
2597:
2577:
2485:
2449:
2429:
2409:
2376:
2349:
2323:
2294:
2274:
2254:
2225:
2189:
2114:
2094:
2074:
2054:
1995:
1975:
1943:
1923:
1831:
1796:
1767:
1747:
1720:
1700:
1680:
1657:
1607:
1587:
1555:
1525:
1473:
1439:
1406:
1373:
1344:
1324:
1301:
1277:
1241:
1212:
1185:
1156:
1136:
1116:
1092:
1029:
1009:
982:
962:
942:
922:
887:
865:
833:
808:
783:
756:
712:
688:
657:
621:
586:
551:
523:
417:
336:
266:
246:
224:
196:
167:
103:
80:
4202:
1644:{\displaystyle (G^{\mathbb {N} },\mathbb {P} _{m})}
69:
4387:
4226:subgroup of a semisimple Lie group, for example a
4173:
4128:
4101:
4062:
4036:
4000:
3980:
3922:
3801:
3781:
3745:
3722:
3613:
3584:
3546:
3514:
3490:
3466:
3430:
3331:
3311:
3274:
3235:
3209:
3157:
3056:
3030:
3001:
2912:
2860:
2840:
2820:
2800:
2758:
2719:
2675:
2655:
2635:
2603:
2583:
2560:
2464:
2435:
2415:
2395:
2362:
2335:
2309:
2280:
2260:
2240:
2211:
2175:
2100:
2080:
2060:
2024:
1981:
1961:
1929:
1906:
1814:
1782:
1753:
1733:
1706:
1686:
1663:
1643:
1593:
1573:
1537:
1511:
1459:
1425:
1392:
1359:
1330:
1307:
1283:
1263:
1227:
1198:
1171:
1142:
1122:
1098:
1078:
1015:
995:
968:
948:
928:
895:
873:
844:
819:
794:
769:
734:
694:
670:
632:
603:
572:
545:. One way to interpret this is that the functions
537:
501:
403:
319:
252:
232:
202:
182:
153:
86:
1545:(the two trajectories have the same "tail"). The
4472:
4102:{\displaystyle 0\leq \lambda \leq \lambda _{0}}
2176:{\displaystyle \sum _{h\in G}f(hg)\mu (h)=f(g)}
3275:{\displaystyle {\mathcal {K}}(\cdot ,\gamma )}
2268:obtained by taking the limit of the values of
4392:. Vol. 6, no. 3. pp. 301–313.
976:, which will be used to define a random walk
604:{\displaystyle \xi \in \partial \mathbb {D} }
314:
278:
148:
112:
4453:
4430:
4404:
4371:
4359:
4335:
4323:
4311:
4287:
4275:
4263:
1989:, satisfying the additional condition that
2766:be a random walk on a discrete group. Let
3396:
2532:
2301:
1770:
1628:
1617:
1453:
1351:
1215:
1163:
889:
867:
838:
813:
788:
626:
597:
531:
385:
361:
288:
271:
226:
122:
105:
46:. It is an object designed to encode the
4012:Martin boundary of a Riemannian manifold
3339:such that a Poisson-like formula holds:
2611:. In particular the Poisson boundary of
750:of possible (infinite) trajectories for
62:on the space via generalisations of the
3243:is usually represented by the notation
14:
4473:
4458:. 2. Vol. 152. pp. 659–692.
2730:
2043:
1783:{\displaystyle \mathbb {P} _{\theta }}
4424:Compactifications of symmetric spaces
4409:. 2. Vol. 77. pp. 335–386.
2920:. The Green kernel is by definition:
2686:
2571:This establishes a bijection between
845:{\displaystyle \partial \mathbb {D} }
820:{\displaystyle \partial \mathbb {D} }
795:{\displaystyle \partial \mathbb {D} }
633:{\displaystyle \partial \mathbb {D} }
4233:
1079:{\displaystyle p(x,y)=\mu (xy^{-1})}
640:leads to the more general notion of
4193:
2736:Martin boundary of a discrete group
2443:is either positive or bounded then
2396:{\displaystyle \theta =\delta _{x}}
1460:{\displaystyle n,m\in \mathbb {N} }
24:
4051:
4031:
3946:
3891:
3860:
3819:
3753:the radius of convergence of this
3637:
3370:
3326:
3252:
3236:{\displaystyle \gamma \in \Gamma }
3230:
3181:
3132:
3108:
3074:
3064:and define the Martin kernel by:
2932:
2913:{\displaystyle \mu ^{*n}(x^{-1}y)}
2598:
2506:
2255:
1748:
1588:
834:
809:
784:
689:
622:
593:
357:
267:
197:
168:
25:
4502:
4207:For random walks on a semisimple
4203:Lie groups and discrete subgroups
4024:of the Laplace–Beltrami operator
2691:The general setting is that of a
1790:. It is a stationary measure for
538:{\displaystyle z\in \mathbb {D} }
52:boundary in the topological sense
4063:{\displaystyle \Delta +\lambda }
2310:{\displaystyle G^{\mathbb {N} }}
1360:{\displaystyle G^{\mathbb {N} }}
1228:{\displaystyle \mathbb {P} _{m}}
1172:{\displaystyle G^{\mathbb {N} }}
644:(which in this case is the full
240:) there exists a unique measure
70:The case of the hyperbolic plane
4348:Guivarc'h, Ji & Taylor 1998
4300:Guivarc'h, Ji & Taylor 1998
3930:. The closure of the embedding
2808:be the probability to get from
2720:{\displaystyle f\mapsto \mu *f}
2025:{\displaystyle (X_{t})_{*}\nu }
1761:obtained as the pushforward of
1512:{\displaystyle x_{t+n}=y_{t+m}}
4491:Compactification (mathematics)
3975:
3963:
3940:
3914:
3902:
3883:
3871:
3848:
3836:
3704:
3692:
3660:
3648:
3579:
3567:
3422:
3416:
3393:
3381:
3358:
3352:
3269:
3257:
3204:
3192:
3175:
3149:
3137:
3125:
3113:
3097:
3085:
2993:
2981:
2949:
2937:
2907:
2888:
2795:
2783:
2705:
2630:
2618:
2552:
2546:
2529:
2523:
2517:
2495:
2489:
2456:
2232:
2206:
2193:
2170:
2164:
2155:
2149:
2143:
2134:
2010:
1996:
1956:
1944:
1898:
1892:
1876:
1870:
1864:
1845:
1809:
1797:
1734:{\displaystyle \nu _{\theta }}
1638:
1608:
1568:
1556:
1420:
1407:
1387:
1374:
1073:
1054:
1045:
1033:
729:
716:
573:{\displaystyle K(\cdot ,\xi )}
567:
555:
486:
471:
458:
449:
433:
421:
398:
392:
382:
370:
346:
340:
304:
296:
138:
130:
13:
1:
4381:
3782:{\displaystyle 1\leq r\leq R}
3547:{\displaystyle 0\leq v\leq u}
3502:if for any harmonic function
906:
4439:actions (Warwick, 1993–1994)
4129:{\displaystyle \lambda _{0}}
1651:by the equivalence relation
1601:obtained as the quotient of
896:{\displaystyle \mathbb {D} }
874:{\displaystyle \mathbb {D} }
233:{\displaystyle \mathbb {D} }
7:
4188:
2472:is as well and we have the
2183:. Then the random variable
1581:is then the measured space
1264:{\displaystyle m*\mu ^{*n}}
10:
4507:
4222:The Poisson boundary of a
4174:{\displaystyle \nu _{o,1}}
4008:-Martin compactification.
