1,720,972 research outputs found
Nonautonomous Ornstein-Uhlenbeck operators in weighted spaces of continuous functions
No full textWe consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in RN. We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator Ps,t, which we use to prove optimal Schauder estimates for the solution to some nonhomogeneous parabolic Cauchy problems associated with the Ornstein-Uhlenbeck operator. We also prove that, for any t>s, the evolution operator Ps,t is compact in the previous weighted spaces. © 2013 Springer Science+Business Media New York
Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure
Full text linkIn this paper we show that the realization in L p(X, ν∞) of a nonsymmetric Ornstein-Uhlenbeck operator Lp is sectorial for any p∈ (1 , + ∞) and we provide an explicit sector of analyticity. Here, (X, μ∞, H∞) is an abstract Wiener space, i.e., X is a separable Banach space, μ∞ is a centred nondegenerate Gaussian measure on X and H∞ is the associated Cameron-Martin space. Further, ν∞ is a weighted Gaussian measure, that is, ν∞= e−Uμ∞ where U is a convex function which satisfies some minimal conditions. Our results strongly rely on the theory of nonsymmetric Dirichlet forms and on the divergence form of the realization of L2 in L 2(X, ν∞)
A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control
Full text linkWe obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified
Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures
No full textWe consider the spaces Lp(X,nu;V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\text {L}}<^>p(X,\nu ;V)\end{document}, where X is a separable Banach space, mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} is a centred non-degenerate Gaussian measure, nu:=Ke-U mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\nu :=Ke<^>{-U}\mu \end{document} with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincar & eacute; inequality for functions F is an element of W1,p(X,nu;V)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}F\in W<^>{1,p}(X,\nu ;V)\end{document}, which allows us to show that for every p is an element of(1,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} and every k is an element of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} the norm in Wk,p(X,nu)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}W<^>{k,p}(X,\nu )\end{document} is equivalent to the graph norm of DHk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}D_H<^>{k}\end{document} (the k-th Malliavin derivative) in Lp(X,nu)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}{\text {L}}<^>p(X,\nu )\end{document}. To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup (TV(t))t >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}(T<^>V(t))_{t\ge 0}\end{document}, defined in Section 2.6, as t goes to infinity.Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck (T(t))t >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}, and pointwise estimates for |DHT(t)f|Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}|D_HT(t)f|_H<^>p\end{document} by means of both T(t)|DHf|Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}T(t)|D_Hf|<^>p_H\end{document} and T(t)|f|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}T(t)|f|<^>p\end{document}
BV functions on open domains: The wiener case and a fomin differentiable case
No full textWe provide three different characterizations of the space BV (O, γ) of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure γ on open domains O in Wiener spaces. Throughout these different characterizations we deduce a sufficient condition in order to belong to BV (O, γ) by means of the Ornstein-Uhlenbeck semigroup and we provide an explicit formula for one-dimensional sections of functions of bounded variation. Finally, we apply our techniques to Fomin differentiable probability measures ν on a Hilbert space X, and we infer a characterization of the space BV (O, ν) of the functions of bounded variation with respect to ν on open domains O ⊆ X
Young equations with singularities
No full textIn this paper we prove existence and uniqueness of a mild solution to the Young equation dy(t)=Ay(t)dt+σ(y(t))dx(t), t∈[0,T], y(0)=ψ. Here, A is an unbounded operator which generates a semigroup of bounded linear operators (S(t))t≥0 on a Banach space X, x is a real-valued η-Hölder continuous. Our aim is to reduce, in comparison to Gubinelli et al. (2006) and Addona et al. (2022) (see also Deya et al. (2012) and Gubinelli and Tindel, (2010)), the regularity requirement on the initial datum ψ eventually dropping it. The main tool is the definition of a sewing map for a new class of increments which allows the construction of a Young convolution integral in a general interval [a,b]⊂R when the Xα-norm of the function under the integral sign blows up approaching a and Xα is an intermediate space between X and D(A)
On integration by parts formula on open convex sets in Wiener spaces
No full textIn Euclidean space, it is well known that any integration by parts formula for a set of finite perimeter Ω is expressed by the integration with respect to a measure P(Ω , ·) which is equivalent to the one-codimensional Hausdorff measure restricted to the reduced boundary of Ω. The same result has been proved in an abstract Wiener space, typically an infinite-dimensional space, where the surface measure considered is the one-codimensional spherical Hausdorff–Gauss measure S∞-1 restricted to the measure-theoretic boundary of Ω. In this paper, we consider an open convex set Ω and we provide an explicit formula for the density of P(Ω , ·) with respect to S∞-1. In particular, the density can be written in terms of the Minkowski functional p of Ω with respect to an inner point of Ω. As a consequence, we obtain an integration by parts formula for open convex sets in Wiener spaces
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
A nonlinear Bismut–Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces
Full text linkWe consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes (Y, Z), with generator with quadratic growth with respect to Z. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut–Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to Z. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations in Banach spaces for the unknown v, with nonlinear term with quadratic growth with respect to ∇ v and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth
A nonlinear Bismut–Elworthy formula for HJB equations with quadratic Hamiltonian in Banach spaces
Full text linkWe consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes (Y, Z), with generator with quadratic growth with respect to Z. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut–Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to Z. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations in Banach spaces for the unknown v, with nonlinear term with quadratic growth with respect to ∇ v and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth
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