4204:. 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
4235:
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.
4231:, 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
4187:
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.
3917:
870:. 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
3152:
3425:
2555:
398:
3717:
3274:
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
314:
1901:
2996:
496:
148:
3975:
39:
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
3204:
1638:
4096:
2170:
3269:
598:
1777:
839:
814:
789:
627:
1073:
2390:
1454:
3230:
2907:
532:
4057:
2304:
1354:
1222:
1166:
2714:
2019:
1506:
1728:
567:
3776:
3541:
4123:
890:
868:
227:
3801:
1258:
4168:
3461:
3306:
2795:
2459:
2235:
177:
2630:
2357:
2206:
1956:
1809:
1568:
729:
4031:
3326:
2598:
2255:
1748:
1681:
1588:
1532:
1420:
1387:
689:
197:
3051:
2753:
2330:
3579:
1658:
1325:
1193:
990:
764:
665:
3608:
2650:
2578:
2075:
1976:
1701:
1093:
943:
247:
3025:
791:
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
3995:
3796:
3740:
3509:
3485:
3056:
2855:
2835:
2815:
2670:
2430:
2410:
2275:
2095:
2055:
1924:
1302:
1278:
1137:
1117:
1010:
963:
923:
81:
3334:
2471:
4420:
Kaimanovich, Vadim A. (1996). "Boundaries of invariant Markov operators: the identification problem". In
Pollicott, Mark; Schmidt, Klaus (eds.).
3206:
has a relatively compact image for the topology of pointwise convergence, and the Martin compactification is the closure of this image. A point
4125:
is the bottom of the spectrum. Examples where this construction can be used to define a compactification are bounded domains in the plane and
322:
3619:
252:
1817:
2915:
4479:
4208:) the full Martin boundary is the same as the minimal Martin boundary (the situation in higher-rank groups is more complicated).
403:
89:
3922:
4377:
Ballmann, Werner; Ledrappier, François (1994). "The
Poisson boundary for rank one manifolds and their cocompact lattices".
3157:
1593:
766:. Thus the Poisson formula identifies this measured space with the Martin boundary constructed above, and ultimately to
4200:(with step distribution absolutely continuous with respect to the Haar measure) the Poisson boundary is equal to the
4062:
2100:
3235:
572:
43:. However, in the case where the random walk is on a topological space the Poisson boundary can be related to the
841:
as the space of trajectories for a Markov process is a special case of the construction of the
Poisson boundary.
732:
40:
4474:
200:
2716:
associated to a random walk. Much of the theory can be developed in this abstract and very general setting.
1753:
844:
Finally, the constructions above can be discretised, i.e. restricted to the random walks on the orbits of a
47:, which is an analytic construction yielding a genuine topological boundary. Both boundaries are related to
819:
794:
769:
607:
1015:
4430:. London Math. Soc. Lecture Note Ser. Vol. 228. Cambridge Univ. Press, Cambridge. pp. 127–176.
4216:
3467:
Martin boundary, whose elements can also be characterised by being minimal. A positive harmonic function
2362:
1425:
3209:
2860:
4469:
509:
4036:
2280:
1330:
1198:
1142:
4174:
on the Martin boundary. With this measure the Martin boundary is isomorphic to the
Poisson boundary.
4033:. In this case there is again a whole family of Martin compactifications associated to the operators
2687:
1981:
1459:
3912:{\displaystyle {\mathcal {K}}_{o,r}(x,y)={\frac {{\mathcal {G}}_{r}(x,y)}{{\mathcal {G}}_{r}(o,y)}}}
3613:
There is actually a whole family of Martin compactifications. Define the Green generating series as
1706:
537:
3749:
3514:
4101:
873:
851:
210:
2022:
1227:
4140:
3433:
3278:
2758:
2435:
2211:
153:
2603:
2335:
2175:
1929:
1782:
1541:
698:
4016:
3311:
2583:
2240:
1733:
1666:
1573:
1511:
1392:
1359:
674:
182:
28:
3030:
2732:
2309:
4452:
4443:
Kaimanovich, Vadim A. (2000). "The
Poisson formula for groups with hyperbolic properties".
4435:
4403:
4386:
4201:
3546:
1643:
1310:
1305:
1171:
968:
742:
643:
3584:
3147:{\displaystyle {\mathcal {K}}_{o}(x,y)={\frac {{\mathcal {G}}(x,y)}{{\mathcal {G}}(o,y)}}}
2635:
2563:
2060:
1961:
1686:
1078:
928:
232:
8:
4010:
4006:
3004:
604:
in the cone of nonnegative harmonic functions. This analytical interpretation of the set
36:
4009:
the Martin boundary is constructed, when it exists, in the same way as above, using the
4232:
3980:
3781:
3725:
3494:
3470:
2840:
2820:
2800:
2655:
2415:
2395:
2260:
2080:
2040:
1909:
1287:
1263:
1122:
1102:
995:
948:
908:
66:
204:
1906:
It is possible to give an implicit definition of the
Poisson boundary as the maximal
48:
4228:
4212:
4205:
4448:
4431:
4399:
4382:
4126:
692:
845:
668:
52:
3420:{\displaystyle f(x)=\int {\mathcal {K}}_{o}(x,\gamma )\,d\nu _{o,f}(\gamma ).}
4463:
601:
2580:-harmonic bounded functions and essentially bounded measurable functions on
2550:{\displaystyle f(x)=\int _{\Gamma }{\hat {f}}(\gamma )\,d\nu _{x}(\gamma ).}
900:
4394:
Furstenberg, Harry (1963). "A Poisson formula for semi-simple Lie groups".
3743:
2208:
is a discrete-time martingale and so it converges almost surely. Denote by
736:
4282:
4132:
1281:
32:
20:
2632:
is trivial, that is reduced to a point, if and only if the only bounded
393:{\displaystyle f(z)=\int _{\partial \mathbb {D} }K(z,\xi )\,d\mu (\xi )}
3712:{\displaystyle {\mathcal {G}}_{r}(x,y)=\sum _{n\geq 1}p_{n}(x,y)r^{n}.}
2026:
4197:
84:
2684:
on a measured space, a notion which generalises the Markov operator
1683:
is the initial distribution of a random walk with step distribution
63:
The
Poisson formula states that given a positive harmonic function
1012:(a discrete-time Markov process whose transition probabilities are
695:
on the disc with the
Poincaré Riemannian metric), then the process
4354:
4342:
4306:
4294:
4270:
4258:
1139:, which will be the initial state for the random walk. The space
309:{\displaystyle \partial \mathbb {D} =\{z\in \mathbb {C} :|z|=1\}}
4330:
3001:
If the walk is transient then this series is convergent for all
1896:{\displaystyle \int _{G}\nu (g^{-1}A)\mu (g)=\nu _{\theta }(A).}
640:
This fact can also be interpreted in a probabilistic manner. If
1284:
of measures; this is the distribution of the random walk after
735:, and as such converges almost everywhere to a function on the
2991:{\displaystyle {\mathcal {G}}(x,y)=\sum _{n\geq 1}p_{n}(x,y).}
4000:
491:{\displaystyle K(z,\xi )={\frac {1-|z|^{2}}{|\xi -z|^{2}}}}
143:{\displaystyle \mathbb {D} =\{z\in \mathbb {C} :|z|<1\}}
4219:, is also equal to the Furstenberg boundary of the group.