3467:{\displaystyle \nu _{o,f}}
3312:{\displaystyle \nu _{o,f}}
2801:{\displaystyle p_{n}(x,y)}
2465:{\displaystyle {\hat {f}}}
2317:and shift-invariant). Let
2241:{\displaystyle {\hat {f}}}
1206:is endowed with a measure
827:). This interpretation of
611:are up to scaling all the
183:{\displaystyle \Delta f=0}
1315:steps). There is also an
1110:for the random walk. Let
956:a probability measure on
212:Laplace–Beltrami operator
4250:
2636:{\displaystyle (G,\mu )}
2363:{\displaystyle \nu _{x}}
2212:{\displaystyle f(X_{t})}
1962:{\displaystyle (G,\mu )}
1815:{\displaystyle (G,\mu )}
1574:{\displaystyle (G,\mu )}
936:be a discrete group and
735:{\displaystyle f(W_{t})}
327:such that the equality
4037:{\displaystyle \Delta }
3332:{\displaystyle \Gamma }
2663:-harmonic functions on
2604:{\displaystyle \Gamma }
2261:{\displaystyle \Gamma }
1754:{\displaystyle \Gamma }
1687:{\displaystyle \theta }
1594:{\displaystyle \Gamma }
1538:{\displaystyle t\geq 0}
1426:{\displaystyle (y_{t})}
1393:{\displaystyle (x_{t})}
695:{\displaystyle \Delta }
642:minimal Martin boundary
203:{\displaystyle \Delta }
4238:For random walks on a
4175:
4130:
4103:
4064:
4038:
4002:
3982:
3924:
3803:
3783:
3747:
3724:
3615:
3586:
3548:
3516:
3492:
3468:
3432:
3333:
3313:
3276:
3237:
3211:
3159:
3058:
3057:{\displaystyle o\in G}
3032:
3003:
2914:
2862:
2842:
2822:
2802:
2760:
2759:{\displaystyle G,\mu }
2721:
2677:
2657:
2637:
2605:
2585:
2562:
2466:
2437:
2417:
2397:
2364:
2337:
2336:{\displaystyle x\in G}
2311:
2282:
2262:
2242:
2213:
2177:
2102:
2088:-harmonic function on
2082:
2062:
2026:
1983:
1963:
1931:
1908:
1816:
1784:
1755:
1735:
1708:
1688:
1665:
1645:
1595:
1575:
1539:
1513:
1461:
1427:
1394:
1361:
1332:
1309:
1285:
1265:
1229:
1200:
1173:
1144:
1130:be another measure on
1124:
1100:
1080:
1017:
997:
970:
950:
930:
897:
875:
846:
821:
796:
771:
736:
696:
672:
634:
605:
574:
539:
503:
405:
321:
254:
234:
204:
184:
155:
88:
4176:
4140:of non-compact type.
