
Sergio POLIDORO
Professore Ordinario Dipartimento di Scienze Fisiche, Informatiche e Matematiche sede exMatematica

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2023
 A Yosida's parametrix approach to Varadhan's estimates for a degenerate diffusion under the weak Hörmander condition
[Articolo su rivista]
Pagliarani, S.; Polidoro, S.
abstract
We adapt and extend Yosida's parametrix method, originally introduced for the construction of the fundamental solution to a parabolic operator on a Riemannian manifold, to derive Varadhantype asymptotic estimates for the transition density of a degenerate diffusion under the weak Hörmander condition. This diffusion process, widely studied by Yor in a series of papers, finds direct application in the study of a class of pathdependent financial derivatives known as Asian options. We obtain a Varadhantype formula which relates the transition density p of the stochastic process with the optimal cost Ψ of a deterministic control problem associated to the diffusion. We provide a partial proof of this formula, and present numerical evidence to support the validity of an intermediate inequality that is required to complete the proof. We also derive an asymptotic expansion of the cost function Ψ, expressed in terms of elementary functions, which is useful in order to design efficient approximation formulas for the transition density.
2023
 Hölder Continuity and Boundedness Estimates for Nonlinear Fractional Equations in the Heisenberg Group
[Articolo su rivista]
Manfredini, Maria; Palatucci, Giampiero; Piccinini, Mirco; Polidoro, Sergio
abstract
We extend the celebrate De GiorgiNashMoser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integrodifferential operators, whose model is the fractional pLaplacian operator on the HeisenbergWeyl group Hn. Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppolitype estimates with tail and logarithmictype estimates.
2023
 Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non‐smooth coefficients
[Articolo su rivista]
Malagoli, Emanuele; Pallara, Diego; Polidoro, Sergio
abstract
We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in divergence form. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C^1boundary, the other stated for sets with finite perimeter.
2022
 Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups
[Articolo su rivista]
Pallara, Diego; Polidoro, Sergio
abstract
We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.
2022
 Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients
[Articolo su rivista]
Polidoro, Sergio; Rebucci, Annalaura; Stroffolini, Bianca
abstract
We study the regularity properties of the second order linear degenerate parabolic operators. We prove that, if the operator Lsatisfies Hörmander’s hypoellipticity condition, and f is a Dini continuous function, then the second order derivatives of the solution u to the equation L u = f are Dini continuous functions as well. We also consider the case of Dini continuous coefficients of the secondo order derivatives. A key step in our proof is a Taylor formula for classical solutions to L u = f that we establish under minimal regularity assumptions on u.
2021
 A Compactness Result for the Sobolev Embedding via Potential Theory
[Capitolo/Saggio]
Camellini, Filippo; Eleuteri, Michela; Polidoro, Sergio
abstract
In this note we give a proof of the Sobolev and Morrey embedding theorems based on the representation of functions in terms of the fundamental solution of suitable partial differential operators. We also prove the compactness of the Sobolev embedding. We first describe this method in the classical setting, where the fundamental solution of the Laplace equation is used, to recover the classical Sobolev and Morrey theorems. We next consider degenerate Kolmogorov equations.
In this case, the fundamental solution is invariant with respect to a nonEuclidean translation group and the usual convolution is replaced by an operation that is defined in accordance with this geometry. We recover some known embedding results and we prove the compactness of the Sobolev embedding. We finally apply our regularity results to a kinetic equation.
2021
 Existence of a fundamental solution of partial differential equations associated to Asian options
[Articolo su rivista]
Anceschi, Francesca; Muzzioli, Silvia; Polidoro, Sergio
abstract
We prove the existence and uniqueness of the fundamental solution for Kolmogorov operators associated to some stochastic processes, that arise in the Black & Scholes setting for the pricing problem relevant to path dependent options. We improve previous results in that we provide a closed form expression for the solution of the Cauchy problem under weak regularity assumptions on the coefficients of the differential operator. Our method is based on a limiting procedure, whose convergence relies on some barrier arguments and uniform a priori estimates recently discovered.
2020
 A survey on the classical theory for Kolmogorov equation
[Articolo su rivista]
Polidoro, Sergio; Anceschi, Francesca
abstract
We present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a strong maximum principle.
