Stochastic calculus and degenerate boundary value problems

Patrick Cattiaux

Annales de l'institut Fourier (1992)

  • Volume: 42, Issue: 3, page 541-624
  • ISSN: 0373-0956

Abstract

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Consider the boundary value problem (L.P): ( h - A ) u = f in D , ( v - Γ ) u = g on D where A is written as A = 1 / 2 i = 1 m Y i 2 + Y 0 , and Γ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields Y i ( 0 i m ), for regular open sets D with a non-characteristic boundary.Our study lies on the stochastic representation of u and uses the stochastic calculus of variations for the ( A , Γ ) -diffusion process in D . Applications to the decomposition of C ( D ) , and to invariant measures are also discussed.

How to cite

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Cattiaux, Patrick. "Stochastic calculus and degenerate boundary value problems." Annales de l'institut Fourier 42.3 (1992): 541-624. <http://eudml.org/doc/74966>.

@article{Cattiaux1992,
abstract = {Consider the boundary value problem (L.P):$(h-A)u=f$ in $D$, $(v- \Gamma )u=g$ on $\partial D$ where $A$ is written as $A= \{1/ 2\} \sum ^ m_\{i=1\} Y^ 2_ i+Y_ 0$, and $\Gamma $ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_ i$ ($0\le i\le m$), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A,\Gamma )$-diffusion process in $\overline\{D\}$. Applications to the decomposition of $C^ \infty (\overline\{D\})$, and to invariant measures are also discussed.},
author = {Cattiaux, Patrick},
journal = {Annales de l'institut Fourier},
keywords = {diffusions with a boundary condition; hypoellipticity; resolvent; diffusion process; Venttsel's condition; oblique derivative; existence; uniqueness; smoothness; stochastic representation; stochastic calculus of variations; invariant measures},
language = {eng},
number = {3},
pages = {541-624},
publisher = {Association des Annales de l'Institut Fourier},
title = {Stochastic calculus and degenerate boundary value problems},
url = {http://eudml.org/doc/74966},
volume = {42},
year = {1992},
}

TY - JOUR
AU - Cattiaux, Patrick
TI - Stochastic calculus and degenerate boundary value problems
JO - Annales de l'institut Fourier
PY - 1992
PB - Association des Annales de l'Institut Fourier
VL - 42
IS - 3
SP - 541
EP - 624
AB - Consider the boundary value problem (L.P):$(h-A)u=f$ in $D$, $(v- \Gamma )u=g$ on $\partial D$ where $A$ is written as $A= {1/ 2} \sum ^ m_{i=1} Y^ 2_ i+Y_ 0$, and $\Gamma $ is a general Venttsel’s condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander’s condition on the Lie brackets of the vector fields $Y_ i$ ($0\le i\le m$), for regular open sets $D$ with a non-characteristic boundary.Our study lies on the stochastic representation of $u$ and uses the stochastic calculus of variations for the $(A,\Gamma )$-diffusion process in $\overline{D}$. Applications to the decomposition of $C^ \infty (\overline{D})$, and to invariant measures are also discussed.
LA - eng
KW - diffusions with a boundary condition; hypoellipticity; resolvent; diffusion process; Venttsel's condition; oblique derivative; existence; uniqueness; smoothness; stochastic representation; stochastic calculus of variations; invariant measures
UR - http://eudml.org/doc/74966
ER -

