591:
1200:
769:
288:
1024:
1439:
1340:
1484:
The great achievement of Hörmander's 1967 paper was to show that a smooth fundamental solution exists under a considerably weaker assumption: the parabolic version of the condition that now bears his name.
1004:
195:
1630:
903:
interpretation of the SDE. Hörmander's theorem asserts that if the SDE above satisfies the parabolic Hörmander condition, then its solutions admit a smooth density with respect to
Lebesgue measure.
293:
655:
897:
870:
1541:
818:
586:{\displaystyle {\begin{aligned}&A_{j_{0}}(x)~,\\&~,\\&,A_{j_{2}}(x)]~,\\&\quad \vdots \quad \end{aligned}}\qquad 0\leq j_{0},j_{1},\ldots ,j_{n}\leq n}
1195:{\displaystyle {\begin{cases}{\dfrac {\partial u}{\partial t}}(t,x)=Fu(t,x),&t>0,x\in \mathbf {R} ^{d};\\u(t,\cdot )\to f,&{\text{as }}t\to 0;\end{cases}}}
631:
1636:
1355:
1243:
1762:
924:
92:
1757:
1009:
An important problem in the theory of partial differential equations is to determine sufficient conditions on the vector fields
1553:
1687:
764:{\displaystyle \operatorname {d} x=A_{0}(x)\operatorname {d} t+\sum _{i=1}^{n}A_{i}(x)\circ \operatorname {d} W_{i}}
646:
33:
879:
29:
823:
83:
1345:
satisfies the Cauchy problem above. It had been known for some time that a smooth solution exists in the
1644:
1500:
777:
1033:
1648:
912:
900:
1547:. Assuming that these vector fields satisfy Hörmander's condition, then the control system
209:
1742:
1697:
1206:
609:
225:
8:
1660:
1730:
1683:
1478:
1346:
1704:
43:
1720:
1739:
1694:
76:
55:
1434:{\displaystyle A_{i}=\sum _{j=1}^{d}a_{ji}{\frac {\partial }{\partial x_{j}}},}
1751:
1734:
40:
25:
1335:{\displaystyle u(t,x)=\int _{\mathbf {R} ^{d}}p(t,x,y)f(y)\,\mathrm {d} y}
1665:
596:
17:
1725:
1708:
73:
640:
28:
that, if satisfied, has many useful consequences in the theory of
999:{\displaystyle F={\frac {1}{2}}\sum _{i=1}^{n}A_{i}^{2}+A_{0}.}
37:
190:{\displaystyle (x)=\mathrm {D} V(x)W(x)-\mathrm {D} W(x)V(x),}
1188:
1682:. Mineola, NY: Dover Publications Inc. pp. x+113.
1625:{\displaystyle {\dot {x}}=\sum _{i=0}^{n}u_{i}A_{i}(x)}
911:
With the same notation as above, define a second-order
1556:
1503:
1358:
1246:
1037:
1027:
927:
882:
826:
780:
658:
612:
291:
95:
906:
1709:"Hypoelliptic second order differential equations"
1624:
1535:
1433:
1334:
1194:
998:
891:
864:
812:
763:
625:
585:
189:
1488:
1749:
641:Application to stochastic differential equations
606:if the same holds true, but with the index
1213: (0, +∞) ×
1724:
1703:
1323:
892:{\displaystyle \circ \operatorname {d} }
820:are assumed to have bounded derivative,
1750:
865:{\displaystyle (W_{1},\dotsc ,W_{n})}
1677:
36:. The condition is named after the
1536:{\displaystyle A_{0},\dotsc ,A_{n}}
813:{\displaystyle A_{0},\dotsc ,A_{n}}
13:
1412:
1408:
1325:
1048:
1040:
886:
745:
690:
659:
156:
124:
86:, another vector field defined by
14:
1774:
1763:Stochastic differential equations
907:Application to the Cauchy problem
876:-dimensional Brownian motion and
34:stochastic differential equations
1275:
1122:
647:stochastic differential equation
1229:, ·, ·) is smooth on
602:. They are said to satisfy the
528:
523:
519:
224:, which can be thought of as a
1758:Partial differential equations
1639:in any time at every point of
1619:
1613:
1489:Application to control systems
1320:
1314:
1308:
1290:
1262:
1250:
1209:, i.e. a real-valued function
1176:
1157:
1154:
1142:
1094:
1082:
1070:
1058:
859:
827:
739:
733:
687:
681:
503:
500:
494:
471:
468:
462:
439:
433:
413:
410:
394:
391:
385:
362:
356:
336:
320:
314:
228:that is applied to the vector
181:
175:
169:
163:
149:
143:
137:
131:
117:
111:
108:
96:
1:
1671:
604:parabolic Hörmander condition
49:
633:taking only values in 1,...,
270:. They are said to satisfy
7:
1654:
1543:be smooth vector fields on
10:
1779:
1497:be a smooth manifold and
774:where the vectors fields
1678:Bell, Denis R. (2006).
