Abstract
We consider a second-order elliptic operator k arising from Hamiltonian systems with friction in ℝ2n perturbed by noise. An invariant measure for this operator is μ(dx, dy) = exp(−2H(x, y))dx dy, where H is the Hamiltonian. We study the realization K: H 2(ℝ2n, μ) ↦ L 2(ℝ2n, μ) of k in L 2(ℝ2n, μ), proving that it is m-dissipative and that it generates an analytic semigroup.
A Philippe avec nos amitiées sincères
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References
M. Bertoldi, S. Fornaro, Gradient estimates in parabolic problems with unbounded coefficients, Studia Math. 165 (2004), 221–254.
M. Bertoldi, L. Lorenzi, Analytic methods for Markov semigroups, CRC Press / Chapman & Hall (2006), to appear.
S. Cerrai, Second-order PDE’s in finite and infinite dimensions. A probabilistic approach, Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin (2001).
G. Da Prato, A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition, J. Diff. Eqns. 198 (2004), 35–52.
M. Freidlin, Some remarks on the Smoluchowski-Kramers approximation, J. Statist. Phys. 117 (2004), 617–634.
A. Lunardi, Schauder theorems for elliptic and parabolic problems with unbounded coefficients in ℝn, Studia Math. 128 (1998), 171–198.
A. Lunardi, G. Metafune, D. Pallara, Dirichlet boundary conditions for elliptic operators with unbounded drift, Proc. Amer. Math. Soc. 133 (2005), 2625–2635.
G. Metafune, D. Pallara, M. Wacker, Feller Semigroups on ℝn, Sem. Forum 65 (2002), 159–205.
G. Metafune, E. Priola, Some classes of non-analytic Markov semigroups, J. Math. Anal. Appl. 294 (2004), 596–613.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Da Prato, G., Lunardi, A. (2006). Kolmogorov Operators of Hamiltonian Systems Perturbed by Noise. In: Koelink, E., van Neerven, J., de Pagter, B., Sweers, G., Luger, A., Woracek, H. (eds) Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol 168. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7601-5_4
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DOI: https://doi.org/10.1007/3-7643-7601-5_4
Publisher Name: Birkhäuser Basel
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