2020
 Fundamental solutions for KolmogorovFokkerPlanck operators with timedepending measurable coefficients
[Articolo su rivista]
Bramanti, Marco; Polidoro, Sergio
abstract
We consider a degenerate KolmogorovFokkerPlanck operator in nondivergence form bounded measurable coefficients. We assume that the drift term is a linear function of the space variable that makes hypoelliptic the corresponding operator with constant coefficients. We construct an explicit fundamental solution Γ for L, study its properties, show a comparison result between Γ and the fundamental solution of some model operators with constant coefficients, and show the unique solvability of the Cauchy problem for L under various assumptions on the initial datum.
2020
 Gaussian lower bounds for nonhomogeneous Kolmogorov equations with measurable coefficients
[Articolo su rivista]
Lanconelli, Alberto; Pascucci, Andrea; Polidoro, Sergio
abstract
We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hörmander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations.
2019
 A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients
[Articolo su rivista]
Anceschi, Francesca; Eleuteri, Michela; Polidoro, Sergio
abstract
We consider weak solutions of secondorder partial differential equations of KolmogorovFokkerPlancktype with measurable coefficients in the form ∂tu + (v,∇xu) = div(A(v,x,t)∇vu) + (b(v,x,t),∇vu) + f, (v,x,t) ϵ2n+1, where A is a symmetric uniformly positive definite matrix with bounded measurable coefficients; f and the components of the vector b are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by Golse et al. As a corollary, we obtain a strong maximum principle.
2019
 Moser's estimates for degenerate Kolmogorov equations with nonnegative divergence lower order coefficients
[Articolo su rivista]
Anceschi, F.; Polidoro, S.; Ragusa, M. A.
abstract
We prove Lloc∞ estimates for positive solutions to the following degenerate second order partial differential equation of Kolmogorov type with measurable coefficients of the form ∑i,j=1m0∂xjavax.xml.bind.JAXBElement@14c7905daij(x,t)∂xjavax.xml.bind.JAXBElement@41a3b5feu(x,t)+∑i,j=1Nbijxj∂xjavax.xml.bind.JAXBElement@21dceba6u(x,t)−∂tu(x,t)++∑i=1m0bi(x,t)∂iu(x,t)−∑i=1m0∂xjavax.xml.bind.JAXBElement@638b72d3ai(x,t)u(x,t)+c(x,t)u(x,t)=0 where (x,t)=(x1,…,xN,t)=z is a point of RN+1, and 1≤m0≤N. (aij) is a uniformly positive symmetric matrix with bounded measurable coefficients, (bij) is a constant matrix. We apply the Moser's iteration method to prove the local boundedness of the solution u under minimal integrability assumption on the coefficients.
2019
 Sharp Estimates for Geman–Yor Processes and applications to Arithmetic Average Asian options
[Articolo su rivista]
Cibelli, Gennaro; Polidoro, Sergio; Rossi, Francesco
abstract
We prove the existence of the fundamental solution of the degenerate second order partial differential equation related to Geman–Yor stochastic processes, that arise in models for option pricing theory in finance. We then prove pointwise lower and upper bounds for such fundamental solution. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. Upper bounds are obtained by the fact that the cost satisfies a specific Hamilton–Jacobi–Bellman equation.
2018
 L’equazione di Laplace: Una riflessione storicoepistemologica
[Articolo su rivista]
Asenova, Miglena; Polidoro, Sergio
abstract
The equation that at present time is known as “Laplace’s equation” achieved considerable visibility thanks to the publication of the famous work of PierreSimon Laplace “Traité de Mécanique Céleste” (1799). Laplace is acknowledged as the developer of an analytical theory to deal with problems of astronomy understood as “celestial mechanics”. In this context, the equation models
the problem of gravitational attraction that a spheroid exerts on a generic material point. However, the equation was already known to Leonhard Euler, who had obtained it in 1752 in a work in which he describes the motion of an incompressible fluid. Adopting an epistemological perspective and comparing the contributions of Euler and Laplace, in this article we discuss the question of whether it is correct to associate only Laplace’s name with the equation we are considering.
2017
 Harnack inequalities and Bounds for Densities of Stochastic Processes
[Relazione in Atti di Convegno]
Cibelli, Gennaro; Polidoro, Sergio
abstract
We consider possibly degenerate parabolic operators in the form of "sum of squares of vector fields plus a drif term" that are naturally associated to a suitable family of stochastic differential equations, and satisfying the Hörmander condition. Note that, under this assumption, the operators considered have a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods to prove analogous upper bounds. We eventually give a list of meaningful examples of operators to which the method applies.
