1,721,820 research outputs found
La figura e la filosofia di Francesco da Prato
No full textL'articolo fornisce una presentazione d'insieme della figura e della filosofia del domenicano italiano Francesco da Prato (XIV secolo). L'articolo contiene inoltre una prima appendice con un elenco degli studi e dei documenti su Francesco da Prato e una seconda appendice con la prima trascrizione delle sue "Quaestiones disputate" dal ms. Siena, Biblioteca Comunale, G.VII.40
Strong Uniqueness for Stochastic Evolution Equations with Unbounded Measurable Drift Term
No full textWe consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term B and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato et al. in Ann Probab 41:3306–3344, 2013) which generalized Veretennikov’s fundamental result to infinite dimensions assuming boundedness of the drift term. As in Da Prato et al. (Ann Probab 41:3306–3344, 2013), pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift B is assumed only to be measurable and bounded and grow more than linearly
A note on regularizing properties of Ornstein-Uhlenbeck semigroups in infinite dimensions
No full texteds G. Da Prato - L. Tubaro, Lecture Notes in Pure and Appl. Math., 227, Dekker, 2002
Existence of strong solutions for stochastic porous media equation under general monotonicity conditions
Full text linkBarbu V, Da Prato G, Röckner M. Existence of strong solutions for stochastic porous media equation under general monotonicity conditions. Annals of Probability. 2009;37(2):428-452.This paper addresses the existence and uniqueness of strong solutions to stochasic porous media equations dX - Delta Psi(X)dt = B(X)dW(t) in bounded domains of R-d with Dirichlet boundary conditions. Here Psi is a maximal monotone graph in R x R (possibly multivalued) with the domain and range all of R. Compared with the existing literature on stochastic porous media equations, no growth condition on Psi is assumed and the diffusion coefficient Psi might be multivalued and discontinuous. The latter case is encountered in stochastic models for self-organized criticality or phase transition
Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift
Full text linkDa Prato G, Flandoli F, Priola E, Röckner M. Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. The Annals Of Probability. 2013;41(5):3306-3344.We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on R-d to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions
Absolutely continuous solutions for continuity equations in Hilbert spaces
Full text linkDa Prato G, Flandoli F, Röckner M. Absolutely continuous solutions for continuity equations in Hilbert spaces. Journal de Mathématiques Pures et Appliquées. 2019;128:42-86.We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure gamma which is Fomin-differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where gamma is the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator D-x is closable with respect to L-p(H, gamma) and a recent formula for the commutator DxPt - PtDx where P-t is the transition semigroup corresponding to the reaction-diffusion equation, [10]. We stress that P-t is not necessarily symmetric in this case. This uniqueness result is an extension to such gamma of that in [12] where gamma was the Gaussian invariant measure of a suitable Ornstein-Uhlenbeck process. (C) 2019 Elsevier Masson SAS. All rights reserved
Da Prato-Zabczyk's maximal inequality revisited. I.
Full text linksummary:Existence, uniqueness and regularity of mild solutions to semilinear nonautonomous stochastic parabolic equations with locally lipschitzian nonlinear terms is investigated. The adopted approach is based on the factorization method due to Da Prato, Kwapień and Zabczyk
Pathwise uniqueness for a class of SDE in Hilbert spaces and applications
No full textWe consider an abstract parabolic equation in Hilbert spaces with cylindrical noise and Holder continuous semilinear part and prove strong uniqueness of solutions by means of an infinite dimensional Kolmogorov equation
Sobolev regularity for a class of second order elliptic PDE's in infinite dimension
Full text linkWe consider an elliptic Kolmogorov equation λu - Ku = f in a separable Hilbert space H. The Kolmogorov operator K is associated to an infinite dimensional convex gradient system: dX = (AX-DU(X)) dt +dW(t), where A is a self-adjoint operator in H, and U is a convex lower semicontinuous function. Under mild assumptions we prove that for λ > 0 and f ∈ L2(H, ν) the weak solution u belongs to the Sobolev space W2,2(H, ν), where ν is the log-concave probability measure of the system.Moreover maximal estimates on the gradient of u are proved. The maximal regularity results are used in the study of perturbed nongradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction-diffusion and Cahn-Hilliard stochastic PDEs
Transition Semigroup
No full textBarbu V, Da Prato G, Röckner M. Transition Semigroup. In: Barbu V, Da Prato G, Röckner M, eds. Stochastic porous media equations. Lecture Notes in Mathematics. Vol 2163. Cham: Springer Int Publishing AG; 2016: 167-195
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