4131:
4104:
4065:
4039:
4003:
3983:
3925:
3804:
3784:
3748:
3725:
3616:
3587:
3585:{\displaystyle c\in }
3549:
3517:
3493:
3474:are supported on the
3469:
3433:
3334:
3314:
3277:
3238:
3212:
3160:
3059:
3033:
3004:
2915:
2863:
2843:
2823:
2803:
2761:
2722:
2678:
2658:
2638:
2606:
2586:
2563:
2467:
2438:
2418:
2398:
2365:
2338:
2312:
2283:
2263:
2243:
2214:
2178:
2103:
2083:
2063:
2027:
1984:
1964:
1932:
1909:
1817:
1785:
1756:
1736:
1709:
1689:
1666:
1664:{\displaystyle \sim }
1646:
1596:
1576:
1540:
1514:
1462:
1428:
1395:
1362:
1333:
1331:{\displaystyle \sim }
1310:
1286:
1266:
1235:whose marginales are
1230:
1201:
1199:{\displaystyle X_{t}}
1174:
1145:
1125:
1101:
1081:
1018:
998:
996:{\displaystyle X_{t}}
971:
951:
931:
898:
876:
847:
822:
797:
772:
770:{\displaystyle W_{t}}
742:is a continuous-time
737:
697:
673:
671:{\displaystyle W_{t}}
635:
606:
575:
540:
504:
406:
322:
255:
235:
205:
185:
156:
89:
4486:Stochastic processes
4213:Furstenberg boundary
4152:
4113:
4074:
4048:
4028:
3992:
3934:
3813:
3793:
3761:
3737:
3631:
3614:{\displaystyle v=cu}
3596:
3558:
3526:
3506:
3482:
3445:
3346:
3323:
3290:
3247:
3221:
3169:
3068:
3042:
3016:
2927:
2872:
2852:
2832:
2812:
2770:
2744:
2699:
2667:
2656:{\displaystyle \mu }
2647:
2615:
2595:
2584:{\displaystyle \mu }
2575:
2483:
2447:
2427:
2407:
2374:
2347:
2321:
2292:
2272:
2252:
2223:
2187:
2112:
2092:
2081:{\displaystyle \mu }
2072:
2052:
1993:
1982:{\displaystyle \nu }
1973:
1969:-stationary measure
1941:
1921:
1829:
1794:
1765:
1745:
1718:
1707:{\displaystyle \mu }
1698:
1678:
1655:
1605:
1585:
1553:
1523:
1471:
1437:
1404:
1371:
1342:
1322:
1317:equivalence relation
1299:
1275:
1239:
1210:
1183:
1179:of trajectories for
1154:
1134:
1114:
1099:{\displaystyle \mu }
1090:
1027:
1007:
980:
960:
949:{\displaystyle \mu }
940:
920:
885:
863:
831:
806:
781:
754:
710:
686:
655:
619:
584:
549:
521:
415:
334:
264:
253:{\displaystyle \mu }
244:
222:
194:
165:
101:
78:
4018:Riemannian manifold
3031:{\displaystyle x,y}
2731:The Martin boundary
2403:(the Dirac mass at
2044:The Poisson formula
1367:, which identifies
4433:Ergodic theory of
4171:
4126:
4099:
4060:
4034:
3998:
3978:
3920:
3809:-Martin kernel by
3799:
3779:
3743:
3720:
3681:
3611:
3582:
3544:
3512:
3488:
3464:
3428:
3329:
3309:
3272:
3233:
3207:
3155:
3054:
3028:
2999:
2970:
2910:
2858:
2838:
2818:
2798:
2756:
2717:
2687:General definition
2673:
2653:
2633:
2601:
2581:
2558:
2462:
2433:
2413:
2393:
2360:
2333:
2307:
2278:
2258:
2238:
2209:
2173:
2130:
2098:
2078:
2058:
2022:
1979:
1959:
1927:
1904:
1812:
1780:
1751:
1731:
1704:
1684:
1661:
1641:
1591:
1571:
1535:
1509:
1457:
1423:
1390:
1357:
1328:
1305:
1281:
1261:
1225:
1196:
1169:
1140:
1120:
1096:
1076:
1013:
993:
966:
946:
926:
893:
871:
842:
817:
792:
767:
732:
692:
668:
630:
601:
570:
535:
499:
401:
317:
250:
230:
214:associated to the
200:
180:
151:
84:
60:harmonic functions
4481:Harmonic analysis
4234:Hyperbolic groups
4217:hyperbolic spaces
4001:{\displaystyle r}
3918:
3802:{\displaystyle r}
3746:{\displaystyle R}
3666:
3515:{\displaystyle v}
3491:{\displaystyle u}
3165:. The embedding
3153:
2955:
2861:{\displaystyle n}
2841:{\displaystyle y}
2821:{\displaystyle x}
2676:{\displaystyle G}
2520:
2459:
2436:{\displaystyle f}
2416:{\displaystyle x}
2281:{\displaystyle f}
2235:
2115:
2101:{\displaystyle G}
2061:{\displaystyle f}
1930:{\displaystyle G}
1714:then the measure
1308:{\displaystyle n}
1284:{\displaystyle *}
1143:{\displaystyle G}
1123:{\displaystyle m}
1108:step distribution
1016:{\displaystyle G}
969:{\displaystyle G}
929:{\displaystyle G}
497:
87:{\displaystyle f}
16:(Redirected from
4498:
4467:
4450:
4427:
4418:
4401:
4375:
4372:Kaimanovich 2000
4369:
4363:
4360:Kaimanovich 2000
4357:
4351:
4345:
4339:
4336:Furstenberg 1963
4333:
4327:
4324:Kaimanovich 1996
4321:
4315:
4312:Kaimanovich 1996
4309:
4303:
4297:
4291:
4288:Kaimanovich 1996
4285:
4279:
4276:Kaimanovich 1996
4273:
4267:
4264:Kaimanovich 1996
4261:
4240:hyperbolic group
4194:Nilpotent groups
4183:harmonic measure