2277:
along a trajectory (this is defined almost everywhere on
901:
The Poisson boundary of a random walk on a discrete group
3970:{\displaystyle y\mapsto {\mathcal {K}}_{o,r}(\cdot ,y)}
2359:
be the measure obtained by the construction above with
4318:
4246:
4133:
The relationship between Martin and Poisson boundaries
2724:
16:
Mathematical measure space associated to a random walk
4411:
Guivarc'h, Yves; Ji, Lizhen; Taylor, John C. (1998).
4410:
4336:
4288:
4170:
corresponding to the constant function is called the
4143:
4104:
4065:
4039:
4019:
3983:
3925:
3804:
3784:
3752:
3728:
3622:
3587:
3549:
3517:
3497:
3473:
3436:
3337:
3314:
3281:
3238:
3212:
3199:{\displaystyle y\mapsto {\mathcal {K}}_{o}(\cdot ,y)}
3160:
3059:
3033:
3007:
2918:
2863:
2843:
2823:
2803:
2761:
2735:
2690:
2658:
2638:
2606:
2586:
2566:
2474:
2438:
2418:
2398:
2365:
2338:
2312:
2283:
2263:
2243:
2214:
2178:
2103:
2083:
2063:
2043:
1984:
1964:
1932:
1912:
1820:
1785:
1756:
1736:
1709:
1689:
1669:
1646:
1596:
1576:
1544:
1514:
1462:
1428:
1395:
1362:
1333:
1313:
1290:
1266:
1230:
1201:
1174:
1145:
1125:
1105:
1081:
1018:
998:
971:
951:
931:
911:
876:
854:
822:
797:
772:
745:
701:
677:
646:
610:
575:
540:
512:
406:
325:
255:
235:
213:
185:
156:
92:
69:
4191:
1633:{\displaystyle (G^{\mathbb {N} },\mathbb {P} _{m})}
58:
4376:
4215:subgroup of a semisimple Lie group, for example a
4162:
4117:
4090:
4051:
4025:
3989:
3969:
3911:
3790:
3770:
3734:
3711:
3602:
3573:
3535:
3503:
3479:
3455:
3419:
3320:
3300:
3263:
3224:
3198:
3146:
3045:
3019:
2990:
2901:
2849:
2829:
2809:
2789:
2747:
2708:
2664:
2644:
2624:
2592:
2572:
2549:
2453:
2424:
2404:
2384:
2351:
2324:
2298:
2269:
2249:
2229:
2200:
2164:
2089:
2069:
2049:
2013:
1970:
1950:
1918:
1895:
1803:
1771:
1742:
1722:
1695:
1675:
1652:
1632:
1582:
1562:
1526:
1500:
1448:
1414:
1381:
1348:
1319:
1296:
1272:
1252:
1216:
1187:
1160:
1131:
1111:
1087:
1067:
1004:
984:
957:
937:
917:
884:
862:
833:
808:
783:
758:
723:
683:
659:
621:
592:
561:
534:. One way to interpret this is that the functions
526:
490:
392:
308:
241:
221:
191:
171:
142:
75:
1534:(the two trajectories have the same "tail"). The
4461:
4091:{\displaystyle 0\leq \lambda \leq \lambda _{0}}
2165:{\displaystyle \sum _{h\in G}f(hg)\mu (h)=f(g)}
3264:{\displaystyle {\mathcal {K}}(\cdot ,\gamma )}
2257:obtained by taking the limit of the values of
4381:. Vol. 6, no. 3. pp. 301–313.
965:, which will be used to define a random walk
593:{\displaystyle \xi \in \partial \mathbb {D} }
303:
267:
137:
101:
4442:
4419:
4393:
4360:
4348:
4324:
4312:
4300:
4276:
4264:
4252:
1978:, satisfying the additional condition that
2755:be a random walk on a discrete group. Let
3385:
2521:
2290:
1759:
1617:
1606:
1442:
1340:
1204:
1152:
878:
856:
827:
802:
777:
615:
586:
520:
374:
350:
277:
260:
215:
111:
94:
35:. It is an object designed to encode the
4001:Martin boundary of a Riemannian manifold
3328:such that a Poisson-like formula holds:
2600:. In particular the Poisson boundary of
739:of possible (infinite) trajectories for
51:on the space via generalisations of the
3232:is usually represented by the notation
4462:
4447:. 2. Vol. 152. pp. 659–692.
2719:
2032:
1772:{\displaystyle \mathbb {P} _{\theta }}
4413:Compactifications of symmetric spaces
4398:. 2. Vol. 77. pp. 335–386.
