1,721,037 research outputs found
A counterexample to L∞-gradient type estimates for Ornstein-Uhlenbeck operators
No full textLet (lambda(k)) be a strictly increasing sequence of positive numbers such thatSigma(infinity)(k=1) 1/lambda(k) 0 is independent of f and m. This is a consequence of generalized Meyer's inequalities [4]. We show that, if lambda(k) similar to k(2), then such estimate does not hold when p = infinity. Indeed we provesup (f is an element of Cb2 (Rm), ||f ||infinity infinity as m -> infinity.This is in contrast to the case of lambda(k) = lambda > 0, k = 1, where a dimension-free bound holds for p = infinity
Global Lipschitz regularizing effects for linear and nonlinear parabolic equations
No full textIn this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in (0, T) x R-N. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their L-infinity-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (2006) [35] (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman-Isaacs equations. We investigate both the classical regularizing effect (at time t > 0) and the possible conservation of Lipschitz regularity from t = 0, and similarly we prove global Holder estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and Holder continuous coefficients. Applications to Liouville type theorems are also given in the paper. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method. (C) 2013 Elsevier Masson SAS. All rights reserved
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
Flow of diffeomorphisms for SDEs with unbounded Hölder continuous drift
No full textWe consider a SDE with a smooth multiplicative non-degenerate noise and a possibly unbounded Hölder continuous drift term. We prove the existence of a global flow of diffeomorphisms by means of a special transformation of the drift of Itô–Tanaka type. The proof requires non-standard elliptic estimates in Hölder spaces. As an application of the stochastic flow, we obtain a Bismut–Elworthy–Li type formula for the first derivatives of the associated diffusion semigroup
A BSDEs approach to pathwise uniqueness for stochastic evolution equations
Full text linkWe prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Hölder continuous. This class includes examples of semilinear stochastic Euler-Bernoulli beam equations which describe elastic systems with structural damping, and semilinear stochastic 3D heat
equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Itô formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Itô-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H ; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions
Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations
No full textThe motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when a generic stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears
Linear Operator Inequality and Null Controllability with Vanishing Energy for Unbounded Control Systems
Full text linkWe consider a linear boundary or point control system on a Hilbert space which is null controllable at some time . To every initial state we associate the minimal ``energy'' needed to transfer to in a time (``energy'' of a control being the square of its norm). Clearly, it decreases with the control time . We shall prove that, under suitable spectral properties of the linear system operator, the minimal energy converges to for $ T\to+\infty
Poisson stochastic process and basic schauder and Sobolev estimates in the theory of parabolic equations (short version)
No full textNorm discontinuity and spectral properties of Ornstein-Uhlenbeck semigroups
No full textDelft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
- …
