Mark Veraar
Delft University of Technology
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Featured researches published by Mark Veraar.
Annals of Probability | 2007
J. M. A. M. van Neerven; Mark Veraar; Lutz Weis
In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent
Journal of Differential Equations | 2008
Zdzisław Brzeźniak; J. M. A. M. van Neerven; Mark Veraar; Lutz Weis
p\in (0,1]
Annals of Probability | 2012
Jan van Neerven; Mark Veraar; Lutz Weis
. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.
Stochastics An International Journal of Probability and Stochastic Processes | 2012
Mark Veraar
Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.
American Mathematical Monthly | 2010
Lutz Dümbgen; Sara van de Geer; Mark Veraar; Jon A. Wellner
In this article we prove a maximal Lp-regularity result for stochastic convolutions, which extends Krylov’s basic mixed Lp(Lq)-inequality for the Laplace operator on ℝd to large classes of elliptic operators, both on ℝd and on bounded domains in ℝd with various boundary conditions. Our method of proof is based on McIntosh’s H∞-functional calculus, R-boundedness techniques and sharp Lp(Lq)-square function estimates for stochastic integrals in Lq-spaces. Under an additional invertibility assumption on A, a maximal space–time Lp-regularity result is obtained as well.
Potential Analysis | 2017
Chiara Gallarati; Mark Veraar
In this paper, we present an elementary and self-contained proof of the stochastic Fubini theorem, which states that one can interchange a Lebesgue integral and a stochastic integral. The integrability conditions we use are weaker and more natural than the usual conditions in the literature. In particular, we do not need integrability in , and we use -integrability instead of -integrability in the additional parameter.
arXiv: Probability | 2015
Jan van Neerven; Mark Veraar; Lutz Weis
Our starting point is the following well-known theorem from probability: Let X1, …, Xn be independent random variables with finite second moments, and let Sn=∑i=1nXi. Then Var(Sn)=∑i=1nVar(Xi). (1) If we suppose that each Xi has mean zero, Xi = 0, then (1) becomes ESn2=∑i=1nEXi2. (2) This equality generalizes easily to vectors in a Hilbert space ℍ with inner product 〈·, ·〉: If the Xis are independent with values in ℍ such that Xi = 0 and ‖Xi‖2 < ∞, then ‖Sn‖2=〈Sn,Sn〉=∑i,j=1n〈Xi,Xj〉, and since 〈Xi, Xj〉 = 0 for i ≠ j by independence, E‖Sn‖2=∑i,j=1nE〈Xi,Xj〉=∑i=1nE‖Xi‖2. (3) What happens if the Xis take values in a (real) Banach space (, ‖ · ‖)? In such cases, in particular when the square of the norm ‖ · ‖ is not given by an inner product, we are aiming at inequalities of the following type: Let X1, X2, …, Xn be independent random vectors with values in (, ‖ · ‖) with Xi = 0 and ‖Xi‖2 < ∞. With Sn≔∑i=1nXi we want to show that E‖Sn‖2≤K∑i=1nE‖Xi‖2 (4) for some constant K depending only on (, ‖ · ‖). For statistical applications, the case (B,‖⋅‖)=lrd≔(ℝd,‖⋅‖r) for some r ∈ [1, ∞] is of particular interest. Here the r-norm of a vector x ∈ ℝd is defined as ‖x‖r≔{(∑j=1d|xj|r)1/rif1≤r≤∞,max1≤j≤d|xj|ifr=∞. (5) An obvious question is how the exponent r and the dimension d enter an inequality of type (4). The influence of the dimension d is crucial, since current statistical research often involves small or moderate “sample size” n (the number of independent units), say on the order of 102 or 104, while the number d of items measured for each independent unit is large, say on the order of 106 or 107. The following two examples for the random vectors Xi provide lower bounds for the constant K in (4): Example 1.1 (A lower bound in lrd) Let b1, b2, …, bd denote the standard basis of ℝd, and let e1, e2, …, ed be independent Rademacher variables, i.e., random variables taking the values +1 and −1 each with probability 1/2. Define Xi≔ eibi for 1 ≤ i ≤ n ≔ d. Then Xi = 0, ‖Xi‖r2=1, and ‖Sn‖r2=d2/r=d2/r−1∑i=1n‖Xi‖r2. Thus any candidate for K in (4) has to satisfy K ≥ d2/r−1.
arXiv: Probability | 2010
Mark Veraar
In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an Lp(Lq)-theory for such equations for p,q∈(1,∞)
Banach Journal of Mathematical Analysis | 2017
Jan Rozendaal; Mark Veraar
p,q\in (1, \infty )
Journal of Fourier Analysis and Applications | 2018
Jan Rozendaal; Mark Veraar
. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.