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33
Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in nondivergence form
 Advances in Differential Equations
, 2006
"... We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound i ..."
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Cited by 27 (4 self)
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We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound is obtained by solving a suitable optimal control problem and using the invariant Harnack inequality.
Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance
 NONLINEAR PROBLEMS IN MATHEMATICAL PHYSICS AND RELATED TOPICS VOL. II IN HONOR OF PROFESSOR O.A. LADYZHENSKAYA”. INTERNATIONAL MATHEMATICAL SERIES
, 2002
"... This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorovtype operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, ar ..."
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Cited by 24 (15 self)
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This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorovtype operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, are also announced.
On the Regularity of Solutions to a Nonlinear Ultraparabolic Equation Arising in Mathematical Finance
 in mathematical finance, Differential Integral Equations 14 (6
, 2001
"... We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:... ..."
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Cited by 19 (13 self)
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We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance:...
Regularity Properties of Viscosity Solutions of a NonHörmander Degenerate Equation
 J. Math. Pures Appl
, 2001
"... We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity ..."
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Cited by 17 (11 self)
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We study the interior regularity properties of the solutions of a nonlinear degenerate equation arising in mathematical finance. We set the problem in the framework of Hrmander type operators without assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions. 2001 ditions scientifiques et mdicales Elsevier SAS Keywords: Nonlinear degenerate Kolmogorov equation, Interior regularity, Hrmander operators RSUM.  Nous tudions la rgularit intrieure des solutions de viscosit d'une quation non linaire du second ordre dgnre que l'on rencontre en finance mathmatique. Nous tudions le problme par la thorie des oprateurs de Hrmander sans aucune hypothse sur la dgnerescence de l'algbre de Lie engendre. Nous montrons que la solution de viscosit est une solution classique. 2001 ditions scientifiques et mdicales Elsevier SAS 1.
The Moser's Iterative Method for a Class of Ultraparabolic Equations
 Commun. Contemp. Math
, 2004
"... We adapt the iterative scheme by Moser... ..."
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Optimal Blowup Rates for the Minimal Energy Null Control for the Structurally Damped Abstract Wave Equation
, 2002
"... The null controllability problem for a structurally damped abstract wave equation–a socalled elastic model–is considered with a view towards obtain optimal rates of blowup for the associated minimal energy function Emin(T), as terminal time T ↓ 0. Key use is made of the underlying analyticity of the ..."
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Cited by 8 (2 self)
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The null controllability problem for a structurally damped abstract wave equation–a socalled elastic model–is considered with a view towards obtain optimal rates of blowup for the associated minimal energy function Emin(T), as terminal time T ↓ 0. Key use is made of the underlying analyticity of the elastic generator A, as well as of the explicit characterization of its domain of definition. We ultimately find that the blowup rate for Emin(T), as T goes to zero, depends on the extent of structural damping. 1
Formulae for the derivatives of degenerate diffusion semigroups
 J. Evol. Equ
"... Abstract: We consider the diffusion semigroup Pt associated to a class of degenerate elliptic operators A on Rn. This class includes the hypoelliptic OrnsteinUhlenbeck operator but does not satisfy in general the well known Hörmander condition on commutators for sums of squares of vector fields. ..."
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Cited by 8 (3 self)
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Abstract: We consider the diffusion semigroup Pt associated to a class of degenerate elliptic operators A on Rn. This class includes the hypoelliptic OrnsteinUhlenbeck operator but does not satisfy in general the well known Hörmander condition on commutators for sums of squares of vector fields. We establish probabilistic formulae for the spatial derivatives of Ptf up to the third order. We obtain L∞estimates for the derivatives of Ptf and show the existence of a classical bounded solution for the parabolic Cauchy problem involving A and having f ∈ Cb(Rn) as initial datum. 1
Lunardi: Invariant measures and maximal L 2 regularity for nonautonomous OrnsteinUhlenbeck equations
 J. Lond. Math. Soc
"... Abstract. We characterize the domain of the realizations of the linear parabolic operator G defined by (1.4) in L 2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L 2 regularity results for evolution equations wi ..."
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Cited by 6 (1 self)
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Abstract. We characterize the domain of the realizations of the linear parabolic operator G defined by (1.4) in L 2 spaces with respect to a suitable measure, that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L 2 regularity results for evolution equations with timedepending OrnsteinUhlenbeck operators. 1.
A Gaussian upper bound for the fundamental solutions of a class of ultraparabolic equations
, 2003
"... We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations. ..."
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Cited by 5 (5 self)
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We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolic equations. These estimates are independent of the modulus of continuity of the coefficients and generalize the classical upper bounds by Aronson for uniformly parabolic equations.
Pointwise estimates for a class of nonhomogeneous Kolmogorov Equations
 MATHEMATISCHE ANNALEN
, 2006
"... We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously b ..."
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Cited by 5 (1 self)
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We consider a class of ultraparabolic differential equations that satisfy the Hörmander’s hypoellipticity condition and we prove that the weak solutions to the equation with measurable coefficients are locally bounded functions. The method extends the Moser’s iteration procedure and has previously been employed in the case of operators verifying a further homogeneity assumption. Here we remove that assumption by proving some potential estimates and some ad hoc Sobolev type inequalities for solutions.