2016
 Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators
[Articolo su rivista]
Kogoj, Alessia E; Polidoro, Sergio
abstract
We consider nonnegative solutions (Formula presented.) of second order hypoelliptic equations(Formula presented.) where Ω is a bounded open subset of (Formula presented.) and x denotes the point of Ω. For any fixed x0 ∈ Ω, we prove a Harnack inequality of this type(Formula presented.) where K is any compact subset of the interior of the (Formula presented.)propagation set ofx0 and the constant CK does not depend on u.
2016
 KolmogorovFokkerPlanck equations: Comparison principles near Lipschitz type boundaries
[Articolo su rivista]
Nyström, K; Polidoro, Sergio
abstract
We prove several new results concerning the boundary behavior of nonnegative solutions to the equation Ku=0, where. K:=∑i=1m∂xixi+∑i=1mxi∂yi∂t. Our results are established near the noncharacteristic part of the boundary of certain local LipKdomains, where the latter is a class of local Lipschitz type domains adapted to the geometry of K. Generalizations to more general operators of KolmogorovFokkerPlanck type are also discussed. On démontre plusieurs nouveaux résultats sur le comportement au bord des solutions non négatives de l'équation Ku=0, où. K:=∑i=1m∂xixi+∑i=1mxi∂yi∂t. Les résultats sont établis dans un voisinage de la partie noncaractéristique du bord de certains domaines locaux LipK, qui sont des domains localement lipschitziens adaptés à la géométrie de K. On discute aussi des généralisations à d'autres opérateurs plus généraux de type KolmogorovFokkerPlanck.
2016
 On Liouvilletype theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators
[Articolo su rivista]
Kogoj, Alessia E.; Pinchover, Yehuda; Polidoro, Sergio
abstract
The paper contains a representation formula for positive solutions of linear degenerate secondorder equations of the form of "sum of squares of vector fields plus a drift term" where the vector fields X_j's satisfy the Hörmander condition. It is assumed that X_j's are invariant under left translations of a Lie group and the corresponding paths satisfy a local admissibility criterion. The representation formula is established by an analytic approach based on Choquet theory. As a consequence we obtain Liouvilletype theorems and uniqueness results for the positive Cauchy problem.
2015
 Geometric Methods in PDE’s
[Curatela]
Citti, Giovanna; Manfredini, Maria; Morbidelli, Daniele; Polidoro, Sergio; Uguzzoni, Francesco
abstract
The analysis of PDEs is a prominent discipline in mathematics research, both in terms of its theoretical aspects and its relevance in applications. In recent years, the geometric properties of linear and nonlinear second order PDEs of elliptic and parabolic type have been extensively studied by many outstanding researchers. This book collects contributions from a selected group of leading experts who took part in the INdAM meeting "Geometric methods in PDEs", on the occasion of the 70th birthday of Ermanno Lanconelli. They describe a number of new achievements and/or the state of the art in their discipline of research, providing readers an overview of recent progress and future research trends in PDEs. In particular, the volume collects significant results for subelliptic equations, potential theory and diffusion equations, with an emphasis on comparing different methodologies and on their implications for theory and applications.
2015
 The role of fundamental solution in Potential and Regularity Theory for subelliptic PDE
[Capitolo/Saggio]
Bonfiglioli, Andrea; Citti, Giovanna; Cupini, Giovanni; Manfredini, Maria; Montanari, Annamaria; Morbidelli, Daniele; Pascucci, Andrea; Polidoro, Sergio; Uguzzoni, Francesco
abstract
In this survey we consider a general Hormander type operator, represented as a sum of squares of vector fields plus a drift and we outline the central role of the fundamental solution in developing Potential and Regularity Theory for solutions of related PDEs. After recalling the Gaussian behavior at infinity of the kernel, we show some mean value formulas on the level sets of the fundamental solution, which are the starting point to obtain a comprehensive parallel of the classical Potential Theory. Then we show that a precise knowledge of the fundamental solution leads to global regularity results, namely estimates at the boundary or on the whole space. Finally in the problem of regularity of non linear differential equations we need an ad hoc modification of the parametrix method, based on the properties of the fundamental solution of an approximating problem.
2015
 Twosided bounds for degenerate processes with densities supported in subsets of R^N
[Articolo su rivista]
Chiara, Cinti; Stephane, Menozzi; Polidoro, Sergio
abstract
We obtain twosided bounds for the density of stochastic processes satisfying a weak H"ormander condition. In particular we consider the cases when the support of the density is not the whole space and when the density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in H"ormander's theorem).
