Bounds on short cylinders and uniqueness results for degenerate Kolmogorov equation

We consider the Cauchy problem for hypoelliptic Kolmogorov equations in both divergence and non divergence form. We prove that, if |u(x,t)| < M exp(a(t^{-\beta}+|x|^2)) for some positive constants a, M, \beta in ]0,1[ and u(x,0) = 0, then u(x,t) = 0 for positive t. The proof of the main result is based on some previous uniqueness result and on the application of some estimates in short cylinders, first introduced by Safonov in the study of uniformly parabolic operators

Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term

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 operator and V belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem

Harnack inequality and no-arbitrage bounds for self-financing portfolios

We give a direct proof of the Harnack inequality for a class of Kolmogorov operators associated with a linear SDE and we find the explicit expression of the optimal Harnack constant. We discuss some possible implication of the Harnack inequality in finance: specifically we infer no-arbitrage bounds for the value of self-financing portfolios in terms of the initial wealth.Harnack inequality; no-arbitrage principle; self-financing portfolio; Kolmogorov equation; linear stochastic equation

A survey on the classical theory for Kolmogorov equation

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

A survey on the classical theory for Kolmogorov equation

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

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

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

Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators

We consider non-negative solutions (Formula presented.) of second order hypoelliptic equations(Formula presented.) where \u3a9 is a bounded open subset of (Formula presented.) and x denotes the point of \u3a9. For any fixed x0 08 \u3a9, 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

Gaussian lower bounds for non-homogeneous Kolmogorov equations with measurable coefficients

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

Disuguaglianze di Harnack per operatori di evoluzione ipoellittici: aspetti geometrici ed applicazioni

We consider linear second order Partial Differential Equations in the form of "sum of squares of Hörmander vector fields plus a drift term" on a given domain. We prove that an Harnack inequality holds for every compact subset of the interior of the attainable set defined in terms of the vector fields that define the Partial Differential Equation considered. We then ally Harnack's inequalities to prove asymptotic lower bounds of the joint density of a wide class of stochastic processes. Analogous upper bound are proved by Mallilavin's calculus.Consideriamo Equazioni alle Derivate Parziali lineari del secondo ordine in forma di "somma di quadrati di campi vettoriali di Hörmander piu un termine di drift" in un dominio assegnato. Dimostriamo che una disuguaglianza di Harnack vale in ogni sottoinsieme compatto dell'insieme raggiungibile denito in termini dei compi vettoiali che definiscono l'Equazione alle Derivate Parziali considerata. Applichiamo quindi le disuguaglianze di Harnack per dimostrare stime asintotiche dal basso per la densità congiunta di un'ampia classe di processi stocastici. Analoghe stime dall'alto sono dimostrate per mezzo del Calcolo di Malliavin