2909:. The Green kernel is by definition:
2675:
2560:This establishes a bijection between
834:{\displaystyle \partial \mathbb {D} }
809:{\displaystyle \partial \mathbb {D} }
784:{\displaystyle \partial \mathbb {D} }
622:{\displaystyle \partial \mathbb {D} }
4222:
1068:{\displaystyle p(x,y)=\mu (xy^{-1})}
629:leads to the more general notion of
4182:
2725:Martin boundary of a discrete group
2432:is either positive or bounded then
2385:{\displaystyle \theta =\delta _{x}}
1449:{\displaystyle n,m\in \mathbb {N} }
13:
4040:
4020:
3935:
3880:
3849:
3808:
3742:the radius of convergence of this
3626:
3359:
3315:
3241:
3225:{\displaystyle \gamma \in \Gamma }
3219:
3170:
3121:
3097:
3063:
3053:and define the Martin kernel by:
2921:
2902:{\displaystyle \mu ^{*n}(x^{-1}y)}
2587:
2495:
2244:
1737:
1577:
823:
798:
773:
678:
611:
582:
346:
256:
186:
157:
14:
4491:
4196:For random walks on a semisimple
4192:Lie groups and discrete subgroups
4013:of the Laplace–Beltrami operator
2680:The general setting is that of a
1779:. It is a stationary measure for
527:{\displaystyle z\in \mathbb {D} }
41:boundary in the topological sense
4052:{\displaystyle \Delta +\lambda }
2299:{\displaystyle G^{\mathbb {N} }}
1349:{\displaystyle G^{\mathbb {N} }}
1217:{\displaystyle \mathbb {P} _{m}}
1161:{\displaystyle G^{\mathbb {N} }}
633:(which in this case is the full
229:) there exists a unique measure
59:The case of the hyperbolic plane
4337:Guivarc'h, Ji & Taylor 1998
4289:Guivarc'h, Ji & Taylor 1998
3919:. The closure of the embedding
2797:be the probability to get from
2709:{\displaystyle f\mapsto \mu *f}
2014:{\displaystyle (X_{t})_{*}\nu }
1750:obtained as the pushforward of
1501:{\displaystyle x_{t+n}=y_{t+m}}
4480:Compactification (mathematics)
3964:
3952:
3929:
3903:
3891:
3872:
3860:
3837:
3825:
3693:
3681:
3649:
3637:
3568:
3556:
3411:
3405:
3382:
3370:
3347:
3341:
3258:
3246:
3193:
3181:
3164:
3138:
3126:
3114:
3102:
3086:
3074:
2982:
2970:
2938:
2926:
2896:
2877:
2784:
2772:
2694:
2619:
2607:
2541:
2535:
2518:
2512:
2506:
2484:
2478:
2445:
2221:
2195:
2182:
2159:
2153:
2144:
2138:
2132:
2123:
1999:
1985:
1945:
1933:
1887:
1881:
1865:
1859:
1853:
1834:
1798:
1786:
1723:{\displaystyle \nu _{\theta }}
1627:
1597:
1557:
1545:
1409:
1396:
1376:
1363:
1062:
1043:
1034:
1022:
718:
705:
562:{\displaystyle K(\cdot ,\xi )}
556:
544:
475:
460:
447:
438:
422:
410:
387:
381:
371:
359:
335:
329:
293:
285:
127:
119:
1:
4370:
3771:{\displaystyle 1\leq r\leq R}
3536:{\displaystyle 0\leq v\leq u}
3491:if for any harmonic function
895:
4428:actions (Warwick, 1993–1994)
4118:{\displaystyle \lambda _{0}}
1640:by the equivalence relation
1590:obtained as the quotient of
885:{\displaystyle \mathbb {D} }
863:{\displaystyle \mathbb {D} }
222:{\displaystyle \mathbb {D} }
7:
4177:
2461:is as well and we have the
2172:. Then the random variable
1570:is then the measured space
1253:{\displaystyle m*\mu ^{*n}}
10:
4496:
4211:The Poisson boundary of a
4163:{\displaystyle \nu _{o,1}}
3997:-Martin compactification.
3456:{\displaystyle \nu _{o,f}}
3301:{\displaystyle \nu _{o,f}}
2790:{\displaystyle p_{n}(x,y)}
2454:{\displaystyle {\hat {f}}}
2306:and shift-invariant). Let
2230:{\displaystyle {\hat {f}}}
1195:is endowed with a measure
816:). This interpretation of
600:are up to scaling all the
172:{\displaystyle \Delta f=0}
1304:steps). There is also an
1099:for the random walk. Let
945:a probability measure on
201:Laplace–Beltrami operator
4239:
2625:{\displaystyle (G,\mu )}
2352:{\displaystyle \nu _{x}}
2201:{\displaystyle f(X_{t})}
1951:{\displaystyle (G,\mu )}
1804:{\displaystyle (G,\mu )}
1563:{\displaystyle (G,\mu )}
925:be a discrete group and
724:{\displaystyle f(W_{t})}
316:such that the equality
4026:{\displaystyle \Delta }
3321:{\displaystyle \Gamma }
2652:-harmonic functions on
2593:{\displaystyle \Gamma }
2250:{\displaystyle \Gamma }
1743:{\displaystyle \Gamma }
1676:{\displaystyle \theta }
1583:{\displaystyle \Gamma }
1527:{\displaystyle t\geq 0}
1415:{\displaystyle (y_{t})}
1382:{\displaystyle (x_{t})}
684:{\displaystyle \Delta }
631:minimal Martin boundary
192:{\displaystyle \Delta }
4227:For random walks on a
4164:
4119:
4092:
4053:
4027:
3991:
3971:
3913:
3792:
3772:
3736:
3713:
3604:
3575:
3537:
3505:
3481:
3457:
3421:
3322:
3302:
3265:
3226:
3200:
3148:
3047:
3046:{\displaystyle o\in G}
3021:
2992:
2903:
2851:
2831:
2811:
2791:
2749:
2748:{\displaystyle G,\mu }
2710:
2666:
2646:
2626:
2594:
2574:
2551:
2455:
2426:
2406:
2386:
2353:
2326:
2325:{\displaystyle x\in G}
2300:
2271:
2251:
2231:
2202:
2166:
2091:
2077:-harmonic function on
2071:
2051:
2015:
1972:
1952:
1920:
1897:
1805:
1773:
1744:
1724:
1697:
1677:
1654:
1634:
1584:
1564:
1528:
1502:
1450:
1416:
1383:
1350:
1321:
1298:
1274:
1254:
1218:
1189:
1162:
1133:
1119:be another measure on
1113:
1089:
1069:
1006:
986:
959:
939:
919:
886:
864:
835:
810:
785:
760:
725:
685:
661:
623:
594:
563:
528:
492:
394:
310:
243:
223:
193:
173:
144:
77:
4165:
4129:of non-compact type.