The proofs of our lower bounds are based on Harnack inequalities for positive solutions of PDEs whereas the upper bounds are derived from the probabilistic representation of the density given by the Malliavin calculus.
2014
 Multiplicity of solutions for laminar, fullydeveloped natural convection in inclined, parallelplate channels
[Articolo su rivista]
Piller, Marzio; Polidoro, Sergio; Stalio, Enrico
abstract
Natural convection in inclined channels is a rather common flow configuration: it occurs in solar energy systems, ventilated roofs as well as in many industrial applications and chemical processes. Analytical solutions for laminar, fullydeveloped natural convection in inclined parallelplate channels are presented in this paper. The Boussinesq approximation is applied and viscous energy dissipation is neglected. One specific thermal configuration is addressed, where one wall is perfectly insulated and a constant, uniform heat flux is released to the fluid from the other wall. The resulting set of governing equations is nonlinear, as the mean velocity is not assigned a priori but determined as part of the solution. Depending on the channel inclination angle and on the imposed heat flux conditions, either no solution, one solution, multiple or infinite solutions exist. Under restrictive assumptions velocity profiles are selfsimilar with respect to the channel inclination, while the temperature profile is independent of the inclination. The twodimensional, hydraulically and thermallydeveloping natural convection channel flow is simulated numerically for some combinations of channel inclination angle and heating intensity to identify the most physical between the many solutions.
2013
 A Carlesontype estimate in Lipschitz type domains for nonnegative solutions to Kolmogorov operators
[Articolo su rivista]
C., Cinti; K., Nystrom; Polidoro, Sergio
abstract
We prove bondary Harnack estimates, in Lipschitz domains, for nonnegative solutions to a class of second order degenerate dierential operators of Kolmogorov type. Our estimate is scaleinvariant and generalizes previous results, valid for second order uniformly parabolic equations, to the class of operators considered.
2013
 A Revised Approach for OneDimensional TimeDependent Heat Conduction in a Slab
[Articolo su rivista]
Caffagni, A.; Angeli, Diego; Barozzi, Giovanni Sebastiano; Polidoro, Sergio
abstract
Classical Green’s and Duhamel’s integral formulas are enforced for the solution of one
dimensional heat conduction in a slab, under general boundary conditions of the first
kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.
2013
 Geometric Methods in PDE ́s: I.N.d.A.M. Meeting on the occasion of the 70 th birthday of Ermanno Lanconelli, Cortona, 2731 maggio 2013,
[Altro]
Bonfiglioli, Andrea; Citti, Giovanna; Cupini, Giovanni; Manfredini, Maria; Montanari, Annamaria; Morbidelli, Daniele; Pascucci, Andrea; Polidoro, Sergio; Uguzzoni, Francesco
abstract
The scope of this conference is to celebrate the 70th birthday of Ermanno Lanconelli and to bring together Italian and foreign Mathematicians to favour the discussion in the areas of research where Ermanno Lanconelli has been particularly active:
 Second order linear and nonlinear partial differential equations with nonnegative characteristic form;
 Geometric problems related to the underlying algebraic, geometrical or topological structure;
 Application to complex geometry and CR manifolds.
These fields are the objects of active research and development, and possess a remarkable degree of interrelation in their pure and applied aspects.
2012
 A boundary estimate for nonnegative solutions to Kolmogorov operators in nondivergence form
[Articolo su rivista]
C., Cinti; K., Nystrom; Polidoro, Sergio
abstract
We consider nonnegative solutions to a class of second order degenerate Kolmogorov operators L in nondivergence form, defined in a bounded open domain Omega contained in R^{N+1}. Let K be a compact subset of the closure of Omega, let z be a point of Omega, and let Sigma be a subset of the boundary of Omega. We give sufficient geometric conditions for the validity of the following Carleson type estimate: There exists a positive constant C, depending only on the Kolmogorov operator L, on Omega, Sigma, K and z, such that sup_K u < C u(z), for every nonnegative solution u of Lu = 0 in Omega such that u vanishes on Sigma.
2010
 A note on Harnack inequalities and propagation set for a class of hypoelliptic operators
[Articolo su rivista]
C., Cinti; K., Nystrom; Polidoro, Sergio
abstract
We consider a class of second order hypoelliptic ultraparabolic partial differential equations, a typical example of which is the FokkerPlank equation. We find a sufficient condition on compact sets K such that Harnack's inequality holds on K for all nonnegative solutions. This condition is geometric and it is described in terms of paths connecting couples of points.The proof uses a local invariant Harnack inequality proved by Kogoj and Lanconelli on cylinders and a construction with chains ofcylinders. These results are related with the corresponding ones appearing in potential theory. It is also shown for an explicit operator of the type considered that the condition on the compact sets is optimal.
