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A survey on the classical theory for Kolmogorov equation
Full text linkWe 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
Sull'equazione di Kolmogorov: teoria della regolarità ed applicazioni
No full textL’equazione di Kolmogorov è stata introdotta nel 1934 come ingrediente fondamentale di un modello cinetico per lo studio della densità di un sistema di N particelle di gas nello spazio delle fasi. Kolmogorov osservò che tale operatore è fortemente degenere in quanto la dimensione dello spazio delle fasi è 2N, mentre il termine di diffusione agisce sulla variabile velocità di dimensione N. Nonostante ciò, egli fornì l’espressione esplicita della soluzione fondamentale per tale operatore, una funzione differenziabile infinite volte, così dimostrando che l’operatore è ipoellittico. Nella mia tesi mi occupo prevalentemente di equazioni di Kolmogorov degeneri in forma di divergenza, per le quali la teoria della regolarità classica è stata ampiamente sviluppata nel corso degli anni e viene ad oggi considerata completa. Nel Capitolo 1 presento i principali risultati di tale teoria per operatori di Kolmogorov a coefficienti costanti o continui. Nel Capitolo 2 considero un’applicazione dell’equazione di Kolmgorov in ambito finanziario, dove la teoria di Black & Scholes si applica al pricing problem per le opzioni Asiatiche. Il prezzo di un’opzione si calcola risolvendo un problema di Cauchy, il cui dato iniziale rappresenta il payoff dell’opzione e la EDP associata è un’equazione di tipo Kolmogorov a coefficienti localmente Hölderiani. Attraverso una procedura di limite, la cui convergenza è assicurata da stime di tipo Schauder, si dimostrano l’esistenza e l’unicità della soluzione per l’ODP associato ed un risultato di unicità per la soluzione del problema di Cauchy. Nel Capitolo 3 considero un’ulteriore applicazione dell’equazione di Kolmogorov alla teoria cinetica. In particolare, introduco un modello cinetico non omogeneo associato ad un operatore non lineare di tipo Kolmogorov-Fokker-Planck (KFP) e studio la teoria della regolarità classica per il problema di Cauchy associato.
La seconda parte della mia tesi è dedicata alla teoria della regolarità per soluzioni deboli dell’equazione di Kolmogorov a coefficienti misurabili, argomento su cui è prevalentemente concentrata la comunità scientifica oggigiorno. Gli sviluppi più recenti in questa direzione sono stati ottenuti nel caso particolare dell'equazione di KFP. Nel Capitolo 4 dimostro un enunciato di tipo geometrico per la disuguaglianza di Harnack provata da Golse, Imbert, Mouhot e Vasseur nel 2017 per le soluzioni deboli dell’equazione di KFP a coefficienti misurabili, basandomi sul concetto di catene di Harnack e insieme ammissibile. Per quanto riguarda invece l’equazione di Kolmogorov in forma di divergenza nella sua forma più generale, il Capitolo 5 è dedicato all’estensione dell’iterazione di Moser alle soluzioni deboli per l’equazione di Kolmogorov sotto ipotesi minimali di integrabilità per i coefficienti di ordine inferiore nel caso non invariante per dilatazioni.The Kolmogorov equation was firstly introduced in 1934 as a fundamental ingredient of a kinetic model for the study of the density of a system of N particles of gas in the phase space. Kolmogorov pointed out that, although the dimension of the phase space is 2N and the diffusion term acts on the velocity variable, whose dimension is N, the differential operator is strongly degenerate. Nevertheless, Kolmogorov exhibited the explicit expression of the fundamental solution of the operator and pointed out that it is a smooth function, in fact proving that the operator is hypoelliptic. Throughout this work, we are mainly concerned with degenerate Kolmogorov equations in divergence form, for which the regularity theory for classical solutions had widely been developed during the years. Chapter 1 of this work is devoted to a survey of results on the classical regularity theory for Kolmogorov operators with constant or continuous coefficients, which can nowadays be considered complete. In Chapter 2 we consider an application of the Kolmogorov equation in finance, where the Black and Scholes theory is applied to the pricing problem for Asian options. The price of the option is computed by solving a Cauchy problem, where the initial data represents the payoff of the option and the associated PDE is a Kolmogorov type equation with local Hölder continuous coefficients. The existence and uniqueness of the fundamental solution of the associated PDO are proved, alongside with a uniqueness result for the solution of the Cauchy problem, through a limiting procedure whose convergence is ensured by Schauder type estimates. Furthermore, in Chapter 3 we consider an application of the Kolmogorov equation to the kinetic theory. Specifically, we introduce a space inhomogeneous kinetic model associated to a nonlinear Kolmogorov-Fokker-Planck (KFP) operator and we investigate the classical theory for the associated Cauchy problem.
The second part of my thesis is devoted to the regularity theory for weak solutions to the Kolmogorov equation with measurable coefficients, which is nowadays the main focus of the research community. It has been developed during the last decade, and the most advanced achievements in this framework have been established in the particular case of the KFP equation. In Chapter 4 we give proof of a geometric statement for the Harnack inequality for weak solutions to the KFP equation proved by Golse, Imbert, Mouhot and Vasseur in 2017, based on the concepts of Harnack chains and attainable set. As far as we are concerned with the more general Kolmogorov equation in divergence form, Chapter 5 is devoted to the extension of the Moser’s iterative procedure for weak solutions to the Kolmogorov equation under minimal integrability assumptions for the lower order coefficients in the non dilation invariant case
On a spatially inhomogeneous nonlinear Fokker–Planck equation : Cauchy problem and diffusion asymptotics
Full text linkWe investigate the Cauchy problem and the diffusion asymptotics for a spatially inhomogeneous kinetic model associated to a nonlinear Fokker–Planck operator. We derive the global well-posedness result with instantaneous smoothness effect, when the initial data lies below a Maxwellian. The proof relies on the hypoelliptic analog of classical parabolic theory, as well as a positivity-spreading result based on the Harnack inequality and barrier function methods. Moreover, the scaled equation leads to the fast diffusion flow under the low field limit. The relative phi-entropy method enables us to see the connection between the overdamped dynamics of the nonlinearly coupled kinetic model and the correlated fast diffusion. The global-in-time quantitative diffusion asymptotics is then derived by combining entropic hypocoercivity, relative phi-entropy, and barrier function methods
Existence results for singular nonlinear BVPs in the critical regime
Full text linkWe study the existence of solutions for a class of boundary value problems on the half line, associated to a third order ordinary differential equation governed by the
Φ-Laplacian operator. The equation contains a Carathéodory function satisfying a weak growth condition of Winter-Nagumo type which is assumed to be
continuous and it may vanish in a subset of zero Lebesgue measure, so that the problem
can be singular. The approach we follow is based on fixed point techniques
combined with the upper and lower solutions method
Existence of a fundamental solution of partial differential equations associated to Asian options
Full text linkWe 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
A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients
Full text linkWe consider weak solutions of second-order partial differential equations of Kolmogorov-Fokker-Planck-type 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
Boundary value problems for integro-differential and singular higher-order differential equations
Full text linkWe investigate third-order strongly nonlinear differential equations of the type ( Phi ( k ( t ) u '' ( t ) ) ) ' = f ( t , u ( t ) , u ' ( t ) , u '' ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}{{\prime\prime} }\left(t))){\prime} =f\left(t,u\left(t),u{\prime} \left(t),{u}{{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Phi \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Phi ( k ( t ) v ' ( t ) ) ) ' = f t , integral 0 t v ( s ) d s , v ( t ) , v ' ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v{\prime} \left(t))){\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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