4120:
4093:
4054:
4028:
3992:
3972:
3914:
3793:
3773:
3737:
3714:
3605:
3576:
3574:{\displaystyle c\in }
3538:
3506:
3482:
3463:are supported on the
3458:
3422:
3323:
3303:
3266:
3227:
3201:
3149:
3048:
3022:
2993:
2904:
2852:
2832:
2812:
2792:
2750:
2711:
2667:
2647:
2627:
2595:
2575:
2552:
2456:
2427:
2407:
2387:
2354:
2327:
2301:
2272:
2252:
2232:
2203:
2167:
2092:
2072:
2052:
2016:
1973:
1953:
1921:
1898:
1806:
1774:
1745:
1725:
1698:
1678:
1655:
1653:{\displaystyle \sim }
1635:
1585:
1565:
1529:
1503:
1451:
1417:
1384:
1351:
1322:
1320:{\displaystyle \sim }
1299:
1275:
1255:
1224:whose marginales are
1219:
1190:
1188:{\displaystyle X_{t}}
1163:
1134:
1114:
1090:
1070:
1007:
987:
985:{\displaystyle X_{t}}
960:
940:
920:
887:
865:
836:
811:
786:
761:
759:{\displaystyle W_{t}}
731:is a continuous-time
726:
686:
662:
660:{\displaystyle W_{t}}
624:
595:
564:
529:
493:
395:
311:
244:
224:
194:
174:
145:
78:
4475:Stochastic processes
4202:Furstenberg boundary
4141:
4102:
4063:
4037:
4017:
3981:
3923:
3802:
3782:
3750:
3726:
3620:
3603:{\displaystyle v=cu}
3585:
3547:
3515:
3495:
3471:
3434:
3335:
3312:
3279:
3236:
3210:
3158:
3057:
3031:
3005:
2916:
2861:
2841:
2821:
2801:
2759:
2733:
2688:
2656:
2645:{\displaystyle \mu }
2636:
2604:
2584:
2573:{\displaystyle \mu }
2564:
2472:
2436:
2416:
2396:
2363:
2336:
2310:
2281:
2261:
2241:
2212:
2176:
2101:
2081:
2070:{\displaystyle \mu }
2061:
2041:
1982:
1971:{\displaystyle \nu }
1962:
1958:-stationary measure
1930:
1910:
1818:
1783:
1754:
1734:
1707:
1696:{\displaystyle \mu }
1687:
1667:
1644:
1594:
1574:
1542:
1512:
1460:
1426:
1393:
1360:
1331:
1311:
1306:equivalence relation
1288:
1264:
1228:
1199:
1172:
1168:of trajectories for
1143:
1123:
1103:
1088:{\displaystyle \mu }
1079:
1016:
996:
969:
949:
938:{\displaystyle \mu }
929:
909:
874:
852:
820:
795:
770:
743:
699:
675:
644:
608:
573:
538:
510:
404:
323:
253:
242:{\displaystyle \mu }
233:
211:
183:
154:
90:
67:
4007:Riemannian manifold
3020:{\displaystyle x,y}
2720:The Martin boundary
2392:(the Dirac mass at
2033:The Poisson formula
1356:, which identifies
4422:Ergodic theory of
4160:
4115:
4088:
4049:
4023:
3987:
3967:
3909:
3798:-Martin kernel by
3788:
3768:
3732:
3709:
3670:
3600:
3571:
3533:
3501:
3477:
3453:
3417:
3318:
3298:
3261:
3222:
3196:
3144:
3043:
3017:
2988:
2959:
2899:
2847:
2827:
2807:
2787:
2745:
2706:
2676:General definition
2662:
2642:
2622:
2590:
2570:
2547:
2451:
2422:
2402:
2382:
2349:
2322:
2296:
2267:
2247:
2227:
2198:
2162:
2119:
2087:
2067:
2047:
2011:
1968:
1948:
1916:
1893:
1801:
1769:
1740:
1720:
1693:
1673:
1650:
1630:
1580:
1560:
1524:
1498:
1446:
1412:
1379:
1346:
1317:
1294:
1270:
1250:
1214:
1185:
1158:
1129:
1109:
1085:
1065:
1002:
982:
955:
935:
915:
882:
860:
831:
806:
781:
756:
721:
681:
657:
619:
590:
559:
524:
488:
390:
306:
239:
219:
203:associated to the
189:
169:
140:
73:
49:harmonic functions
4470:Harmonic analysis
4223:Hyperbolic groups
4206:hyperbolic spaces
3990:{\displaystyle r}
3907:
3791:{\displaystyle r}
3735:{\displaystyle R}
3655:
3504:{\displaystyle v}
3480:{\displaystyle u}
3154:. The embedding
3142:
2944:
2850:{\displaystyle n}
2830:{\displaystyle y}
2810:{\displaystyle x}
2665:{\displaystyle G}
2509:
2448:
2425:{\displaystyle f}
2405:{\displaystyle x}
2270:{\displaystyle f}
2224:
2104:
2090:{\displaystyle G}
2050:{\displaystyle f}
1919:{\displaystyle G}
1703:then the measure
1297:{\displaystyle n}
1273:{\displaystyle *}
1132:{\displaystyle G}
1112:{\displaystyle m}
1097:step distribution
1005:{\displaystyle G}
958:{\displaystyle G}
918:{\displaystyle G}
486:
76:{\displaystyle f}
4487:
4456:
4439:
4416:
4407:
4390:
4364:
4361:Kaimanovich 2000
4358:
4352:
4349:Kaimanovich 2000
4346:
4340:
4334:
4328:
4325:Furstenberg 1963
4322:
4316:
4313:Kaimanovich 1996
4310:
4304:
4301:Kaimanovich 1996
4298:
4292:
4286:
4280:
4277:Kaimanovich 1996
4274:
4268:
4265:Kaimanovich 1996
4262:
4256:
4253:Kaimanovich 1996
4250:
4229:hyperbolic group
4183:Nilpotent groups
4172:harmonic measure
4169:
4167:
4166:
4161:
4159:
4158:
4127:symmetric spaces
4124:
4122:
4121:
4116:
4114:
4113:
4097:
4095:
4094:
4089:
4087:
4086:
4058:
4056:
4055:
4050:
4032:
4030:
4029:
4024:
3996:
3994:
3993:
3988:
3976:
3974:
3973:
3968:
3951:
3950:
3939:
3938:
3918:
3916:
3915:
3910:
3908:
3906:
3890:
3889:
3884:
3883:
3875:
3859:
3858:
3853:
3852:
3844:
3824:
3823:
3812:
3811:
3797:
3795:
3794:
3789:
3777:
3775:
3774:
3769:
3741:
3739:
3738:
3733:
3718:
3716:
3715:
3710:
3705:
3704:
3680:
3679:
3669:
3636:
3635:
3630:
3629:
3609:
3607:
3606:
3601:
3580:
3578:
3577:
3572:
3542:
3540:
3539:
3534:
3510:
3508:
3507:
3502:
3486:
3484:
3483:
3478:
3462:
3460:
3459:
3454:
3452:
3451:
3426:
3424:
3423:
3418:
3404:
3403:
3369:
3368:
3363:
3362:
3327:
3325:
3324:
3319:
3307:
3305:
3304:
3299:
3297:
3296:
3270:
3268:
3267:
3262:
3245:
3244:
3231:
3229:
3228:
3223:
3205:
3203:
3202:
3197:
3180:
3179:
3174:
3173:
3153:
3151:
3150:
3145:
3143:
3141:
3125:
3124:
3117:
3101:
3100:
3093:
3073:
3072:
3067:
3066:
3052:
3050:
3049:
3044:
3026:
3024:
3023:
3018:
2997:
2995:
2994:
2989:
2969:
2968:
2958:
2925:
2924:
2908:
2906:
2905:
2900:
2892:
2891:
2876:
2875:
2856:
2854:
2853:
2848:
2836:
2834:
2833:
2828:
2816:
2814:
2813:
2808:
2796:
2794:
2793:
2788:
2771:
2770:
2754:
2752:
2751:
2746:
2715:
2713:
2712:
2707:
2671:
2669:
2668:
2663:
2651:
2649:
2648:
2643:
2631:
2629:
2628:
2623:
2599:
2597:
2596:
2591:
2579:
2577:
2576:
2571:
2556:
2554:
2553:
2548:
2534:
2533:
2511:
2510:
2502:
2499:
2498:
2460:
2458:
2457:
2452:
2450:
2449:
2441:
2431:
2429:
2428:
2423:
2411:
2409:
2408:
2403:
2391:
2389:
2388:
2383:
2381:
2380:
2358:
2356:
2355:
2350:
2348:
2347:
2331:
2329:
2328:
2323:
2305:
2303:
2302:
2297:
2295:
2294:
2293:
2276:
2274:
2273:
2268:
2256:
2254:
2253:
2248:
2237:the function on
2236:
2234:
2233:
2228:
2226:
2225:
2217:
2207:
2205:
2204:
2199:
2194:
2193:
2171:
2169:
2168:
2163:
2118:
2096:
2094:
2093:
2088:
2076:
2074:
2073:
2068:
2056:
2054:
2053:
2048:
2023:weakly converges
2020:
2018:
2017:
2012:
2007:
2006:
1997:
1996:
1977:
1975:
1974:
1969:
1957:
1955:
1954:
1949:
1925:
1923:
1922:
1917:
1902:
1900:
1899:
1894:
1880:
1879:
1849:
1848:
1830:
1829:
1811:, meaning that
1810:
1808:
1807:
1802:
1778:
1776:
1775:
1770:
1768:
1767:
1762:
1749:
1747:
1746:
1741:
1729:
1727:
1726:
1721:
1719:
1718:
1702:
1700:
1699:
1694:
1682:
1680:
1679:
1674:
1659:
1657:
1656:
1651:
1639:
1637:
1636:
1631:
1626:
1625:
1620:
1611:
1610:
1609:
1589:
1587:
1586:
1581:
1569:
1567:
1566:
1561:
1536:Poisson boundary
1533:
1531:
1530:
1525:
1507:
1505:
1504:
1499:
1497:
1496:
1478:
1477:
1455:
1453:
1452:
1447:
1445:
1422:if there exists
1421:
1419:
1418:
1413:
1408:
1407:
1388:
1386:
1385:
1380:
1375:
1374:
1355:
1353:
1352:
1347:
1345:
1344:
1343:
1326:
1324:
1323:
1318:
1303:
1301:
1300:
1295:
1279:
1277:
1276:
1271:
1259:
1257:
1256:
1251:
1249:
1248:
1223:
1221:
1220:
1215:
1213:
1212:
1207:
1194:
1192:
1191:
1186:
1184:
1183:
1167:
1165:
1164:
1159:
1157:
1156:
1155:
1138:
1136:
1135:
1130:
1118:
1116:
1115:
1110:
1094:
1092:
1091:
1086:
1074:
1072:
1071:
1066:
1061:
1060:
1011:
1009:
1008:
1003:
991:
989:
988:
983:
981:
980:
964:
962:
961:
956:
944:
942:
941:
936:
924:
922:
921:
916:
891:
889:
888:
883:
881:
869:
867:
866:
861:
859:
840:
838:
837:
832:
830:
815:
813:
812:
807:
805:
790:
788:
787:
782:
780:
765:
763:
762:
757:
755:
754:
730:
728:
727:
722:
717:
716:
690:
688:
687:
682:
666:
664:
663:
658:
656:
655:
628:
626:
625:
620:
618:
599:
597:
596:
591:
589:
568:
566:
565:
560:
533:
531:
530:
525:
523:
497:
495:
494:
489:
487:
485:
484:
483:
478:
463:
457:
456:
455:
450:
441:
429:
399:
397:
396:
391:
355:
354:
353:
315:
313:
312:
307:
296:
288:
280:
263:
249:on the boundary
248:
246:
245:
240:
228:
226:
225:
220:
218:
198:
196:
195:
190:
178:
176:
175:
170:
149:
147:
146:
141:
130:
122:
114:
97:
82:
80:
79:
74:
31:associated to a
25:Poisson boundary
4495:
4494:
4490:
4489:
4488:
4486:
4485:
4484:
4460:
4459:
4373:
4368:
4367:
4359:
4355:
4351:, Theorem 10.7.
4347:
4343:
4335:
4331:
4323:
4319:
4311:
4307:
4299:
4295:
4287:
4283:
4275:
4271:
4263:
4259:
4251:
4247:
4242:
4225:
4194:
4185:
4180:
4148:
4144:
4142:
4139:
4138:
4135:
4109:
4105:
4103:
4100:
4099:
4082:
4078:
4064:
4061:
4060:
4038:
4035:
4034:
4018:
4015:
4014:
4003:
3982:
3979:
3978:
3940:
3934:
3933:
3932:
3924:
3921:
3920:
3885:
3879:
3878:
3877:
3876:
3854:
3848:
3847:
3846:
3845:
3843:
3813:
3807:
3806:
3805:
3803:
3800:
3799:
3783:
3780:
3779:
3751:
3748:
3747:
3746:and define for
3727:
3724:
3723:
3700:
3696:
3675:
3671:
3659:
3631:
3625:
3624:
3623:
3621:
3618:
3617:
3586:
3583:
3582:
3548:
3545:
3544:
3516:
3513:
3512:
3496:
3493:
3492:
3472:
3469:
3468:
3441:
3437:
3435:
3432:
3431:
3393:
3389:
3364:
3358:
3357:
3356:
3336:
3333:
3332:
3313:
3310:
3309:
3286:
3282:
3280:
3277:
3276:
3240:
3239:
3237:
3234:
3233:
3211:
3208:
3207:
3175:
3169:
3168:
3167:
3159:
3156:
3155:
3120:
3119:
3118:
3096:
3095:
3094:
3092:
3068:
3062:
3061:
3060:
3058:
3055:
3054:
3032:
3029:
3028:
3006:
3003:
3002:
2964:
2960:
2948:
2920:
2919:
2917:
2914:
2913:
2884:
2880:
2868:
2864:
2862:
2859:
2858:
2842:
2839:
2838:
2822:
2819:
2818:
2802:
2799:
2798:
2766:
2762:
2760:
2757:
2756:
2734:
2731:
2730:
2727:
2722:
2689:
2686:
2685:
2682:Markov operator
2678:
2657:
2654:
2653:
2637:
2634:
2633:
2605:
2602:
2601:
2585:
2582:
2581:
2565:
2562:
2561:
2529:
2525:
2501:
2500:
2494:
2490:
2473:
2470:
2469:
2463:Poisson formula
2440:
2439:
2437:
2434:
2433:
2417:
2414:
2413:
2397:
2394:
2393:
2376:
2372:
2364:
2361:
2360:
2343:
2339:
2337:
2334:
2333:
2311:
2308:
2307:
2289:
2288:
2284:
2282:
2279:
2278:
2262:
2259:
2258:
2242:
2239:
2238:
2216:
2215:
2213:
2210:
2209:
2189:
2185:
2177:
2174:
2173:
2108:
2102:
2099:
2098:
2097:, meaning that
2082:
2079:
2078:
2062:
2059:
2058:
2042:
2039:
2038:
2035:
2002:
1998:
1992:
1988:
1983:
1980:
1979:
1963:
1960:
1959:
1931:
1928:
1927:
1911:
1908:
1907:
1875:
1871:
1841:
1837:
1825:
1821:
1819:
1816:
1815:
1784:
1781:
1780:
1763:
1758:
1757:
1755:
1752:
1751:
1735:
1732:
1731:
1714:
1710:
1708:
1705:
1704:
1688:
1685:
1684:
1668:
1665:
1664:
1645:
1642:
1641:
1621:
1616:
1615:
1605:
1604:
1600:
1595:
1592:
1591:
1575:
1572:
1571:
1543:
1540:
1539:
1513:
1510:
1509:
1486:
1482:
1467:
1463:
1461:
1458:
1457:
1441:
1427:
1424:
1423:
1403:
1399:
1394:
1391:
1390:
1370:
1366:
1361:
1358:
1357:
1339:
1338:
1334:
1332:
1329:
1328:
1312:
1309:
1308:
1289:
1286:
1285:
1265:
1262:
1261:
1241:
1237:
1229:
1226:
1225:
1208:
1203:
1202:
1200:
1197:
1196:
1179:
1175:
1173:
1170:
1169:
1151:
1150:
1146:
1144:
1141:
1140:
1124:
1121:
1120:
1104:
1101:
1100:
1080:
1077:
1076:
1075:); the measure
1053:
1049:
1017:
1014:
1013:
997:
994:
993:
976:
972:
970:
967:
966:
950:
947:
946:
930:
927:
926:
910:
907:
906:
903:
898:
877:
875:
872:
871:
855:
853:
850:
849:
826:
821:
818:
817:
801:
796:
793:
792:
776:
771:
768:
767:
750:
746:
744:
741:
740:
712:
708:
700:
697:
696:
693:Brownian motion
676:
673:
672:
651:
647:
645:
642:
641:
635:Martin boundary
614:
609:
606:
605:
585:
574:
571:
570:
539:
536:
535:
519:
511:
508:
507:
479:
474:
473:
459:
458:
451:
446:
445:
437:
430:
428:
405:
402:
401:
349:
345:
341:
324:
321:
320:
292:
284:
276:
259:
254:
251:
250:
234:
231:
230:
214:
212:
209:
208:
205:Poincaré metric
184:
181:
180:
155:
152:
151:
126:
118:
110:
93:
91:
88:
87:
68:
65:
64:
61:
53:Poisson formula
45:Martin boundary
17:
12:
11:
5:
4493:
4483:
4482:
4477:
4472:
4458:
4457:
4440:
4417:
4408:
4391:
4372:
4369:
4366:
4365:
4363:, Theorem 7.4.
