Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients

Kim, Doyoon

Kim, Doyoon

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 55-76.

Résumé

We prove the unique solvability of parabolic equations with discontinuous leading coefficients in $W_{p}^{1, 2} ((0, T) \times ℝ^{d})$ . Using this result, we establish the uniqueness of diffusion processes with time-dependent discontinuous coefficients.

MR Zbl

Classification : 35K10, 35R05, 60G44, 60H10

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     title = {Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {55--76},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {1},
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Kim, Doyoon. Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 1, pp. 55-76. http://www.numdam.org/item/ASNSP_2006_5_5_1_55_0/

Bibliographie
Cité par

[1] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. 35 (2000), 161-177. Dedicated to the memory of Branko Najman. | MR | Zbl

[2] R. F. Bass and É. Pardoux, Uniqueness for diffusions with piecewise constant coefficients, Probab. Theory Related Fields 76(1987), 557-572. | MR | Zbl

[3] M. C. Cerutti, L. Escauriaza and E. B. Fabes, Uniqueness for some diffusions with discontinuous coefficients, Ann. Probab. 19 (1991), 525-537. | MR | Zbl

[4] I. Karatzas and S. E. Shreve, “Brownian Motion and Stochastic Calculus”, Graduate Texts in Mathematics, Vol. 113, Springer-Verlag, New York, second edition, 1991. | MR | Zbl

[5] D. Kim, Second order elliptic equations in $ℝ^{d}$ with piecewise continuous coefficients, Potential Anal., 2005, submitted. | MR | Zbl

[6] N. V. Krylov, “Controlled Diffusion Processes”, Applications of Mathematics, Vol. 14, Springer-Verlag, New York, 1980. Translated from the Russian by A. B. Aries. | MR | Zbl

[7] N. V. Krylov, On weak uniqueness for some diffusions with discontinuous coefficients, Stochastic Process. Appl. 113 (2004), 37-64. | MR | Zbl

[8] O. A. Ladyženskaja, V. A. Solonnikov and N. N. UralʼCeva, “Linear and Quasilinear Equations of Parabolic Type”. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. | MR | Zbl

[9] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. | MR | Zbl

[10] A. Lorenzi, On elliptic equations with piecewise constant coefficients, Appl. Anal. 2 (1972), 79-96. | MR | Zbl

[11] A. Lorenzi, On elliptic equations with piecewise constant coefficients, II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 839-870. | Numdam | MR | Zbl

[12] S. Salsa, Un problema di Cauchy per un operatore parabolico con coefficienti costanti a tratti, Matematiche 31 (1977), 126-146. | MR | Zbl

[13] L. G. Softova, Quasilinear parabolic operators with discontinuous ingredients, Nonlinear Anal. 52 (2003), 1079-1093. | MR

[14] E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. | MR | Zbl

[15] D. W. Stroock and S. R. Srinivasa Varadhan, “Multidimensional Diffusion Processes”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 233, Springer-Verlag, Berlin, 1979. | MR | Zbl

[16] H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland Mathematical Library, Vol. 18, North-Holland Publishing Co., Amsterdam, 1978. | MR | Zbl

[17] P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_{p}$ -norm, Electron. Res. Announc. Amer. Math. Soc. (electronic), 8 (2002), 47-51. | MR | Zbl