We prove the unique solvability of parabolic equations with discontinuous leading coefficients in . Using this result, we establish the uniqueness of diffusion processes with time-dependent discontinuous coefficients.
@article{ASNSP_2006_5_5_1_55_0, author = {Kim, Doyoon}, 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}, year = {2006}, mrnumber = {2240183}, zbl = {1107.35051}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2006_5_5_1_55_0/} }
TY - JOUR AU - Kim, Doyoon TI - Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2006 SP - 55 EP - 76 VL - 5 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2006_5_5_1_55_0/ LA - en ID - ASNSP_2006_5_5_1_55_0 ER -
%0 Journal Article %A Kim, Doyoon %T Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2006 %P 55-76 %V 5 %N 1 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2006_5_5_1_55_0/ %G en %F ASNSP_2006_5_5_1_55_0
Kim, Doyoon. Second order parabolic equations and weak uniqueness of diffusions with discontinuous coefficients. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 1, pp. 55-76. http://archive.numdam.org/item/ASNSP_2006_5_5_1_55_0/
[1] 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] Uniqueness for diffusions with piecewise constant coefficients, Probab. Theory Related Fields 76(1987), 557-572. | MR | Zbl
and ,[3] Uniqueness for some diffusions with discontinuous coefficients, Ann. Probab. 19 (1991), 525-537. | MR | Zbl
, and ,[4] “Brownian Motion and Stochastic Calculus”, Graduate Texts in Mathematics, Vol. 113, Springer-Verlag, New York, second edition, 1991. | MR | Zbl
and ,[5] Second order elliptic equations in with piecewise continuous coefficients, Potential Anal., 2005, submitted. | MR | Zbl
,[6] “Controlled Diffusion Processes”, Applications of Mathematics, Vol. 14, Springer-Verlag, New York, 1980. Translated from the Russian by A. B. Aries. | MR | Zbl
,[7] On weak uniqueness for some diffusions with discontinuous coefficients, Stochastic Process. Appl. 113 (2004), 37-64. | MR | Zbl
,[8] “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
, and ,[9] “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. | MR | Zbl
,[10] On elliptic equations with piecewise constant coefficients, Appl. Anal. 2 (1972), 79-96. | MR | Zbl
,[11] On elliptic equations with piecewise constant coefficients, II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 26 (1972), 839-870. | Numdam | MR | Zbl
,[12] Un problema di Cauchy per un operatore parabolico con coefficienti costanti a tratti, Matematiche 31 (1977), 126-146. | MR | Zbl
,[13] Quasilinear parabolic operators with discontinuous ingredients, Nonlinear Anal. 52 (2003), 1079-1093. | MR
,[14] “Singular Integrals and Differentiability Properties of Functions”, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. | MR | Zbl
,[15] “Multidimensional Diffusion Processes”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 233, Springer-Verlag, Berlin, 1979. | MR | Zbl
and ,[16] “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland Mathematical Library, Vol. 18, North-Holland Publishing Co., Amsterdam, 1978. | MR | Zbl
,[17] Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed -norm, Electron. Res. Announc. Amer. Math. Soc. (electronic), 8 (2002), 47-51. | MR | Zbl
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