Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

Kunze, Markus; Lorenzi, Luca; Lunardi, Alessandra

doi:10.1090/S0002-9947-09-04738-2

Nonautonomous Kolmogorov parabolic equations with unbounded coefficients
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by Markus Kunze, Luca Lorenzi and Alessandra Lunardi

Trans. Amer. Math. Soc. 362 (2010), 169-198

DOI: https://doi.org/10.1090/S0002-9947-09-04738-2

Published electronically: August 3, 2009

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Abstract:

We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times \mathbb {R}^d$ for some unbounded interval $I\subset \mathbb {R}$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot )=f\in C_b(\mathbb {R}^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family $\{G(t,s)\}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $\{G(t,s)\}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.

References

Paolo Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations 1 (1988), no. 4, 433–457. MR 945820
Serge Bernstein, Sur la généralisation du problème de Dirichlet, Math. Ann. 62 (1906), no. 2, 253–271 (French). MR 1511375, DOI 10.1007/BF01449980
Luca Lorenzi and Marcello Bertoldi, Analytical methods for Markov semigroups, Pure and Applied Mathematics (Boca Raton), vol. 283, Chapman & Hall/CRC, Boca Raton, FL, 2007. MR 2313847
Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999. MR 1707332, DOI 10.1090/surv/070
Giuseppe Da Prato and Michael Röckner, A note on evolution systems of measures for time-dependent stochastic differential equations, Seminar on Stochastic Analysis, Random Fields and Applications V, Progr. Probab., vol. 59, Birkhäuser, Basel, 2008, pp. 115–122. MR 2401953, DOI 10.1007/978-3-7643-8458-6_{7}
Giuseppe Da Prato and Alessandra Lunardi, Ornstein-Uhlenbeck operators with time periodic coefficients, J. Evol. Equ. 7 (2007), no. 4, 587–614. MR 2369672, DOI 10.1007/s00028-007-0321-z
E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der mathematischen Wissenschaften, Band 121, vol. 122, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR 0193671
S. Fornaro, G. Metafune, and E. Priola, Gradient estimates for Dirichlet parabolic problems in unbounded domains, J. Differential Equations 205 (2004), no. 2, 329–353. MR 2092861, DOI 10.1016/j.jde.2004.06.019
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Matthias Geissert and Alessandra Lunardi, Invariant measures and maximal $L^2$ regularity for nonautonomous Ornstein-Uhlenbeck equations, J. Lond. Math. Soc. (2) 77 (2008), no. 3, 719–740. MR 2418301, DOI 10.1112/jlms/jdn009
M. Geissert, A. Lunardi, Asymptotic behavior in nonautonomous Ornstein-Uhlenbeck equations, J. London Math. Soc. (2) 79 (2009), no 1, 85–106.
Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940, DOI 10.1007/978-1-4612-0949-2
Hiroshi Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, vol. 24, Cambridge University Press, Cambridge, 1997. Reprint of the 1990 original. MR 1472487
Nobuyuki Ikeda and Shinzo Watanabe, Stochastic differential equations and diffusion processes, 2nd ed., North-Holland Mathematical Library, vol. 24, North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989. MR 1011252
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). Translated from the Russian by S. Smith. MR 0241822
Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
G. Metafune, D. Pallara, and M. Wacker, Feller semigroups on $\mathbf R^N$, Semigroup Forum 65 (2002), no. 2, 159–205. MR 1911723, DOI 10.1007/s002330010129
D.W. Strook, S.R.S. Varadhan, Multidimensional diffusion processes, Classics in Mathematics, Springer-Verlag, Berlin, 2006.