References

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  1. [1] G. BEN AROUS, S. KUSUOKA, D. STROOCK, The Poisson kernel for certain degenerate elliptic operators, J. Func. Anal., 56 (1984), 171-209. Zbl0556.35036MR85k:35093
  2. [2] J. M. BISMUT, Mécanique aléatoire, Lect. Notes in Math., 866, Springer, Berlin, 1981. Zbl0457.60002MR84a:70002
  3. [3] J. M. BISMUT, Martingales, the Malliavin's calculus and hypoellipticity under general Hörmander's conditions, Z. Wahrsch., 56 (1981), 469-506. Zbl0445.60049MR82k:60134
  4. [4] J. M. BISMUT, The calculus of boundary processes, Ann. Scient. École Normale Sup., 17 (1984), 507-622. Zbl0561.60081MR86d:60087
  5. [5] J. M. BISMUT, Last exist decompositions and regularity at the boundary of transition probabilities, Z. Wahrsch., 69 (1985), 65-98. Zbl0551.60077MR86i:60192
  6. [6] J. M. BISMUT, Large deviations and the Malliavin calculus, Progress in Math., 45, Birkhaüser, Boston, 1984. Zbl0537.35003MR86f:58150
  7. [7] J. M. BONY, Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés, Ann. Inst. Fourier, 19-1 (1969), 277-304. Zbl0176.09703MR41 #7486
  8. [8] J. M. BONY, P. COURRÈGE, P. PRIOURET, Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites... Ann. Inst. Fourier, 18-2 (1968), 369-521. Zbl0181.11704MR39 #6397
  9. [9] P. CATTIAUX, Thèse de 3e cycle, Univ. Paris XI, 1984. 
  10. [10] P. CATTIAUX, Hypoellipticité et hypoellipticité partielle pour les diffusions avec une condition frontière, Ann. Inst., H. Poincaré, 22 (1986), 67-112. Zbl0595.60059MR87k:60180
  11. [11] P. CATTIAUX, Régularité au bord pour les densités et les densités conditionnelles d'une diffusion réfléchie hypoelliptique, Stochastics, 20 (1987), 309-340. Zbl0637.60092MR88h:60153
  12. [12] P. CATTIAUX, Time reversal of diffusion processes with a boundary condition, Stochastic processes and their Applications, 28 (1988), 275-292. Zbl0652.60083MR89g:60243
  13. [13] P. CATTIAUX, Calcul stochastique et opérateurs dégénérés du second ordre. I Résolvantes, théorème de Hörmander et application, Bull. Sc. Math., 114 (1990), 421-462. Zbl0715.60064
  14. [14] P. CATTIAUX, Calcul stochastique et opérateurs dégénérés du second ordre. II Problème de Dirichlet, Bull. Sc. Math., 115 (1991), 81-122. Zbl0790.60048MR92f:60099
  15. [15] J. CHAZARAIN, A. PIRIOU, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier Villars, Paris, 1981. Zbl0446.35001MR82i:35001
  16. [16] M. DERRIDJ, Un problème aux limites pour une classe d'opérateurs du second ordre hypoelliptiques, Ann. Inst. Fourier, 21-4 (1971), 99-148. Zbl0215.45405MR58 #29139
  17. [17] H. DOSS, P. PRIOURET, Petites perturbations de systèmes dynamiques avec réflexion, Séminaire de Probas. 17, Lect. Notes in Math., 986 (1983), 353-370. Zbl0529.60061MR86g:60036
  18. [18] N. EL KAROUI, Processus de diffusion associé à un opérateur elliptique dégénéré et à une condition frontière, Thèse, Université Paris VI, 1971. 
  19. [19] C. GRAHAM, Thèse 3e cycle, Université Paris VI, 1985. 
  20. [20] L. HÖRMANDER, Pseudo differential operators and non elliptic boundary problems, Ann. of Math., 83 (1986), 129-209. Zbl0132.07402MR38 #1387
  21. [21] L. HÖRMANDER, Hypoelliptic second order differential operators, Acta. Math., 119 (1967), 147-171. Zbl0156.10701
  22. [22] HSU-PEI, Probabilistic approach to the Neumann problem, Comm. on Pure and Applied Math., 38 (1985), 445-472. Zbl0587.60077MR86k:35033
  23. [23] N. IKEDA, S. WATANABE, Stochastic differential equations and diffusion processes, North Holland, Amsterdam, 1981. Zbl0495.60005MR84b:60080
  24. [24] H. KUNITA, Stochastic differential equations and stochastic flows of diffeomorphisms, Lect. Notes in Math., 1097 (1984), 144-305. Zbl0554.60066MR87m:60127
  25. [25] R. LEANDRE, Minoration en temps petit de la densité d'une diffusion dégénérée, preprint, 1987. Zbl0637.58034MR88k:60147
  26. [26] P. MALLIAVIN, * Stochastic calculus of variations and hypoelliptic operators, Proc. Intern. Symp. S.D.E. of Kyoto, K-Ito ed., (1976), 195-263. Zbl0411.60060MR81f:60083
  27. P. MALLIAVIN, * Ck hypoellipticity with degeneracy. In "Stochastic Analysis", A. Friedman, M. Pinsky ed., (1978), 199-214 & 327-340. Zbl0449.58022MR80i:58045a
  28. [27] MA ZHIMING, On the probabilistic approach to boundary value problems, preprint, 1986. 
  29. [28] B. P. PANEYAKH, Some boundary value problems for elliptic equations and the Lie algebras associated with them, Math. USSR Sbornik, 54 (1986), 207-237. Zbl0656.35035
  30. [29] P. PRIOURET, Processus de Markov sur une variété à bord compacte, Ann. Inst. H. Poincaré, 4 (1968), 193-253. Zbl0177.45703MR39 #3599
  31. [30] D. STROOCK, Some applications of stochastic calculus to partial differential equations, Lect. Notes in Math., 976 (1983), 267-382. Zbl0494.60060MR85c:60088
  32. [31] D. STROOCK, S. R. S. VARADHAN, Diffusion processes with boundary conditions, Comm. on Pure an Applied Math., 24 (1971), 147-225. Zbl0227.76131MR43 #2774
  33. [32] D. STROOCK, S. R. S. VARADHAN, On degenerate elliptic-parabolic operators of second order and their associated diffusions, Comm. on Pure and Applied Math., 25 (1972), 651-713. Zbl0344.35041MR52 #8651
  34. [33] K. TAIRA, Semi-groups and boundary value problems, Duke Math. Journal, 49 (1982), 287-320. Zbl0504.47042MR84e:47057a
  35. [34] F. TRÈVES, Topological vector spaces, Distributions and Kernels, Academic Press, New York-London, 1967. Zbl0171.10402MR37 #726
  36. [35] A. D. VENTCEL, On boundary conditions for multidimensional processes, Theory Prob. Appl., 4 (1959). Zbl0089.13404MR22 #12585
  37. [36] S. WATANABE, Construction of diffusion processes with Wentzell's boundary conditions... Prob. Theory, Banach Center Pub. 5. Polish Sci. Publ., Warsaw, (1979), 255-271. Zbl0442.60076MR82a:60119
  38. [37] K. YOSIDA, Functional Analysis, Springer, Berlin, 1965. Zbl0126.11504MR31 #5054
  39. [38] Temps locaux. J. AZEMA, M. YOR, éd. Astérisque SMF, 52-53 (1978). Zbl0385.60063
  40. [39] Géodésiques et diffusions en temps petit, Astérisque SMF, 84-85 (1981). Zbl0458.00008

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