1645:Chow–Rashevskii theorem
1643:. This is known as the
1457:), 1 ≤
1018:for the Cauchy problem
1700:(See the introduction)
1680:The Malliavin calculus
1649:Orbit (control theory)
1626:
1592:
1537:
1465:, 1 ≤
1435:
1392:
1336:
1196:
1000:
964:
893:
866:
814:
765:
722:
627:
587:
191:
1627:
1572:
1538:
1436:
1372:
1337:
1197:
1001:
944:
913:differential operator
901:Stratonovich integral
894:
867:
815:
766:
702:
628:
626:{\displaystyle j_{0}}
588:
272:Hörmander's condition
192:
22:Hörmander's condition
1637:locally controllable
1554:
1501:
1356:
1244:
1207:fundamental solution
1025:
925:
880:
824:
778:
656:
610:
289:
274:if, for every point
266:be vector fields on
93:
82:, let denote their
1469: ≤
1461: ≤
979:
278: ∈
220: ∈
1726:10.1007/BF02392081
1661:Malliavin calculus
1622:
1533:
1431:
1332:
1192:
1187:
1056:
996:
965:
889:
862:
810:
761:
623:
583:
526:
210:Fréchet derivative
187:
1566:
1479:invertible matrix
1477:is everywhere an
1426:
1205:to have a smooth
1171:
1055:
942:
508:
399:
325:
24:is a property of
1770:
1738:
1728:
1693:
1631:
1629:
1628:
1623:
1612:
1611:
1602:
1601:
1591:
1586:
1568:
1567:
1559:
1542:
1540:
1539:
1534:
1532:
1531:
1513:
1512:
1440:
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1407:
1405:
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1391:
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1368:
1367:
1341:
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1338:
1333:
1328:
1286:
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1283:
1278:
1201:
1199:
1198:
1193:
1191:
1190:
1172:
1169:
1131:
1130:
1125:
1057:
1054:
1046:
1038:
1005:
1003:
1002:
997:
992:
991:
978:
973:
963:
958:
943:
935:
898:
896:
895:
890:
871:
869:
868:
863:
858:
857:
839:
838:
819:
817:
816:
811:
809:
808:
790:
789:
770:
768:
767:
762:
760:
759:
732:
731:
721:
716:
680:
679:
632:
630:
629:
624:
622:
621:
592:
590:
589:
584:
576:
575:
557:
556:
544:
543:
527:
515:
506:
493:
492:
491:
490:
461:
460:
459:
458:
432:
431:
430:
429:
406:
397:
384:
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382:
381:
355:
354:
353:
352:
332:
323:
313:
312:
311:
310:
295:
196:
194:
193:
188:
159:
127:
1778:
1777:
1773:
1772:
1771:
1769:
1768:
1767:
1748:
1747:
1705:Hörmander, Lars
1690:
1674:
1657:
1607:
1603:
1597:
1593:
1587:
1576:
1558:
1557:
1555:
1552:
1551:
1527:
1523:
1508:
1504:
1502:
1499:
1498:
1491:
1456:
1444:and the matrix
1419:
1415:
1411:
1406:
1397:
1393:
1387:
1376:
1363:
1359:
1357:
1354:
1353:
1349:case, in which
1324:
1279:
1274:
1273:
1272:
1268:
1245:
1242:
1241:
1186:
1185:
1168:
1166:
1136:
1135:
1126:
1121:
1120:
1100:
1047:
1039:
1036:
1029:
1028:
1026:
1023:
1022:
1017:
987:
983:
974:
969:
959:
948:
934:
926:
923:
922:
909:
899:stands for the
881:
878:
877:
872:the normalized
853:
849:
834:
830:
825:
822:
821:
804:
800:
785:
781:
779:
776:
775:
755:
751:
727:
723:
717:
706:
675:
671:
657:
654:
653:
643:
617:
613:
611:
608:
607:
571:
567:
552:
548:
539:
535:
525:
524:
513:
512:
486:
482:
481:
477:
454:
450:
449:
445:
425:
421:
420:
416:
404:
403:
377:
373:
372:
368:
348:
344:
343:
339:
330:
329:
306:
302:
301:
297:
292:
290:
287:
286:
265:
256:
249:
155:
123:
94:
91:
90:
77:Euclidean space
52:
12:
11:
5:
1776:
1766:
1765:
1760:
1746:
1745:
1701:
1688:
1673:
1670:
1669:
1668:
1663:
1656:
1653:
1633:
1632:
1621:
1618:
1615:
1610:
1606:
1600:
1596:
1590:
1585:
1582:
1579:
1575:
1571:
1565:
1562:
1530:
1526:
1522:
1519:
1516:
1511:
1507:
1490:
1487:
1452:
1448: = (
1442:
1441:
1430:
1422:
1418:
1414:
1410:
1403:
1400:
1396:
1390:
1385:
1382:
1379:
1375:
1371:
1366:
1362:
1343:
1342:
1331:
1327:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1282:
1277:
1271:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1203:
1202:
1189:
1184:
1181:
1178:
1175:
1167:
1165:
1162:
1159:
1156:
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1150:
1147:
1144:
1141:
1138:
1137:
1134:
1129:
1124:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1053:
1050:
1045:
1042:
1035:
1034:
1032:
1013:
1007:
1006:
995:
990:
986:
982:
977:
972:
968:
962:
957:
954:
951:
947:
941:
938:
933:
930:
908:
905:
888:
885:
861:
856:
852:
848:
845:
842:
837:
833:
829:
807:
803:
799:
796:
793:
788:
784:
772:
771:
758:
754:
750:
747:
744:
741:
738:
735:
730:
726:
720:
715:
712:
709:
705:
701:
698:
695:
692:
689:
686:
683:
678:
674:
670:
667:
664:
661:
642:
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620:
616:
594:
593:
582:
579:
574:
570:
566:
563:
560:
555:
551:
547:
542:
538:
534:
531:
522:
518:
516:
514:
511:
505:
502:
499:
496:
489:
485:
480:
476:
473:
470:
467:
464:
457:
453:
448:
444:
441:
438:
435:
428:
424:
419:
415:
412:
409:
407:
405:
402:
396:
393:
390:
387:
380:
376:
371:
367:
364:
361:
358:
351:
347:
342:
338:
335:
333:
331:
328:
322:
319:
316:
309:
305:
300:
296:
294:
282:, the vectors
261:
254:
247:
208:) denotes the
198:
197:
186:
183:
180:
177:
174:
171:
168:
165:
162:
158:
154:
151:
148:
145:
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133:
130:
126:
122:
119:
116:
113:
110:
107:
104:
101:
98:
51:
48:
44:Lars Hörmander
9:
6:
4:
3:
2:
1775:
1764:
1761:
1759:
1756:
1755:
1753:
1744:
1741:
1736:
1732:
1727:
1722:
1718:
1714:
1710:
1706:
1702:
1699:
1696:
1691:
1689:0-486-44994-7
1685:
1681:
1676:
1675:
1667:
1664:
1662:
1659:
1658:
1652:
1650:
1646:
1642:
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1608:
1604:
1598:
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1588:
1583:
1580:
1577:
1573:
1569:
1563:
1560:
1550:
1549:
1548:
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1528:
1524:
1520:
1517:
1514:
1509:
1505:
1496:
1486:
1482:
1480:
1476:
1473:is such that
1472:
1468:
1464:
1460:
1455:
1451:
1447:
1428:
1420:
1416:
1401:
1398:
1394:
1388:
1383:
1380:
1377:
1373:
1369:
1364:
1360:
1352:
1351:
1350:
1348:
1329:
1317:
1311:
1305:
1302:
1299:
1296:
1293:
1287:
1280:
1269:
1265:
1259:
1256:
1253:
1247:
1240:
1239:
1238:
1236:
1232:
1228:
1224:
1220:
1217: →
1216:
1212:
1208:
1182:
1179:
1173:
1163:
1160:
1151:
1148:
1145:
1139:
1132:
1127:
1117:
1114:
1111:
1108:
1105:
1102:
1097:
1091:
1088:
1085:
1079:
1076:
1073:
1067:
1064:
1061:
1051:
1043:
1030:
1021:
1020:
1019:
1016:
1012:
993:
988:
984:
980:
975:
970:
966:
960:
955:
952:
949:
945:
939:
936:
931:
928:
921:
920:
919:
917:
914:
904:
902:
883:
875:
854:
850:
846:
843:
840:
835:
831:
805:
801:
797:
794:
791:
786:
782:
756:
752:
748:
742:
736:
728:
724:
718:
713:
710:
707:
703:
699:
696:
693:
684:
676:
672:
668:
665:
662:
652:
651:
650:
648:
645:Consider the
638:
636:
618:
614:
605:
601:
598:
580:
577:
572:
568:
564:
561:
558:
553:
549:
545:
540:
536:
532:
529:
520:
517:
509:
497:
487:
483:
478:
474:
465:
455:
451:
446:
442:
436:
426:
422:
417:
408:
400:
388:
378:
374:
369:
365:
359:
349:
345:
340:
334:
326:
317:
307:
303:
298:
285:
284:
283:
281:
277:
273:
269:
264:
260:
253:
246:
241:
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235:
231:
227:
223:
219:
215:
211:
207:
203:
184:
178:
172:
166:
160:
152:
146:
140:
134:
128:
120:
114:
105:
102:
99:
89:
88:
87:
85:
81:
78:
75:
71:
67:
63:
60:
59:vector fields
58:
47:
45:
42:
41:mathematician
39:
35:
31:
27:
26:vector fields
23:
19:
1716:
1712:
1679:
1640:
1634:
1544:
1494:
1492:
1483:
1474:
1470:
1466:
1462:
1458:
1453:
1449:
1445:
1443:
1344:
1234:
1230:
1226:
1222:
1218:
1214:
1210:
1204:
1014:
1010:
1008:
915:
910:
873:
773:
644:
634:
603:
599:
595:
279:
275:
271:
267:
262:
258:
251:
244:
242:
237:
233:
229:
221:
217:
213:
205:
201:
199:
79:
69:
65:
61:
56:
53:
21:
15:
1719:: 147–171.