4180:
4178:
4177:
4172:
4170:
4169:
4138:symmetric spaces
4135:
4133:
4132:
4127:
4125:
4124:
4108:
4106:
4105:
4100:
4098:
4097:
4069:
4067:
4066:
4061:
4043:
4041:
4040:
4035:
4007:
4005:
4004:
3999:
3987:
3985:
3984:
3979:
3962:
3961:
3950:
3949:
3929:
3927:
3926:
3921:
3919:
3917:
3901:
3900:
3895:
3894:
3886:
3870:
3869:
3864:
3863:
3855:
3835:
3834:
3823:
3822:
3808:
3806:
3805:
3800:
3788:
3786:
3785:
3780:
3752:
3750:
3749:
3744:
3729:
3727:
3726:
3721:
3716:
3715:
3691:
3690:
3680:
3647:
3646:
3641:
3640:
3620:
3618:
3617:
3612:
3591:
3589:
3588:
3583:
3553:
3551:
3550:
3545:
3521:
3519:
3518:
3513:
3497:
3495:
3494:
3489:
3473:
3471:
3470:
3465:
3463:
3462:
3437:
3435:
3434:
3429:
3415:
3414:
3380:
3379:
3374:
3373:
3338:
3336:
3335:
3330:
3318:
3316:
3315:
3310:
3308:
3307:
3281:
3279:
3278:
3273:
3256:
3255:
3242:
3240:
3239:
3234:
3216:
3214:
3213:
3208:
3191:
3190:
3185:
3184:
3164:
3162:
3161:
3156:
3154:
3152:
3136:
3135:
3128:
3112:
3111:
3104:
3084:
3083:
3078:
3077:
3063:
3061:
3060:
3055:
3037:
3035:
3034:
3029:
3008:
3006:
3005:
3000:
2980:
2979:
2969:
2936:
2935:
2919:
2917:
2916:
2911:
2903:
2902:
2887:
2886:
2867:
2865:
2864:
2859:
2847:
2845:
2844:
2839:
2827:
2825:
2824:
2819:
2807:
2805:
2804:
2799:
2782:
2781:
2765:
2763:
2762:
2757:
2726:
2724:
2723:
2718:
2682:
2680:
2679:
2674:
2662:
2660:
2659:
2654:
2642:
2640:
2639:
2634:
2610:
2608:
2607:
2602:
2590:
2588:
2587:
2582:
2567:
2565:
2564:
2559:
2545:
2544:
2522:
2521:
2513:
2510:
2509:
2471:
2469:
2468:
2463:
2461:
2460:
2452:
2442:
2440:
2439:
2434:
2422:
2420:
2419:
2414:
2402:
2400:
2399:
2394:
2392:
2391:
2369:
2367:
2366:
2361:
2359:
2358:
2342:
2340:
2339:
2334:
2316:
2314:
2313:
2308:
2306:
2305:
2304:
2287:
2285:
2284:
2279:
2267:
2265:
2264:
2259:
2248:the function on
2247:
2245:
2244:
2239:
2237:
2236:
2228:
2218:
2216:
2215:
2210:
2205:
2204:
2182:
2180:
2179:
2174:
2129:
2107:
2105:
2104:
2099:
2087:
2085:
2084:
2079:
2067:
2065:
2064:
2059:
2034:weakly converges
2031:
2029:
2028:
2023:
2018:
2017:
2008:
2007:
1988:
1986:
1985:
1980:
1968:
1966:
1965:
1960:
1936:
1934:
1933:
1928:
1913:
1911:
1910:
1905:
1891:
1890:
1860:
1859:
1841:
1840:
1822:, meaning that
1821:
1819:
1818:
1813:
1789:
1787:
1786:
1781:
1779:
1778:
1773:
1760:
1758:
1757:
1752:
1740:
1738:
1737:
1732:
1730:
1729:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1670:
1668:
1667:
1662:
1650:
1648:
1647:
1642:
1637:
1636:
1631:
1622:
1621:
1620:
1600:
1598:
1597:
1592:
1580:
1578:
1577:
1572:
1547:Poisson boundary
1544:
1542:
1541:
1536:
1518:
1516:
1515:
1510:
1508:
1507:
1489:
1488:
1466:
1464:
1463:
1458:
1456:
1433:if there exists
1432:
1430:
1429:
1424:
1419:
1418:
1399:
1397:
1396:
1391:
1386:
1385:
1366:
1364:
1363:
1358:
1356:
1355:
1354:
1337:
1335:
1334:
1329:
1314:
1312:
1311:
1306:
1290:
1288:
1287:
1282:
1270:
1268:
1267:
1262:
1260:
1259:
1234:
1232:
1231:
1226:
1224:
1223:
1218:
1205:
1203:
1202:
1197:
1195:
1194:
1178:
1176:
1175:
1170:
1168:
1167:
1166:
1149:
1147:
1146:
1141:
1129:
1127:
1126:
1121:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1072:
1071:
1022:
1020:
1019:
1014:
1002:
1000:
999:
994:
992:
991:
975:
973:
972:
967:
955:
953:
952:
947:
935:
933:
932:
927:
902:
900:
899:
894:
892:
880:
878:
877:
872:
870:
851:
849:
848:
843:
841:
826:
824:
823:
818:
816:
801:
799:
798:
793:
791:
776:
774:
773:
768:
766:
765:
741:
739:
738:
733:
728:
727:
701:
699:
698:
693:
677:
675:
674:
669:
667:
666:
639:
637:
636:
631:
629:
610:
608:
607:
602:
600:
579:
577:
576:
571:
544:
542:
541:
536:
534:
508:
506:
505:
500:
498:
496:
495:
494:
489:
474:
468:
467:
466:
461:
452:
440:
410:
408:
407:
402:
366:
365:
364:
326:
324:
323:
318:
307:
299:
291:
274:
260:on the boundary
259:
257:
256:
251:
239:
237:
236:
231:
229:
209:
207:
206:
201:
189:
187:
186:
181:
160:
158:
157:
152:
141:
133:
125:
108:
93:
91:
90:
85:
42:associated to a
36:Poisson boundary
21:
4506:
4505:
4501:
4500:
4499:
4497:
4496:
4495:
4471:
4470:
4384:
4379:
4378:
4370:
4366:
4362:, Theorem 10.7.
