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, p. 55-76

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

Classification:  35K10,  35R05,  60G44,  60H10
@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},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 5},
number = {1},
year = {2006},
pages = {55-76},
zbl = {1107.35051},
mrnumber = {2240183},
language = {en},
url = {http://www.numdam.org/item/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://www.numdam.org/item/ASNSP_2006_5_5_1_55_0/

[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 1783238 | Zbl 0997.46029

[2] R. F. Bass and É. Pardoux, Uniqueness for diffusions with piecewise constant coefficients, Probab. Theory Related Fields 76(1987), 557-572. | MR 917679 | Zbl 0617.60075

[3] M. C. Cerutti, L. Escauriaza and E. B. Fabes, Uniqueness for some diffusions with discontinuous coefficients, Ann. Probab. 19 (1991), 525-537. | MR 1106274 | Zbl 0737.60038

[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 1121940 | Zbl 0734.60060

[5] D. Kim, Second order elliptic equations in ${ℝ}^{d}$ with piecewise continuous coefficients, Potential Anal., 2005, submitted. | MR 2276531 | Zbl 1152.35341

[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 601776 | Zbl 0459.93002

[7] N. V. Krylov, On weak uniqueness for some diffusions with discontinuous coefficients, Stochastic Process. Appl. 113 (2004), 37-64. | MR 2078536 | Zbl 1073.60064

[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 241822 | Zbl 0174.15403

[9] G. M. Lieberman, “Second Order Parabolic Differential Equations”, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. | MR 1465184 | Zbl 0884.35001

[10] A. Lorenzi, On elliptic equations with piecewise constant coefficients, Appl. Anal. 2 (1972), 79-96. | MR 296490 | Zbl 0228.35031

[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 382836 | Zbl 0252.35019

[12] S. Salsa, Un problema di Cauchy per un operatore parabolico con coefficienti costanti a tratti, Matematiche 31 (1977), 126-146. | MR 499720 | Zbl 0383.35029

[13] L. G. Softova, Quasilinear parabolic operators with discontinuous ingredients, Nonlinear Anal. 52 (2003), 1079-1093. | MR 1941247 | Zbl pre01879998

[14] E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, N.J., 1970. | MR 290095 | Zbl 0207.13501

[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 532498 | Zbl 0426.60069

[16] H. Triebel, “Interpolation Theory, Function Spaces, Differential Operators”, North-Holland Mathematical Library, Vol. 18, North-Holland Publishing Co., Amsterdam, 1978. | MR 503903 | Zbl 0387.46032

[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 1945779 | Zbl 1015.35036