1666:Lie algebra
84:Lie bracket
74:dimensional
18:mathematics
1752:Categories
1672:References
1221:such that
238:vice versa
54:Given two
50:Definition
1735:0001-5962
1713:Acta Math
1574:∑
1564:˙
1518:…
1413:∂
1409:∂
1374:∑
1270:∫
1233:for each
1177:→
1158:→
1152:⋅
1118:∈
1049:∂
1041:∂
946:∑
884:∘
844:…
795:…
749:
743:∘
704:∑
694:
663:
578:≤
562:…
533:≤
521:⋮
153:−
1707:(1967).
1655:See also
1347:elliptic
1170:as
200:where D
1743:0222474
1698:2250060
1647:. See
236:), and
38:Swedish
30:partial
1733:
1686:
649:(SDE)
507:
398:
324:
257:, ...
226:matrix
1731:ISSN
1684:ISBN
1493:Let
1237:and
1106:>
597:span
243:Let
64:and
32:and
1721:doi
1717:119
1635:is
918:by
216:at
212:of
68:on
16:In
1754::
1740:MR
1729:.
1715:.
1711:.
1695:MR
1651:.
1481:.
1475:AA
1454:ji
637:.
250:,
240:.
46:.
20:,
1737:.
1723::
1692:.
1641:M
1620:)
1617:x
1614:(
1609:i
1605:A
1599:i
1595:u
1589:n
1584:0
1581:=
1578:i
1570:=
1561:x
1545:M
1529:n
1525:A
1521:,
1515:,
1510:0
1506:A
1495:M
1471:n
1467:i
1463:d
1459:j
1450:a
1446:A
1429:,
1421:j
1417:x
1402:i
1399:j
1395:a
1389:d
1384:1
1381:=
1378:j
1370:=
1365:i
1361:A
1330:y
1326:d
1321:)
1318:y
1315:(
1312:f
1309:)
1306:y
1303:,
1300:x
1297:,
1294:t
1291:(
1288:p
1281:d
1276:R
1266:=
1263:)
1260:x
1257:,
1254:t
1251:(
1248:u
1235:t
1231:R
1227:t
1225:(
1223:p
1219:R
1215:R
1211:p
1183:;
1180:0
1174:t
1164:,
1161:f
1155:)
1149:,
1146:t
1143:(
1140:u
1133:;
1128:d
1123:R
1115:x
1112:,
1109:0
1103:t
1098:,
1095:)
1092:x
1089:,
1086:t
1083:(
1080:u
1077:F
1074:=
1071:)
1068:x
1065:,
1062:t
1059:(
1052:t
1044:u
1031:{
1015:i
1011:A
994:.
989:0
985:A
981:+
976:2
971:i
967:A
961:n
956:1
953:=
950:i
940:2
937:1
932:=
929:F
916:F
887:d
874:n
860:)
855:n
851:W
847:,
841:,
836:1
832:W
828:(
806:n
802:A
798:,
792:,
787:0
783:A
757:i
753:W
746:d
740:)
737:x
734:(
729:i
725:A
719:n
714:1
711:=
708:i
700:+
697:t
691:d
688:)
685:x
682:(
677:0
673:A
669:=
666:x
660:d
635:n
619:0
615:j
600:R
581:n
573:n
569:j
565:,
559:,
554:1
550:j
546:,
541:0
537:j
530:0
510:,
504:]
501:)
498:x
495:(
488:2
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479:A
475:,
472:]
469:)
466:x
463:(
456:1
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437:x
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414:[
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401:,
395:]
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386:(
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350:0
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337:[
327:,
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280:R
276:x
268:R
263:n
259:A
255:1
252:A
248:0
245:A
234:x
232:(
230:W
222:R
218:x
214:V
206:x
204:(
202:V
185:,
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179:x
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173:V
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167:x
164:(
161:W
157:D
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147:x
144:(
141:W
138:)
135:x
132:(
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121:=
118:)
115:x
112:(
109:]
106:W
103:,
100:V
97:[
80:R
72:-
70:d
66:W
62:V
57:C
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