4353:
4341:
4329:
4317:
4315:, Section 2.8.
4305:
4303:, Section 1.5.
4293:
4281:
4279:, Section 1.2.
4269:
4267:, Section 2.7.
4257:
4244:
4243:
4241:
4238:
4224:
4221:
4193:
4190:
4184:
4181:
4179:
4176:
4157:
4154:
4151:
4147:
4134:
4131:
4112:
4108:
4085:
4081:
4077:
4074:
4071:
4068:
4048:
4045:
4042:
4022:
4011:Green function
4002:
3999:
3986:
3977:is called the
3966:
3963:
3960:
3957:
3954:
3949:
3946:
3943:
3937:
3931:
3928:
3905:
3902:
3899:
3896:
3893:
3888:
3882:
3874:
3871:
3868:
3865:
3862:
3857:
3851:
3842:
3839:
3836:
3833:
3830:
3827:
3822:
3819:
3816:
3810:
3787:
3767:
3764:
3761:
3758:
3755:
3731:
3720:
3719:
3708:
3703:
3699:
3695:
3692:
3689:
3686:
3683:
3678:
3674:
3668:
3665:
3662:
3658:
3654:
3651:
3648:
3645:
3642:
3639:
3634:
3628:
3599:
3596:
3593:
3590:
3570:
3567:
3564:
3561:
3558:
3555:
3552:
3532:
3529:
3526:
3523:
3520:
3500:
3487:is said to be
3476:
3450:
3447:
3444:
3440:
3428:
3427:
3416:
3413:
3410:
3407:
3402:
3399:
3396:
3392:
3388:
3384:
3381:
3378:
3375:
3372:
3367:
3361:
3355:
3352:
3349:
3346:
3343:
3340:
3317:
3295:
3292:
3289:
3285:
3260:
3257:
3254:
3251:
3248:
3243:
3221:
3218:
3215:
3195:
3192:
3189:
3186:
3183:
3178:
3172:
3166:
3163:
3140:
3137:
3134:
3131:
3128:
3123:
3116:
3113:
3110:
3107:
3104:
3099:
3091:
3088:
3085:
3082:
3079:
3076:
3071:
3065:
3042:
3039:
3036:
3027:. Fix a point
3016:
3013:
3010:
2999:
2998:
2987:
2984:
2981:
2978:
2975:
2972:
2967:
2963:
2957:
2954:
2951:
2947:
2943:
2940:
2937:
2934:
2931:
2928:
2923:
2898:
2895:
2890:
2887:
2883:
2879:
2874:
2871:
2867:
2846:
2826:
2806:
2786:
2783:
2780:
2777:
2774:
2769:
2765:
2744:
2741:
2738:
2726:
2723:
2721:
2718:
2705:
2702:
2699:
2696:
2693:
2677:
2674:
2672:are constant.
2661:
2641:
2621:
2618:
2615:
2612:
2609:
2589:
2569:
2558:
2557:
2546:
2543:
2540:
2537:
2532:
2528:
2524:
2520:
2517:
2514:
2508:
2505:
2497:
2493:
2489:
2486:
2483:
2480:
2477:
2447:
2444:
2421:
2401:
2379:
2375:
2371:
2368:
2346:
2342:
2321:
2318:
2315:
2292:
2287:
2266:
2246:
2223:
2220:
2197:
2192:
2188:
2184:
2181:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2117:
2114:
2111:
2107:
2086:
2066:
2046:
2034:
2031:
2021:almost surely
2010:
2005:
2001:
1995:
1991:
1987:
1967:
1947:
1944:
1941:
1938:
1935:
1915:
1904:
1903:
1892:
1889:
1886:
1883:
1878:
1874:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1847:
1844:
1840:
1836:
1833:
1828:
1824:
1800:
1797:
1794:
1791:
1788:
1766:
1761:
1739:
1717:
1713:
1692:
1672:
1649:
1629:
1624:
1619:
1614:
1608:
1603:
1599:
1579:
1559:
1556:
1553:
1550:
1547:
1523:
1520:
1517:
1495:
1492:
1489:
1485:
1481:
1476:
1473:
1470:
1466:
1444:
1440:
1437:
1434:
1431:
1411:
1406:
1402:
1398:
1378:
1373:
1369:
1365:
1342:
1337:
1316:
1293:
1269:
1247:
1244:
1240:
1236:
1233:
1211:
1206:
1182:
1178:
1154:
1149:
1128:
1108:
1095:is called the
1084:
1064:
1059:
1056:
1052:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1001:
979:
975:
954:
934:
914:
902:
899:
897:
894:
880:
858:
846:Fuchsian group
829:
825:
804:
800:
779:
775:
753:
749:
720:
715:
711:
707:
704:
680:
671:associated to
669:Markov process
654:
650:
617:
613:
602:extreme points
588:
584:
581:
578:
558:
555:
552:
549:
546:
543:
522:
518:
515:
506:holds for all
504:
503:
500:Poisson kernel
482:
477:
472:
469:
466:
462:
454:
449:
444:
440:
436:
433:
427:
424:
421:
418:
415:
412:
409:
389:
386:
383:
380:
377:
373:
370:
367:
364:
361:
358:
352:
348:
344:
340:
337:
334:
331:
328:
305:
302:
299:
295:
291:
287:
283:
279:
275:
272:
269:
266:
262:
258:
238:
217:
188:
168:
165:
162:
159:
139:
136:
133:
129:
125:
121:
117:
113:
109:
106:
103:
100:
96:
72:
60:
57:
15:
9:
6:
4:
3:
2:
4492:
4481:
4478:
4476:
4473:
4471:
4468:
4467:
4465:
4454:
4450:
4446:
4441:
4437:
4433:
4429:
4426:
4425:
4418:
4415:. Birkhäuser.