4358:
4354:
4346:
4342:
4334:
4330:
4322:
4318:
4310:
4306:
4298:
4294:
4286:
4282:
4274:
4270:
4262:
4258:
4253:
4236:
4205:
4196:
4191:
4159:
4155:
4153:
4150:
4149:
4146:
4120:
4116:
4114:
4111:
4110:
4093:
4089:
4075:
4072:
4071:
4049:
4046:
4045:
4029:
4026:
4025:
4014:
3993:
3990:
3989:
3951:
3945:
3944:
3943:
3935:
3932:
3931:
3896:
3890:
3889:
3888:
3887:
3865:
3859:
3858:
3857:
3856:
3854:
3824:
3818:
3817:
3816:
3814:
3811:
3810:
3794:
3791:
3790:
3762:
3759:
3758:
3757:and define for
3738:
3735:
3734:
3711:
3707:
3686:
3682:
3670:
3642:
3636:
3635:
3634:
3632:
3629:
3628:
3597:
3594:
3593:
3559:
3556:
3555:
3527:
3524:
3523:
3507:
3504:
3503:
3483:
3480:
3479:
3452:
3448:
3446:
3443:
3442:
3404:
3400:
3375:
3369:
3368:
3367:
3347:
3344:
3343:
3324:
3321:
3320:
3297:
3293:
3291:
3288:
3287:
3251:
3250:
3248:
3245:
3244:
3222:
3219:
3218:
3186:
3180:
3179:
3178:
3170:
3167:
3166:
3131:
3130:
3129:
3107:
3106:
3105:
3103:
3079:
3073:
3072:
3071:
3069:
3066:
3065:
3043:
3040:
3039:
3017:
3014:
3013:
2975:
2971:
2959:
2931:
2930:
2928:
2925:
2924:
2895:
2891:
2879:
2875:
2873:
2870:
2869:
2853:
2850:
2849:
2833:
2830:
2829:
2813:
2810:
2809:
2777:
2773:
2771:
2768:
2767:
2745:
2742:
2741:
2738:
2733:
2700:
2697:
2696:
2693:Markov operator
2689:
2668:
2665:
2664:
2648:
2645:
2644:
2616:
2613:
2612:
2596:
2593:
2592:
2576:
2573:
2572:
2540:
2536:
2512:
2511:
2505:
2501:
2484:
2481:
2480:
2474:Poisson formula
2451:
2450:
2448:
2445:
2444:
2428:
2425:
2424:
2408:
2405:
2404:
2387:
2383:
2375:
2372:
2371:
2354:
2350:
2348:
2345:
2344:
2322:
2319:
2318:
2300:
2299:
2295:
2293:
2290:
2289:
2273:
2270:
2269:
2253:
2250:
2249:
2227:
2226:
2224:
2221:
2220:
2200:
2196:
2188:
2185:
2184:
2119:
2113:
2110:
2109:
2108:, meaning that
2093:
2090:
2089:
2073:
2070:
2069:
2053:
2050:
2049:
2046:
2013:
2009:
2003:
1999:
1994:
1991:
1990:
1974:
1971:
1970:
1942:
1939:
1938:
1922:
1919:
1918:
1886:
1882:
1852:
1848:
1836:
1832:
1830:
1827:
1826:
1795:
1792:
1791:
1774:
1769:
1768:
1766:
1763:
1762:
1746:
1743:
1742:
1725:
1721:
1719:
1716:
1715:
1699:
1696:
1695:
1679:
1676:
1675:
1656:
1653:
1652:
1632:
1627:
1626:
1616:
1615:
1611:
1606:
1603:
1602:
1586:
1583:
1582:
1554:
1551:
1550:
1524:
1521:
1520:
1497:
1493:
1478:
1474:
1472:
1469:
1468:
1452:
1438:
1435:
1434:
1414:
1410:
1405:
1402:
1401:
1381:
1377:
1372:
1369:
1368:
1350:
1349:
1345:
1343:
1340:
1339:
1323:
1320:
1319:
1300:
1297:
1296:
1276:
1273:
1272:
1252:
1248:
1240:
1237:
1236:
1219:
1214:
1213:
1211:
1208:
1207:
1190:
1186:
1184:
1181:
1180:
1162:
1161:
1157:
1155:
1152:
1151:
1135:
1132:
1131:
1115:
1112:
1111:
1091:
1088:
1087:
1086:); the measure
1064:
1060:
1028:
1025:
1024:
1008:
1005:
1004:
987:
983:
981:
978:
977:
961:
958:
957:
941:
938:
937:
921:
918:
917:
914:
909:
888:
886:
883:
882:
866:
864:
861:
860:
837:
832:
829:
828:
812:
807:
804:
803:
787:
782:
779:
778:
761:
757:
755:
752:
751:
723:
719:
711:
708:
707:
704:Brownian motion
687:
684:
683:
662:
658:
656:
653:
652:
646:Martin boundary
625:
620:
617:
616:
596:
585:
582:
581:
550:
547:
546:
530:
522:
519:
518:
490:
485:
484:
470:
469:
462:
457:
456:
448:
441:
439:
416:
413:
412:
360:
356:
352:
335:
332:
331:
303:
295:
287:
270:
265:
262:
261:
245:
242:
241:
225:
223:
220:
219:
216:Poincaré metric
195:
192:
191:
166:
163:
162:
137:
129:
121:
104:
102:
99:
98:
79:
76:
75:
72:
64:Poisson formula
56:Martin boundary
28:
23:
22:
18:Martin boundary
15:
12:
11:
5:
4504:
4494:
4493:
4488:
4483:
4469:
4468:
4451:
4428:
4419:
4402:
4383:
4380:
4377:
4376:
4374:, Theorem 7.4.
4364:
4352:
4340:
4328:
4326:, Section 2.8.
4316:
4314:, Section 1.5.
4304:
4292:
4290:, Section 1.2.
4280:
4278:, Section 2.7.