2010
 Meeting "Kolmogorov Equations in Physics and Finance"
[Altro]
A., Pascucci; Polidoro, Sergio
abstract
Probabilistic and analytical methods are fundamental in the modeling of physical and natural phenomena and in the description of financial markets.The purpose of this meeting is to highlight the new methods, directions and the most recent developments in the theories of Probability and Partial Differential Equations. Special emphasis will be placed on applications to Physics and Mathematical Finance. http://kolmogorov2010.dm.unibo.it/
2010
 Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options
[Articolo su rivista]
M., Frentz; K., Nystrom; A., Pascucci; Polidoro, Sergio
abstract
In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstaclesas well as nonsmooth obstacles. All our proofs follow the same line of thought and are based on blowups, compactness, barriers and arguments by contradiction. This problem arises in financial mathematics, when considering pathdependent derivative contracts with the early exercise feature.
2010
 Regularity near the Initial State in the Obstacle Problem for a class of Hypoelliptic Ultraparabolic Operators
[Articolo su rivista]
K., Nystrom; A., Pascucci; Polidoro, Sergio
abstract
This paper is devoted to a proof of regularity, near the initial state, for solutions to the CauchyDirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smoothobstacles as well as nonsmooth obstacles.
2009
 Bounds on short cylinders and uniqueness in Cauchy problem for degenerate Kolmogorov equations
[Articolo su rivista]
C., Cinti; Polidoro, Sergio
abstract
We consider the Cauchy problem for degenerate Kolmogorov equations in nondivergence form, as well as in its divergence form. We prove that a very mild growth condition of the solution yields the uniqueness.
2009
 Harnack inequality and noarbitrage bounds for selffinancing portfolios
[Articolo su rivista]
A., Carciola; A., Pascucci; Polidoro, Sergio
abstract
We give a direct proof of the Harnack inequality for a class ofKolmogorov operators associated with a linear SDE and we find theexplicit expression of the optimal Harnack constant. We discuss somepossible implication of the Harnack inequality in Finance: specificallywe infer noarbitrage bounds for the value of selfFinancing portfoliosin terms of the initial wealth.
2009
 Non local Harnack inequalities for a class of partial differential equaitons
[Relazione in Atti di Convegno]
U., Boscain; Polidoro, Sergio
abstract
We prove Gausian lower bounds for the Fundamental solution to a class of hypoelliptic PDE's. Our method relies on the repeated application of a Harnack inequality which is invariant woth respect to a suitable Lie group.
2009
 On Some Schroedinger type equations
[Relazione in Atti di Convegno]
Polidoro, Sergio; M. A., Ragusa
abstract
We prove a Harnack type inequality for positive solutions to the equation L u + V u = 0, where L is a degenerate Kolmogorov equation and V is a potential belonging to a StummelKato class
2008
 Geometric Methods in PDE's: a conference on the occasion of the 65th birthday of Ermanno Lanconelli
[Altro]
Citti, G; Dragomir, S; Franchi, B; Gutierrez, C; Montanari, A; Pascucci, A; Polidoro, Sergio
abstract
The scope of this conference is o celebrate the 65th birthday of Ermanno Lanconelli and to bring together Italian and foreign Mathematicians to promote the discussion in the areas of research where Ermanno Lanconelli has been particularly active:second order linear and nonlinear partial differential equations with nonnegative characteristic form; geometric problems related to the underlying algebraic, geometrical or topological structure; application to complex geometry and CR manifoldsThese fields are the objects of active research and development, and possess a remarkable degree of interrelation in their pure and applied aspects. In particular they find natural applications in mathematical finance, in the description of the visual cortex and in image processing.
2008
 Geometric Methods in PDE's: a conference on the occasion of the 65th birthday of Ermanno Lanconelli
[Curatela]
G., Citti; A., Montanari; A., Pascucci; Polidoro, Sergio
abstract
The scope of this conference has been to celebrate the 65th birthday of Ermanno Lanconelli and to bring together italian and foreign Mathematicians to promote the discussion in the areas of research where Ermanno Lanconelli has been particularly active: Second order linear and nonlinear partial differential equations with nonnegative characteristic form Geometric problems related to the underlying algebraic, geometrical or topological structure Application to complex geometry and CR manifoldsThese fields are the objects of active research and development, and possess a remarkable degree of interrelation in their pure and applied aspects. In particular they find natural applications in mathematical finance, in the description of the visual cortex and in image processing.