4414:
4409:
4405:
4401:
4397:
4392:
4388:
4384:
4380:
4375:
4374:
4362:
4357:
4350:
4345:
4338:
4333:
4326:
4321:
4314:
4309:
4302:
4297:
4291:, Chapter VI.
4290:
4285:
4278:
4273:
4266:
4261:
4254:
4249:
4245:
4237:
4234:
4230:
4220:
4218:
4214:
4213:Zariski-dense
4209:
4207:
4203:
4199:
4189:
4175:
4173:
4155:
4152:
4149:
4145:
4130:
4128:
4110:
4106:
4083:
4079:
4075:
4072:
4069:
4066:
4046:
4043:
4012:
4008:
3998:
3984:
3961:
3958:
3955:
3947:
3944:
3941:
3926:
3900:
3897:
3894:
3886:
3869:
3866:
3863:
3855:
3840:
3834:
3831:
3828:
3820:
3817:
3814:
3785:
3765:
3762:
3759:
3756:
3753:
3745:
3729:
3706:
3701:
3697:
3690:
3687:
3684:
3676:
3672:
3666:
3663:
3660:
3656:
3652:
3646:
3643:
3640:
3632:
3616:
3615:
3614:
3611:
3597:
3594:
3591:
3588:
3565:
3562:
3559:
3553:
3550:
3543:there exists
3530:
3527:
3524:
3521:
3518:
3498:
3490:
3474:
3466:
3448:
3445:
3442:
3438:
3430:The measures
3414:
3408:
3400:
3397:
3394:
3390:
3386:
3379:
3376:
3373:
3365:
3353:
3350:
3344:
3338:
3331:
3330:
3329:
3293:
3290:
3287:
3283:
3272:
3255:
3252:
3249:
3216:
3213:
3190:
3187:
3184:
3176:
3161:
3135:
3132:
3129:
3111:
3108:
3105:
3089:
3083:
3080:
3077:
3069:
3040:
3037:
3034:
3014:
3011:
3008:
2985:
2979:
2976:
2973:
2965:
2961:
2955:
2952:
2949:
2945:
2941:
2935:
2932:
2929:
2912:
2911:
2910:
2893:
2888:
2885:
2881:
2872:
2869:
2865:
2844:
2824:
2804:
2781:
2778:
2775:
2767:
2763:
2742:
2739:
2736:
2717:
2703:
2700:
2697:
2691:
2683:
2673:
2659:
2639:
2616:
2613:
2610:
2567:
2544:
2538:
2530:
2526:
2522:
2515:
2503:
2491:
2487:
2481:
2475:
2468:
2467:
2466:
2464:
2442:
2419:
2399:
2377:
2373:
2369:
2366:
2344:
2340:
2319:
2316:
2313:
2285:
2264:
2218:
2190:
2186:
2179:
2156:
2150:
2147:
2141:
2135:
2129:
2126:
2120:
2115:
2112:
2109:
2105:
2084:
2064:
2044:
2030:
2028:
2024:
2008:
2003:
1993:
1989:
1965:
1942:
1939:
1936:
1913:
1890:
1884:
1876:
1872:
1868:
1862:
1856:
1850:
1845:
1842:
1838:
1831:
1826:
1822:
1814:
1813:
1812:
1795:
1792:
1789:
1764:
1715:
1711:
1690:
1670:
1661:
1647:
1622:
1612:
1601:
1554:
1551:
1548:
1537:
1521:
1518:
1515:
1493:
1490:
1487:
1483:
1479:
1474:
1471:
1468:
1464:
1438:
1435:
1432:
1429:
1404:
1400:
1371:
1367:
1335:
1314:
1307:
1291:
1283:
1267:
1245:
1242:
1238:
1234:
1231:
1209:
1180:
1176:
1147:
1126:
1106:
1098:
1082:
1057:
1054:
1050:
1046:
1040:
1037:
1031:
1028:
1025:
1019:
999:
977:
973:
952:
932:
912:
893:
847:
842:
751:
747:
738:
734:
713:
709:
702:
694:
670:
652:
648:
638:
636:
632:
603:
579:
576:
553:
550:
547:
541:
516:
513:
501:
480:
470:
467:
464:
452:
442:
434:
431:
425:
419:
416:
413:
407:
384:
378:
375:
368:
365:
362:
356:
342:
338:
332:
326:
319:
318:
317:
300:
297:
289:
281:
273:
270:
264:
236:
206:
202:
166:
163:
160:
134:
131:
123:
115:
107:
104:
98:
86:
70:
56:
54:
50:
46:
42:
38:
34:
30:
29:measure space
26:
22:
4445:Ann. of Math
4444:
4427:
4423:
4421:
4412:
4396:Ann. of Math
4395:
4378:
4356:
4344:
4332:
4320:
4308:
4296:
4284:
4272:
4260:
4248:
4226:
4210:
4195:
4186:
4171:
4137:The measure
4136:
4004:
3744:power series
3721:
3612:
3488:
3464:
3429:
3273:
3000:
2857:steps, i.e.