4268:
4255:
4254:
4252:
4249:
4235:
4232:
4204:
4201:
4195:
4192:
4190:
4187:
4168:
4165:
4162:
4158:
4145:
4142:
4123:
4119:
4096:
4092:
4088:
4085:
4082:
4079:
4059:
4056:
4053:
4033:
4022:Green function
4013:
4010:
3997:
3988:is called the
3977:
3974:
3971:
3968:
3965:
3960:
3957:
3954:
3948:
3942:
3939:
3916:
3913:
3910:
3907:
3904:
3899:
3893:
3885:
3882:
3879:
3876:
3873:
3868:
3862:
3853:
3850:
3847:
3844:
3841:
3838:
3833:
3830:
3827:
3821:
3798:
3778:
3775:
3772:
3769:
3766:
3742:
3731:
3730:
3719:
3714:
3710:
3706:
3703:
3700:
3697:
3694:
3689:
3685:
3679:
3676:
3673:
3669:
3665:
3662:
3659:
3656:
3653:
3650:
3645:
3639:
3610:
3607:
3604:
3601:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3543:
3540:
3537:
3534:
3531:
3511:
3498:is said to be
3487:
3461:
3458:
3455:
3451:
3439:
3438:
3427:
3424:
3421:
3418:
3413:
3410:
3407:
3403:
3399:
3395:
3392:
3389:
3386:
3383:
3378:
3372:
3366:
3363:
3360:
3357:
3354:
3351:
3328:
3306:
3303:
3300:
3296:
3271:
3268:
3265:
3262:
3259:
3254:
3232:
3229:
3226:
3206:
3203:
3200:
3197:
3194:
3189:
3183:
3177:
3174:
3151:
3148:
3145:
3142:
3139:
3134:
3127:
3124:
3121:
3118:
3115:
3110:
3102:
3099:
3096:
3093:
3090:
3087:
3082:
3076:
3053:
3050:
3047:
3038:. Fix a point
3027:
3024:
3021:
3010:
3009:
2998:
2995:
2992:
2989:
2986:
2983:
2978:
2974:
2968:
2965:
2962:
2958:
2954:
2951:
2948:
2945:
2942:
2939:
2934:
2909:
2906:
2901:
2898:
2894:
2890:
2885:
2882:
2878:
2857:
2837:
2817:
2797:
2794:
2791:
2788:
2785:
2780:
2776:
2755:
2752:
2749:
2737:
2734:
2732:
2729:
2716:
2713:
2710:
2707:
2704:
2688:
2685:
2683:are constant.
2672:
2652:
2632:
2629:
2626:
2623:
2620:
2600:
2580:
2569:
2568:
2557:
2554:
2551:
2548:
2543:
2539:
2535:
2531:
2528:
2525:
2519:
2516:
2508:
2504:
2500:
2497:
2494:
2491:
2488:
2458:
2455:
2432:
2412:
2390:
2386:
2382:
2379:
2357:
2353:
2332:
2329:
2326:
2303:
2298:
2277:
2257:
2234:
2231:
2208:
2203:
2199:
2195:
2192:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2128:
2125:
2122:
2118:
2097:
2077:
2057:
2045:
2042:
2032:almost surely
2021:
2016:
2012:
2006:
2002:
1998:
1978:
1958:
1955:
1952:
1949:
1946:
1926:
1915:
1914:
1903:
1900:
1897:
1894:
1889:
1885:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1858:
1855:
1851:
1847:
1844:
1839:
1835:
1811:
1808:
1805:
1802:
1799:
1777:
1772:
1750:
1728:
1724:
1703:
1683:
1660:
1640:
1635:
1630:
1625:
1619:
1614:
1610:
1590:
1570:
1567:
1564:
1561:
1558:
1534:
1531:
1528:
1506:
1503:
1500:
1496:
1492:
1487:
1484:
1481:
1477:
1455:
1451:
1448:
1445:
1442:
1422:
1417:
1413:
1409:
1389:
1384:
1380:
1376:
1353:
1348:
1327:
1304:
1280:
1258:
1255:
1251:
1247:
1244:
1222:
1217:
1193:
1189:
1165:
1160:
1139:
1119:
1106:is called the
1095:
1075:
1070:
1067:
1063:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1012:
990:
986:
965:
945:
925:
913:
910:
908:
905:
891:
869:
857:Fuchsian group
840:
836:
815:
811:
790:
786:
764:
760:
731:
726:
722:
718:
715:
691:
682:associated to
680:Markov process
665:
661:
628:
624:
613:extreme points
599:
595:
592:
589:
569:
566:
563:
560:
557:
554:
533:
529:
526:
517:holds for all
515:
514:
511:Poisson kernel
493:
488:
483:
480:
477:
473:
465:
460:
455:
451:
447:
444:
438:
435:
432:
429:
426:
423:
420:
400:
397:
394:
391:
388:
384:
381:
378:
375:
372:
369:
363:
359:
355:
351:
348:
345:
342:
339:
316:
313:
310:
306:
302:
298:
294:
290:
286:
283:
280:
277:
273:
269:
249:
228:
199:
179:
176:
173:
170:
150:
147:
144:
140:
136:
132:
128:
124:
120:
117:
114:
111:
107:
83:
71:
68:
26:
9:
6:
4:
3:
2:
4503:
4492:
4489:
4487:
4484:
4482:
4479:
4478:
4476:
4465:
4461:
4457:
4452:
4448:
4444:
4440:
4437:
4436:
4429:
4426:. Birkhäuser.
4425:
4420:
4416:
4412:
4408:
4403:
4399:
4395:
4391:
4386:
4385:
4373:
4368:
4361:
4356:
4349:
4344:
4337:
4332:
4325:
4320:
4313:
4308:
4302:, Chapter VI.
