INVARIANT MEASURES FOR SYSTEMS OF KOLMOGOROV EQUATIONS

Davide Addona; Luciana Angiuli; Luca Lorenzi

doi:10.11948/2018.764

2018 Volume 8 Issue 3

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Davide Addona, Luciana Angiuli, Luca Lorenzi. INVARIANT MEASURES FOR SYSTEMS OF KOLMOGOROV EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 764-804. doi: 10.11948/2018.764

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Davide Addona, Luciana Angiuli, Luca Lorenzi. INVARIANT MEASURES FOR SYSTEMS OF KOLMOGOROV EQUATIONS[J]. Journal of Applied Analysis & Computation, 2018, 8(3): 764-804. doi: 10.11948/2018.764

INVARIANT MEASURES FOR SYSTEMS OF KOLMOGOROV EQUATIONS

1 Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, via N. Machiavelli 30, I-44121 Ferrara, Italy;
2 Dipartimento di Matematica e Fisica "Ennio de Giorgi", Università del Salento, via Per Arnesano s. n. c., I-73100 Lecce, Italy;
3 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Plesso di Matematica e Informatica, Università degli Studi di Parma, Viale Parco Area delle Scienze 53/A, I-43124 Parma, Italy

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Abstract

Abstract

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these measures are absolutely continuous with respect to the Lebesgue measure and study some of their main properties. Finally, we show that they characterize the asymptotic behaviour of the semigroup at infinity.
- Systems of Kolmogorov equations /
- unbounded coefficients /
- gradient estimates /
- systems of invariant measures /
- asymptotic behaviour
MSC: 35B40;35B41;35K45;37L40;47A35

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References

[1]	D. Addona, L. Angiuli, L. Lorenzi, G. Tessitore, On coupled systems of Kolmogorov equations with applications to stochastic differential games, ESAIM-Control, Optimisation and Calculus of Variations, 2017, 23, 937-976. Google Scholar
[2]	D. Addona, L. Angiuli, L. Lorenzi, On invariant measures associated to weakly coupled systems of Kolmogorov equations (submitted), Available on ArXiv at https://arxiv.org/abs/1705.03784. Google Scholar
[3]	L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press, USA, 2000. Google Scholar
[4]	L. Angiuli, L. Lorenzi, On improvement of summability properties in nonautonomous Kolmogorov equations, Commun. Pure Appl. Anal., 2014, 13(3), 1237-1265. Google Scholar
[5]	L. Angiuli, L. Lorenzi, A. Lunardi, Hypercontractivity and asymptotic behavior in nonautonomous Kolmogorov equations, Comm. Partial Differential Equations, 2013, 38(12), 2049-2080. Google Scholar
[6]	L. Angiuli, L. Lorenzi, D. Pallara, Lp-estimates for parabolic systems with unbounded coefficients coupled at zero and first order, J. Math. Anal. Appl., 2016, 444, 110-135. Google Scholar
[7]	V. I. Bogachev, N. Krylov, M. Röckner, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations, 2000, 26, 2037-2080. Google Scholar
[8]	G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture notes, Cambridge University Press, 1996, 229. Google Scholar
[9]	E. B. Davies, B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal., 1984, 59(2), 335-395. Google Scholar
[10]	S. Delmonte, L. Lorenzi, On a class of weakly coupled systems of elliptic operators with unbounded coefficients, Milan J. Math., 2011, 79, 689-727. Google Scholar
[11]	B. Farkas, A. Lunardi, Maximal regularity for Kolmogorov operators in L² spaces with respect to invariant measures, J. Math. Pures Appl., 2006, 86(4), 310-321. Google Scholar
[12]	L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math., 1975, 97(4), 1061-1083. Google Scholar
[13]	R. Z. Has'minskii, Stochastic Stability of Differential Equations, Monographs and Textbooks on Mechanics of Solids and Fluids:Mechanics and Analysis, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980, 7. Google Scholar
[14]	M. Kunze, L. Lorenzi, A. Lunardi, Nonautonomous Kolmogorov parabolic equations with unbounded coefficients, Trans. Amer. Math. Soc., 2010, 362, 169-198. Google Scholar
[15]	G. Kresin, V. G. Maz'ia, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, Mathematical Surveys and Monographs American Mathematical Society, Providence, RI, 2012, 183. Google Scholar
[16]	L. Lorenzi, Analytical Methods for Kolmogorov Equations. Second Edition, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2017. Google Scholar
[17]	L. Lorenzi, A. Lunardi, Elliptic operators with unbounded diffusion coefficients in L² spaces with respect to invariant measures, J. Evol Equ., 2006, 6, 691-709. Google Scholar
[18]	L. Lorenzi, A. Lunardi, R. Schnaubelt, Strong convergence of solutions to nonautonomous Kolmogorov equations, Proc. Amer. Math. Soc., 2016, 144, 3903-3917. Google Scholar
[19]	A. Lunardi, On the Ornstein-Uhlenbeck operator in L² spaces with respect to invariant measures, Trans. Amer. Math. Soc., 1997, 349, 155-169. Google Scholar
[20]	F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2012, 135. Google Scholar
[21]	G. Metafune, D. Pallara, A. Rhandi, Global properties of invariant measures, J. Funct. Anal., 2005, 223(2), 396-424. Google Scholar
[22]	G. Metafune, J. Prüss, A. Rhandi, R. Schnaubelt, The domain of the OrnsteinUhlenbeck operator on an Lp-space with invariant measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 2(2), 471-485. Google Scholar
[23]	G. Metafune, D. Pallara, M. Wacker, Feller semigroups on RN, Semigroup Forum, 2002, 65, 159-205. Google Scholar
[24]	K. Otsuka, On the positivity of the fundamental solutions for parabolic systems, J. Math. Kyoto Univ., 1988, 28, 119-132. Google Scholar
[25]	J, Prüss, A. Rhandi, R. Schnaubelt, The domain of elliptic operators on Lp(Rd) with unbounded drift coefficients, Houston J. Math., 2006, 32, 563-576. Google Scholar
[26]	M. Röckner, F. Y. Wang, Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds, Forum Math., 2003, 15(6), 893-921. Google Scholar
[27]	F. Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Relat. Fields, 1997, 109(3), 417-434. Google Scholar
[28]	K. Yosida, Functional Analysis, Springer-Verlag, New York, 1965. Google Scholar