2728:
2681:
2679:
2559:
2462:
2036:
1926:-set with a
1905:
1662:
1535:
1096:
904:
843:
737:Wiener space
639:
634:
630:
505:
499:
62:
44:
24:
18:
1282:convolution
33:random walk
21:mathematics
4464:Categories
4379:Forum Math
4371:References
3722:Denote by
3581:such that
2027:Dirac mass
1456:such that
896:Definition
848:acting on
733:martingale
691:(i.e. the
150:(that is,
37:asymptotic
4198:Lie group
4146:ν
4107:λ
4080:λ
4076:≤
4073:λ
4070:≤
4047:λ
4041:Δ
4021:Δ
3956:⋅
3930:↦
3763:≤
3757:≤
3664:≥
3657:∑
3554:∈
3528:≤
3522:≤
3439:ν
3409:γ
3391:ν
3380:γ
3354:∫
3316:Γ
3284:ν
3256:γ
3250:⋅
3220:Γ
3217:∈
3214:γ
3185:⋅
3165:↦
3038:∈
2953:≥
2946:∑
2886:−
2870:∗
2866:μ
2743:μ
2701:∗
2698:μ
2695:↦
2640:μ
2617:μ
2588:Γ
2568:μ
2539:γ
2527:ν
2516:γ
2507:^
2496:Γ
2492:∫
2446:^
2374:δ
2367:θ
2341:ν
2317:∈
2245:Γ
2222:^
2136:μ
2113:∈
2106:∑
2065:μ
2009:ν
2004:∗
1966:ν
1943:μ
1877:θ
1873:ν
1857:μ
1843:−
1832:ν
1823:∫
1796:μ
1765:θ
1738:Γ
1716:θ
1712:ν
1691:μ
1671:θ
1648:∼
1578:Γ
1555:μ
1519:≥
1439:∈
1315:∼
1268:∗
1243:∗
1239:μ
1235:∗
1083:μ
1055:−
1041:μ
933:μ
824:∂
799:∂
774:∂
679:Δ
612:∂
583:∂
580:∈
577:ξ
554:ξ
548:⋅
517:∈
468:−
465:ξ
435:−
420:ξ
385:ξ
379:μ
369:ξ
347:∂
343:∫
274:∈
257:∂
237:μ
187:Δ
158:Δ
108:∈
85:unit disc
4178:Examples
2332:and let
1508:for all
1280:denotes
4453:1815698
4436:1411218
4404:0146298
4387:1269841
4217:lattice
3489:minimal
3465:minimal
1260:(where
667:is the
498:is the
199:is the
83:on the
4451:
4434:
4402:
4385:
4098:where
4005:For a
2412:). If
400:where
179:where
23:, the
4240:Notes
3511:with
2057:be a
2025:to a
27:is a
4233:ends
4059:for
3778:the
2729:Let
2037:Let
905:Let
569:for
132:<
3308:on
2837:in
2817:to
1730:on
1663:If
1538:of
1389:to
1327:on
992:on
637:).
207:on
19:In
4466::
4449:MR
4432:MR
4400:MR
4383:MR
3610:.
3271:.
2465::
2029:.
1660:.
892:.
55:.
4455:.
4438:.
4424:Z
4406:.
4389:.
4339:.
4327:.
4255:.
4156:1
4153:,
4150:o
4111:0
4084:0
4067:0
4044:+
3985:r
3965:)
3962:y
3959:,
3953:(
3948:r
3945:,
3942:o
3936:K
3927:y
3904:)
3901:y
3898:,
3895:o
3892:(
3887:r
3881:G
3873:)
3870:y
3867:,
3864:x
3861:(
3856:r
3850:G
3841:=
3838:)
3835:y
3832:,
3829:x
3826:(
3821:r
3818:,
3815:o
3809:K
3786:r
3766:R
3760:r
3754:1
3730:R
3707:.
3702:n
3698:r
3694:)
3691:y
3688:,
3685:x
3682:(
3677:n
3673:p
3667:1
3661:n
3653:=
3650:)
3647:y
3644:,
3641:x
3638:(
3633:r
3627:G
3598:u
3595:c
3592:=
3589:v
3569:]
3566:1
3563:,
3560:0
3557:[
3551:c
3531:u
3525:v
3519:0
3499:v
3475:u
3449:f
3446:,
3443:o
3415:.
3412:)
3406:(
3401:f
3398:,
3395:o
3387:d
3383:)
3377:,
3374:x
3371:(
3366:o
3360:K
3351:=
3348:)
3345:x
3342:(
3339:f
3294:f
3291:,
3288:o
3259:)
3253:,
3247:(
3242:K
3194:)
3191:y
3188:,
3182:(
3177:o
3171:K
3162:y
3139:)
3136:y
3133:,
3130:o
3127:(
3122:G
3115:)
3112:y
3109:,
3106:x
3103:(
3098:G
3090:=
3087:)
3084:y
3081:,
3078:x
3075:(
3070:o
3064:K
3041:G
3035:o
3015:y
3012:,
3009:x
2986:.
2983:)
2980:y
2977:,
2974:x
2971:(
2966:n
2962:p
2956:1
2950:n
2942:=
2939:)
2936:y
2933:,
2930:x
2927:(
2922:G
2897:)
2894:y
2889:1
2882:x
2878:(
2873:n
2845:n
2825:y
2805:x
2785:)
2782:y
2779:,
2776:x
2773:(
2768:n
2764:p
2740:,
2737:G
2704:f
2692:f
2660:G
2620:)
2614:,
2611:G
2608:(
2545:.
2542:)
2536:(
2531:x
2523:d
2519:)
2513:(
2504:f
2488:=
2485:)
2482:x
2479:(
2476:f
2443:f
2420:f
2400:x
2378:x
2370:=
2345:x
2320:G
2314:x
2291:N
2286:G
2265:f
2219:f
2196:)
2191:t
2187:X
2183:(
2180:f
2160:)
2157:g
2154:(
2151:f
2148:=
2145:)
2142:h
2139:(
2133:)
2130:g
2127:h
2124:(
2121:f
2116:G
2110:h
2085:G
2045:f
2000:)
1994:t
1990:X
1986:(
1946:)
1940:,
1937:G
1934:(
1914:G
1891:.
1888:)
1885:A
1882:(
1869:=
1866:)
1863:g
1860:(
1854:)
1851:A
1846:1
1839:g
1835:(
1827:G
1799:)
1793:,
1790:G
1787:(
1760:P
1628:)
1623:m
1618:P
1613:,
1607:N
1602:G
1598:(
1558:)
1552:,
1549:G
1546:(
1522:0
1516:t
1494:m
1491:+
1488:t
1484:y
1480:=
1475:n
1472:+
1469:t
1465:x
1443:N
1436:m
1433:,
1430:n
1410:)
1405:t
1401:y
1397:(
1377:)
1372:t
1368:x
1364:(
1341:N
1336:G
1292:n
1246:n
1232:m
1210:m
1205:P
1181:t
1177:X
1153:N
1148:G
1127:G
1107:m
1063:)
1058:1
1051:y
1047:x
1044:(
1038:=
1035:)
1032:y
1029:,
1026:x
1023:(
1020:p
1000:G
978:t
974:X
953:G
913:G
879:D
857:D
828:D
803:D
778:D
752:t
748:W
719:)
714:t
710:W
706:(
703:f
653:t
649:W
616:D
587:D
557:)
551:,
545:(
542:K
521:D
514:z
502:,
481:2
476:|
471:z
461:|
453:2
448:|
443:z
439:|
432:1
426:=
423:)
417:,
414:z
411:(
408:K
388:)
382:(
376:d
372:)
366:,
363:z
360:(
357:K
351:D
339:=
336:)
333:z
330:(
327:f
304:}
301:1
298:=
294:|
290:z
286:|
282::
278:C
271:z
268:{
265:=
261:D
216:D
167:0
164:=
161:f
138:}
135:1
128:|
124:z
120:|
116::
112:C
105:z
102:{
99:=
95:D
71:f
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