4301:
4296:
4289:
4284:
4277:
4272:
4265:
4260:
4256:
4248:
4245:
4241:
4231:
4229:
4225:
4224:Zariski-dense
4220:
4218:
4214:
4210:
4200:
4186:
4184:
4166:
4163:
4160:
4156:
4141:
4139:
4121:
4117:
4094:
4090:
4086:
4083:
4080:
4077:
4057:
4054:
4023:
4019:
4009:
3995:
3972:
3969:
3966:
3958:
3955:
3952:
3937:
3911:
3908:
3905:
3897:
3880:
3877:
3874:
3866:
3851:
3845:
3842:
3839:
3831:
3828:
3825:
3796:
3776:
3773:
3770:
3767:
3764:
3756:
3740:
3717:
3712:
3708:
3701:
3698:
3695:
3687:
3683:
3677:
3674:
3671:
3667:
3663:
3657:
3654:
3651:
3643:
3627:
3626:
3625:
3622:
3608:
3605:
3602:
3599:
3576:
3573:
3570:
3564:
3561:
3554:there exists
3541:
3538:
3535:
3532:
3529:
3509:
3501:
3485:
3477:
3459:
3456:
3453:
3449:
3441:The measures
3425:
3419:
3411:
3408:
3405:
3401:
3397:
3390:
3387:
3384:
3376:
3364:
3361:
3355:
3349:
3342:
3341:
3340:
3304:
3301:
3298:
3294:
3283:
3266:
3263:
3260:
3227:
3224:
3201:
3198:
3195:
3187:
3172:
3146:
3143:
3140:
3122:
3119:
3116:
3100:
3094:
3091:
3088:
3080:
3051:
3048:
3045:
3025:
3022:
3019:
2996:
2990:
2987:
2984:
2976:
2972:
2966:
2963:
2960:
2956:
2952:
2946:
2943:
2940:
2923:
2922:
2921:
2904:
2899:
2896:
2892:
2883:
2880:
2876:
2855:
2835:
2815:
2792:
2789:
2786:
2778:
2774:
2753:
2750:
2747:
2728:
2714:
2711:
2708:
2702:
2694:
2684:
2670:
2650:
2627:
2624:
2621:
2578:
2555:
2549:
2541:
2537:
2533:
2526:
2514:
2502:
2498:
2492:
2486:
2479:
2478:
2477:
2475:
2453:
2430:
2410:
2388:
2384:
2380:
2377:
2355:
2351:
2330:
2327:
2324:
2296:
2275:
2229:
2201:
2197:
2190:
2167:
2161:
2158:
2152:
2146:
2140:
2137:
2131:
2126:
2123:
2120:
2116:
2095:
2075:
2055:
2041:
2039:
2035:
2019:
2014:
2004:
2000:
1976:
1953:
1950:
1947:
1924:
1901:
1895:
1887:
1883:
1879:
1873:
1867:
1861:
1856:
1853:
1849:
1842:
1837:
1833:
1825:
1824:
1823:
1806:
1803:
1800:
1775:
1726:
1722:
1701:
1681:
1672:
1658:
1633:
1623:
1612:
1565:
1562:
1559:
1548:
1532:
1529:
1526:
1504:
1501:
1498:
1494:
1490:
1485:
1482:
1479:
1475:
1449:
1446:
1443:
1440:
1415:
1411:
1382:
1378:
1346:
1325:
1318:
1302:
1294:
1278:
1256:
1253:
1249:
1245:
1242:
1220:
1191:
1187:
1158:
1137:
1117:
1109:
1093:
1068:
1065:
1061:
1057:
1051:
1048:
1042:
1039:
1036:
1030:
1010:
988:
984:
963:
943:
923:
904:
858:
853:
762:
758:
749:
745:
724:
720:
713:
705:
681:
663:
659:
649:
647:
643:
614:
590:
587:
564:
561:
558:
552:
527:
524:
512:
491:
481:
478:
475:
463:
453:
445:
442:
436:
430:
427:
424:
418:
395:
389:
386:
379:
376:
373:
367:
353:
349:
343:
337:
330:
329:
328:
311:
308:
300:
292:
284:
281:
275:
247:
217:
213:
177:
174:
171:
145:
142:
134:
126:
118:
115:
109:
97:
81:
67:
65:
61:
57:
53:
49:
45:
41:
40:measure space
37:
33:
19:
4456:Ann. of Math
4455:
4438:
4434:
4432:
4423:
4407:Ann. of Math
4406:
4389:
4367:
4355:
4343:
4331:
4319:
4307:
4295:
4283:
4271:
4259:
4237:
4221:
4206:
4197:
4182:
4148:The measure
4147:
4015:
3755:power series
3732:
3623:
3499:
3475:
3440:
3284:
3011:
2868:steps, i.e.
2739:
2692:
2690:
2570:
2473:
2047:
1937:-set with a
1916:
1673:
1546:
1107:
915:
854:
748:Wiener space
650:
645:
641:
516:
510:
73:
55:
35:
29:
1293:convolution
44:random walk
32:mathematics
4475:Categories
4390:Forum Math
4382:References
3733:Denote by
3592:such that
2038:Dirac mass
1467:such that
907:Definition
859:acting on
744:martingale
702:(i.e. the
161:(that is,
48:asymptotic
4209:Lie group
4157:ν
4118:λ
4091:λ
4087:≤
4084:λ
4081:≤
4058:λ
4052:Δ
4032:Δ
3967:⋅
3941:↦
3774:≤
3768:≤
3675:≥
3668:∑
3565:∈
3539:≤
3533:≤
3450:ν
3420:γ
3402:ν
3391:γ
3365:∫
3327:Γ
3295:ν
3267:γ
3261:⋅
3231:Γ
3228:∈
3225:γ
3196:⋅
3176:↦
3049:∈
2964:≥
2957:∑
2897:−
2881:∗
2877:μ
2754:μ
2712:∗
2709:μ
2706:↦
2651:μ
2628:μ
2599:Γ
2579:μ
2550:γ
2538:ν
2527:γ
2518:^
2507:Γ
2503:∫
2457:^
2385:δ
2378:θ
2352:ν
2328:∈
2256:Γ
2233:^
2147:μ
2124:∈
2117:∑
2076:μ
2020:ν
2015:∗
1977:ν
1954:μ
1888:θ
1884:ν
1868:μ
1854:−
1843:ν
1834:∫
1807:μ
1776:θ
1749:Γ
1727:θ
1723:ν
1702:μ
1682:θ
1659:∼
1589:Γ
1566:μ
1530:≥
1450:∈
1326:∼
1279:∗
1254:∗
1250:μ
1246:∗
1094:μ
1066:−
1052:μ
944:μ
835:∂
810:∂
785:∂
690:Δ
623:∂
594:∂
591:∈
588:ξ
565:ξ
559:⋅
528:∈
479:−
476:ξ
446:−
431:ξ
396:ξ
390:μ
380:ξ
358:∂
354:∫
285:∈
268:∂
248:μ
198:Δ
169:Δ
119:∈
96:unit disc
4189:Examples
2343:and let
1519:for all
1291:denotes
4464:1815698
4447:1411218
4415:0146298
4398:1269841
4228:lattice
3500:minimal
3476:minimal
1271:(where
678:is the
509:is the
210:is the
94:on the
4462:
4445:
4413:
4396:
4109:where
4016:For a
2423:). If
411:where
190:where
34:, the
4251:Notes
3522:with
2068:be a
2036:to a
38:is a
4244:ends
4070:for
3789:the
2740:Let
2048:Let
916:Let
580:for
143:<
3319:on
2848:in
2828:to
1741:on
1674:If
1549:of
1400:to
1338:on
1003:on
648:).
