×

zbMATH — the first resource for mathematics

Kernel estimates for nonautonomous Kolmogorov equations. (English) Zbl 1331.35153
The authors prove kernel estimates for transition probabilities associated with nonautonomous evolution equations. They study global regularity properties and pointwise bounds of the density \(\rho\). First they establish global boundedness and Sobolev regularity of \(\rho\) in the case of bounded diffusion coefficients. Then, they use time-dependent Lyapunov functions and an approximation argument based on De Giorgi’s technique to prove global boundedness and pointwise estimates of \(\rho\) in the general case, where the diffusion coefficients are not assumed to be bounded.

MSC:
35K10 Second-order parabolic equations
35K08 Heat kernel
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37L40 Invariant measures for infinite-dimensional dissipative dynamical systems
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aibeche, A.; Laidoune, K.; Rhandi, A., Time dependent Lyapunov functions for some Kolmogorov semigroups perturbed by unbounded potentials, Arch. Math. (Basel), 94, 6, 565-577, (2010) · Zbl 1200.35027
[2] Bogachev, V. I.; Da Prato, G.; Röckner, M., Existence of solutions to weak parabolic equations for measures, Proc. Lond. Math. Soc., 88, 753-774, (2004) · Zbl 1072.35076
[3] Bogachev, V. I.; Da Prato, G.; Röckner, M., On parabolic equations for measures, Comm. Partial Differential Equations, 33, 397-418, (2008) · Zbl 1323.35058
[4] Bogachev, V. I.; Da Prato, G.; Röckner, M.; Stannat, W., Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc., 39, 631-640, (2007) · Zbl 1129.35039
[5] Bogachev, V. I.; Krylov, N. V.; Röckner, M., On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations, 26, 11-12, 2037-2080, (2001) · Zbl 0997.35012
[6] Bogachev, V. I.; Krylov, N. V.; Röckner, M., Elliptic equations for measures: regularity and global bounds of densities, J. Math. Pures Appl., 85, 743-757, (2006) · Zbl 1206.35242
[7] Bogachev, V. I.; Krylov, N. V.; Röckner, M., Elliptic and parabolic equations for measures, Russian Math. Surveys, 64, 6, 973-1078, (2009) · Zbl 1194.35481
[8] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., Global regularity and bounds for solutions of parabolic equations for probability measures, Theory Probab. Appl., 50, 561-581, (2006) · Zbl 1203.60095
[9] Fornaro, S.; Fusco, N.; Metafune, G.; Pallara, D., Sharp upper bounds for the density of some invariant measures, Proc. Roy. Soc. Edinburgh Sect. A, 139, 6, 1145-1161, (2009) · Zbl 1206.47038
[10] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (2001), Springer-Verlag Berlin · Zbl 1042.35002
[11] Giusti, E., Direct methods in the calculus of variations, (2003), World Scientific · Zbl 1028.49001
[12] Hansel, T.; Rhandi, A., The Oseen-Navier-Stokes flow in the exterior of a rotating obstacle: the non-autonomous case, J. Reine Angew. Math., 694, 1-26, (2014) · Zbl 1298.35146
[13] Krylov, N. V., Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, (1996), American Mathematical Society Providence, RI · Zbl 0865.35001
[14] Krylov, N. V., Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal., 183, 1-41, (2001) · Zbl 0980.60091
[15] Kunze, M.; Lorenzi, L.; Lunardi, A., Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 362, 1, 169-198, (2010) · Zbl 1184.35150
[16] Kunze, M.; Lorenzi, L.; Rhandi, A., Kernel estimates for nonautonomous Kolmogorov equations with potential term, (Favini, A.; Fragnelli, G.; Maria Mininni, R., New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Series, vol. 10, (2014), Springer International Publishing), 229-251 · Zbl 1391.35176
[17] Ladyženskaja, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and quasilinear equations of parabolic type, (1968), American Mathematical Society Providence, English transl.:
[18] Laidoune, K.; Metafune, G.; Pallara, D.; Rhandi, A., Global properties of transition kernels associated to second order elliptic operators, Progr. Nonlinear Differential Equations Appl., 60, 415-432, (2011) · Zbl 1254.35049
[19] Lorenzi, L.; Rhandi, A., On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ., 15, 1, 53-88, (2015) · Zbl 06450961
[20] Metafune, G.; Pallara, D.; Rhandi, A., Global properties of invariant measures, J. Funct. Anal., 223, 2, 396-424, (2005) · Zbl 1131.35318
[21] Metafune, G.; Pallara, D.; Rhandi, A., Global properties of transition probabilities of singular diffusions, Theory Probab. Appl., 54, 1, 68-96, (2010) · Zbl 1206.60072
[22] Metafune, G.; Spina, C., Kernel estimates for a class of Schrödinger semigroups, J. Evol. Equ., 7, 719-742, (2007) · Zbl 1148.35016
[23] Metafune, G.; Spina, C.; Tacelli, C., Elliptic operators with unbounded diffusion and drift coefficients in \(L^p\) spaces, Adv. Differential Equations, 19, 5-6, 473-526, (2014) · Zbl 1305.47029
[24] Shaposhnikov, S. V., The Fokker-Planck-Kolmogorov equations with a potential and a non-uniformly elliptic diffusion matrix, Trans. Moscow Math. Soc., 15-29, (2013) · Zbl 1310.35229
[25] Spina, C., Kernel estimates for a class of Kolmogorov semigroups, Arch. Math. (Basel), 91, 3, 265-279, (2008) · Zbl 1161.47028
[26] Stroock, D. W.; Varadhan, S. R.S., Multidimensional diffusion processes, Classics in Mathematics, (2006), Springer-Verlag Berlin, Reprint of the 1997 edition · Zbl 1103.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.