2008
 Harnack inequalities and Lifting Procedure for Evolution Hypoelliptic Equations
[Relazione in Atti di Convegno]
C., Cinti; Polidoro, Sergio
abstract
We consider, for any odd positive integer k, the degenerate Partial Differential Equationu_t = u_xx + x^k u_yand we prove a Harnack inequality which is expressed in terms of the integral curves of the vector fields that occur in the PDE. The novelty of our result is in that, as k>1, we cannot assume the existence of any Lie group in R^3 such that the vector fields are invariant. As an application of the Harnack inequality we prove a lower bond of the fundamental solution.
2008
 Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term
[Articolo su rivista]
Polidoro, Sergio; Ragusa, M. A.
abstract
We prove a Harnack inequality for the positive solutions of ultraparabolic equations of the type L u + V u = 0;where L is a linear second order hypoelliptic differential operator and V belongs to a class of functions of StummelKato type. We also obtain the existence of a Green function and an uniqueness result for the CauchyDirichlet problem.
2008
 Pointwise estimates for a class of nonhomogeneous Kolmogorov equations
[Articolo su rivista]
C., Cinti; A., Pascucci; Polidoro, Sergio
abstract
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 typeinequalities for solutions.
2008
 Pointwise local estimates and Gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators
[Articolo su rivista]
C., Cinti; Polidoro, Sergio
abstract
We consider a set of smooth vector fields X_1,…,X_m and X0−∂t satisfying the Hoermander's hyipoellipticity condition, under the assumption that X_1,…,X_m and X0−∂t are invariant with respect to a suitable homogeneous Lie group. We consider the second order partial differential equations in divergence form, X_i (aij X_j) + X0−∂t, where A=(aij) is a bounded, symmetric and uniformly positive matrix with measurable coefficients, and we prove an L^infty source bound of the solution u in terms of its L^1 norm, by adaptingt the Moser's iterative methods to the nonEuclidean geometry of the Lie group.We then use a technique going back to Aronson to prove a pointwise upper bound of the fundamental solution of the operator X_i (aij X_j) + X0−∂t. The bound is given in terms of the value function of an optimal control problem related to the vector fields X_1,…,X_m and X0−∂t. Finally, by using the upper bound, the existence of a fundamental solution is then established for smooth coefficients aij.
2008
 The obstacle problem for a class of hypoelliptic ultraparabolic equations
[Articolo su rivista]
Marco Di, Francesco; Andrea, Pascucci; Polidoro, Sergio
abstract
We prove that the obstacle problem for a nonuniformly parabolic operator of Kolmogorov type, with Cauchy (or CauchyDirichlet) boundary conditions, has a unique strong solution u. We also show that u is a solution in the viscosity sense.
2007
 Gaussian estimates for hypoelliptic operators via optimal control
[Articolo su rivista]
Ugo, Boscain; Polidoro, Sergio
abstract
We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the valuefunction of a suitable optimal control problem.
2007
 Lower bounds for solutions of degenerate parabolic equations.
[Relazione in Atti di Convegno]
Ugo, Gianazza; Polidoro, Sergio
abstract
We show how pointwise lower bounds for positive weak solutions of degenerate parabolic equations can be derived from the intrinsic Harnack inequality they satisfy. This generalizes a result proved by Moser for linear parabolic equations with bounded and measurable coefficients.
2007
 Meeting on Subelliptic PDE’s and Applications to Geometry and Finance
[Curatela]
Ermanno, Lanconelli; Annamaria, Montanari; Polidoro, Sergio
abstract
The aim of the meeting was to put PDE's specialists together with experts in complex variables and stochastic theory, in order to encourage comparison between different research methods, to present the most recent and important advances, together with perspectives and applications of research. Several speakers gave their contribution in different research topics, and the common feature of the talks was the analysis on analogous problems in Riemannian and Subriemaniann framework. New ideas and results have been stimulated by the different research methods.
2006
 Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
[Articolo su rivista]
Andrea, Pascucci; Polidoro, Sergio
abstract
We prove a nonlocal Harnack inequality for a class of degenerateevolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.