218:on
30:In
4477::
4460:MR
4443:MR
4411:MR
4394:MR
3621:.
3282:.
2476::
2040:.
1671:.
903:.
66:.
4466:.
4449:.
4435:Z
4417:.
4400:.
4350:.
4338:.
4266:.
4167:1
4164:,
4161:o
4122:0
4095:0
4078:0
4055:+
3996:r
3976:)
3973:y
3970:,
3964:(
3959:r
3956:,
3953:o
3947:K
3938:y
3915:)
3912:y
3909:,
3906:o
3903:(
3898:r
3892:G
3884:)
3881:y
3878:,
3875:x
3872:(
3867:r
3861:G
3852:=
3849:)
3846:y
3843:,
3840:x
3837:(
3832:r
3829:,
3826:o
3820:K
3797:r
3777:R
3771:r
3765:1
3741:R
3718:.
3713:n
3709:r
3705:)
3702:y
3699:,
3696:x
3693:(
3688:n
3684:p
3678:1
3672:n
3664:=
3661:)
3658:y
3655:,
3652:x
3649:(
3644:r
3638:G
3609:u
3606:c
3603:=
3600:v
3580:]
3577:1
3574:,
3571:0
3568:[
3562:c
3542:u
3536:v
3530:0
3510:v
3486:u
3460:f
3457:,
3454:o
3426:.
3423:)
3417:(
3412:f
3409:,
3406:o
3398:d
3394:)
3388:,
3385:x
3382:(
3377:o
3371:K
3362:=
3359:)
3356:x
3353:(
3350:f
3305:f
3302:,
3299:o
3270:)
3264:,
3258:(
3253:K
3205:)
3202:y
3199:,
3193:(
3188:o
3182:K
3173:y
3150:)
3147:y
3144:,
3141:o
3138:(
3133:G
3126:)
3123:y
3120:,
3117:x
3114:(
3109:G
3101:=
3098:)
3095:y
3092:,
3089:x
3086:(
3081:o
3075:K
3052:G
3046:o
3026:y
3023:,
3020:x
2997:.
2994:)
2991:y
2988:,
2985:x
2982:(
2977:n
2973:p
2967:1
2961:n
2953:=
2950:)
2947:y
2944:,
2941:x
2938:(
2933:G
2908:)
2905:y
2900:1
2893:x
2889:(
2884:n
2856:n
2836:y
2816:x
2796:)
2793:y
2790:,
2787:x
2784:(
2779:n
2775:p
2751:,
2748:G
2715:f
2703:f
2671:G
2631:)
2625:,
2622:G
2619:(
2556:.
2553:)
2547:(
2542:x
2534:d
2530:)
2524:(
2515:f
2499:=
2496:)
2493:x
2490:(
2487:f
2454:f
2431:f
2411:x
2389:x
2381:=
2356:x
2331:G
2325:x
2302:N
2297:G
2276:f
2230:f
2207:)
2202:t
2198:X
2194:(
2191:f
2171:)
2168:g
2165:(
2162:f
2159:=
2156:)
2153:h
2150:(
2144:)
2141:g
2138:h
2135:(
2132:f
2127:G
2121:h
2096:G
2056:f
2011:)
2005:t
2001:X
1997:(
1957:)
1951:,
1948:G
1945:(
1925:G
1902:.
1899:)
1896:A
1893:(
1880:=
1877:)
1874:g
1871:(
1865:)
1862:A
1857:1
1850:g
1846:(
1838:G
1810:)
1804:,
1801:G
1798:(
1771:P
1639:)
1634:m
1629:P
1624:,
1618:N
1613:G
1609:(
1569:)
1563:,
1560:G
1557:(
1533:0
1527:t
1505:m
1502:+
1499:t
1495:y
1491:=
1486:n
1483:+
1480:t
1476:x
1454:N
1447:m
1444:,
1441:n
1421:)
1416:t
1412:y
1408:(
1388:)
1383:t
1379:x
1375:(
1352:N
1347:G
1303:n
1257:n
1243:m
1221:m
1216:P
1192:t
1188:X
1164:N
1159:G
1138:G
1118:m
1074:)
1069:1
1062:y
1058:x
1055:(
1049:=
1046:)
1043:y
1040:,
1037:x
1034:(
1031:p
1011:G
989:t
985:X
964:G
924:G
890:D
868:D
839:D
814:D
789:D
763:t
759:W
730:)
725:t
721:W
717:(
714:f
664:t
660:W
627:D
598:D
568:)
562:,
556:(
553:K
532:D
525:z
513:,
492:2
487:|
482:z
472:|
464:2
459:|
454:z
450:|
443:1
437:=
434:)
428:,
425:z
422:(
419:K
399:)
393:(
387:d
383:)
377:,
374:z
371:(
368:K
362:D
350:=
347:)
344:z
341:(
338:f
315:}
312:1
309:=
305:|
301:z
297:|
293::
289:C
282:z
279:{
276:=
272:D
227:D
178:0
175:=
172:f
149:}
146:1
139:|
135:z
131:|
127::
123:C
116:z
113:{
110:=
106:D
82:f
20:)
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