2006
 Meeting on Subelliptic PDE's and applications to Geometry and Finance
[Altro]
E., Lanconelli; A., Montanari; Polidoro, Sergio
abstract
The aim of the meeting is to put PDE's specialists together with experts in complex variables and stochastic theory, in order to encourage comparison between different research methods, to present the most recent and important advances, together with perspectives and applications of research. Several speakers gave their contribution in different research topics, and the common feature of the talks was the analysis on analogous problems in Riemannian and Subriemaniann framework. New ideas and results have been stimulated by the different research methods.
2006
 Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in nondivergence form
[Articolo su rivista]
M., DI FRANCESCO; Polidoro, Sergio
abstract
We prove 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.
2005
 Exponential decay for Maxwell equations with a boundary memory condition
[Articolo su rivista]
R., Nibbi; Polidoro, Sergio
abstract
We study the asymptotic behavior of the solution of the Maxwell equations with a boundary condition of memory type. We consider a `Graffi' type free energy and we prove that, if the memory kernel satisfies an exponential decay condition, and the domain is strongly star shaped, then the energy of the solution exponentially decays. We also prove that the exponential decay of the kernel is a necessary condition for the exponential decay of the solution.
2005
 Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
[Relazione in Atti di Convegno]
Polidoro, Sergio
abstract
We announce some results obtained in the recent paper“Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators”, concerning a general class of hypoelliptic evolution operators in R^{N+1}. A Gaussian lower bound for the fundamental solution and a global Harnack inequality are given.
2004
 On the Harnack inequality for a class of hypoelliptic evolution equations
[Articolo su rivista]
Pascucci, A.; Polidoro, Sergio
abstract
We give a direct proof of the Harnack inequality for a class ofdegenerate evolution operators which contains the linearized prototypes of the Kolmogorov and FokkerPlanck operators. We also improve the known results in that we find the optimal constant of the inequality.
2004
 Recent results on Kolmogorov equations and applications.
[Relazione in Atti di Convegno]
Polidoro, Sergio
abstract
The paper contains a survey of results on a class of linear and non linear Kolmogorovtype operators. Applications to Finance are discussed in details.
2004
 The Moser's iterative method for a class of ultraparabolic equations
[Articolo su rivista]
Pascucci, A.; Polidoro, Sergio
abstract
We adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solutions
2003
 A Gaussian upper bound for the fundamental solution of a class of ultraparabolic equations
[Articolo su rivista]
Pascucci, A.; Polidoro, Sergio
abstract
We prove Gaussian estimates from above of the fundamental solutions to a class of ultraparabolicequations. These estimates are independent of the modulus of continuity of the coefficients and generalizethe classical upper bounds by Aronson for uniformly parabolic equations.
2003
 A nonlinear PDE in mathematical finance
[Abstract in Atti di Convegno]
Polidoro, Sergio
abstract
We study a nonlinear degenerate partial Differential Equation arising in Mathematical Finance. We prove the existence of a solution to the Cauchy problem by using the PDE's regularity theory on Lie groups
2003
 On the Cauchy problem for a non linear Kolmogorov equation
[Articolo su rivista]
Pascucci, A.; Polidoro, Sergio
abstract
We consider the Cauchy problem related to the partial differential equationLu ≡ Δ_x u + h(u)∂_y u − ∂_t u = f(·, u),where (x, y, t) ∈ R^N × R × ]0, T[, which arises in mathematical finance and in the theory of diffusion processes. We study the regularity of solutions regarding L as a perturbation of an operatorof Kolmogorov type. We prove the existence of local classical solutions and give some sufficient conditions for global existence.
2002
 A Green function and regularity results for an ultraparabolic equation with a singular potential
[Articolo su rivista]
Polidoro, Sergio; Ragusa, M. A.
abstract
We prove a Harnack inequality for the positive solutions ofa Schroedinger type equationL_0 u + V u = 0;where L_0 is an operator satisfying the Hoermander's condition and V belongs to a class of functions of StummelKato type. We also obtain the existence of a Green function and an uniqueness result for the CauchyDirichlet problem.
2002
 Asymptotic decay for some differential system with fading memory
[Articolo su rivista]
Fabrizio, M.; Polidoro, Sergio
abstract
We study the large time behavior of the solution u to an initial andboundary value problem related to an integrodifferential equationwith a damping term. We prove that the solution u exponentially decays only if the relaxation kernel does.
2002
 Counterexample to the exponential decay for systems with memory
[Abstract in Atti di Convegno]
Polidoro, Sergio
abstract
We prove the asymptotic devcay of the solution u to an initial andboundary value problem related to an integrodiffeerentialequation with a memory kernel. We prove that the exponential decay may occur only if the memory kernel exponentially decays.
2002
 Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance
[Capitolo/Saggio]
Ermanno, Lanconelli; Andrea, Pascucci; Polidoro, Sergio
abstract
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 propertiesand pointwise estimates of solutions, are also announced.
2001
 Hölder regularity for solutions of an ultraparabolic equations in divergence form
[Articolo su rivista]
Polidoro, Sergio; Ragusa, M. A.
abstract
We prove the Holder regularity for the solutions to Kolmogorov Partial Differential Equations in divergence form. The coefficients of the second order part of the operator is assumed in the Vanishing Mean Oscillation space.
2001
 On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance
[Articolo su rivista]
Polidoro, Sergio
abstract
We consider the Cauchy problem relate to a nonlinear degenerate partial differential equation. We give sufficient conditions for the existence of a regular classical solution
2001
 Regularity properties of viscosity solutions of a nonHörmander degenerate equation
[Articolo su rivista]
Citti, G.; Pascucci, A.; Polidoro, Sergio
abstract
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 Hörmander type operatorswithout assuming any hypothesis on the degeneracy of the associated Lie algebra. We prove that the viscosity solutions are indeed classical solutions.
2001
 Some Results on Partial Differential Equations and Asian Options
[Articolo su rivista]
Barucci, E.; Polidoro, Sergio
abstract
We consider the Partial Differential Equation describing the price of an Asian Options in the Black & Scholes model. We prove the existence, the uniqueness and the regularity of the solution. We give explicit and implicit numerical methods to construct numerical solutions
1998
 Interior regularity for weak solutions of ultraparabolic equations in divergence form with discontinuous coefficients
[Articolo su rivista]
Maria, Manfredini; Polidoro, Sergio
abstract
We prove some interior regularity results for the weak solutions of Kolmogorov equations in divergence form
1998
 SobolevMorrey spaces related to an ultraparabolic equation
[Articolo su rivista]
Polidoro, Sergio; Maria Alessandra, Ragusa
abstract
We prove interior regularity results for strng solutions to Kolmogorov equations in nondivergence form
1997
 A global lower bound for the fundamental solution of KolmogorovFokkerPlanck equations
[Articolo su rivista]
Polidoro, Sergio
abstract
We prove a lower bound for the fundamental solution to a class of Kolmogorov equations. Our method relies on an invariant Harnack inequality
1995
 A finite difference method for a boundary value problem related to the Kolmogorov equation
[Articolo su rivista]
Polidoro, Sergio; C., Mogavero
abstract
We prove stability, consistence and convergence for an implicit and an explicite numerical scheme for a class of Kolmogorov equations
1995
 Uniqueness and representation theorems for solutions of KolmogorovFokkerPlanck equations
[Articolo su rivista]
Polidoro, Sergio
abstract
We prove the uniqueness of the solution of the Cauchy problem for a class of Kolmogorov equations
1994
 On a class of hypoelliptic evolution operators
[Articolo su rivista]
E., Lanconelli; Polidoro, Sergio
abstract
We show that Hypoelliptic Kolmogorov equations are invariant with respect to a suitable Lie group structure. We prove a Harnack inequality which is invariant with respet to the Lie group structure. We give a characterization of the Kolmogorov operators that are related to a homogeneous Lie group
1994
 On a class of ultraparabolic operators of KolmogorovFokkerPlanck type
[Articolo su rivista]
Polidoro, Sergio
abstract
We prove the existence of a fundamental solution for a Kolmogorov equation with Holder continuous coefficients. We also prove some mean value formulas and an invariant Hanrnack inequality
1993
 Nontrivial solutions for MongeAmp\`ere type operators in convex domains
[Articolo su rivista]
Bruno, Franchi; Nicolai, Kutev; Polidoro, Sergio
abstract
We prove some existence results for MongeAmpère type PDEs
1992
 Bounded global solutions for a class of elliptic quasilinear equations
[Articolo su rivista]
Polidoro, Sergio
abstract
We prove the existence of positive solutions by nonvariational methods
1991
 Existence of positive solutions of quasilinear elliptic equations through nonvariational methods
[Articolo su rivista]
Polidoro, Sergio
abstract
We prve the existence of a positive solution by a nonvarational method.