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25
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 19 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Sufficient Conditions For Exponential Stability And Dichotomy Of Evolution Equations
 Forum Math
, 1998
"... . We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X . Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform const ..."
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Cited by 13 (7 self)
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. We present several sufficient conditions for exponential stability and dichotomy of solutions of the evolution equation u 0 (t) = A(t)u(t) () on a Banach space X . Our main theorem says that if the operators A(t) generate analytic semigroups on X having exponential dichotomy with uniform constants and A(\Delta) has a sufficiently small Holder constant, then () has exponential dichotomy. We further study robustness of exponential dichotomy under time dependent unbounded Miyaderatype perturbations. Our main tool is a characterization of exponential dichotomy of evolution families by means of the spectra of the socalled evolution semigroup on C 0 (R; X) or L 1 (R; X). 1. Introduction and preliminaries Exponential dichotomy is one of the fundamental asymptotic properties of solutions of the linear Cauchy problem (CP ) ae d dt u(t) = A(t)u(t); t ? s; u(s) = x in a Banach space X. It also plays an important role in the investigation of qualitative properties of nonlinear evolut...
(Nonsymmetric) Dirichlet Operators On L¹: Existence, Uniqueness And Associated Markov Processes
"... Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitel ..."
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Cited by 9 (2 self)
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Let L be a nonsymmetric operator of type Lu = P a ij @ i @ j u + P b i @ i u on an arbitrary subset U ae R d . We analyse L as an operator on L 1 (U; ¯) where ¯ is an invariant measure, i.e., a possibly infinite measure ¯ satisfying the equation L ¯ = 0 (in the weak sense). We explicitely construct, under mild regularity assumptions, extensions of L generating subMarkovian C0 semigroups on L 1 (U; ¯) as well as associated diffusion processes. We give sufficient conditions on the coefficients so that there exists only one extension of L generating a C0 semigroup and apply the results to prove uniqueness of the invariant measure ¯. Our results imply in particular that if ' 2 H 1;2 loc (R d ; dx), ' 6= 0 dxa.e., the generalized Schrödinger operator (\Delta + 2' \Gamma1 r' \Delta r;C 1 0 (R d )) has exactly one extension generating a C0 semigroup if and only if the Friedrich's extension is conservative. We also give existence and uniqueness results for ...
A Spectral Characterization of Exponentially Dichotomic and Hyperbolic Evolution Families
, 1994
"... : We characterize hyperbolic evolution families on a Banach space X by means of spectral properties of the induced semigroup on the spaces C 0 (IR; X) and L p (IR; X), 1 p ! 1. This improves previous results of R. Rau [Ra1], [Ra2] and Y. Latushkin and S. MontgomerySmith [LM1], [LM2]. 1. Expo ..."
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Cited by 6 (3 self)
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: We characterize hyperbolic evolution families on a Banach space X by means of spectral properties of the induced semigroup on the spaces C 0 (IR; X) and L p (IR; X), 1 p ! 1. This improves previous results of R. Rau [Ra1], [Ra2] and Y. Latushkin and S. MontgomerySmith [LM1], [LM2]. 1. Exponentially dichotomic and hyperbolic evolution families Given a nonautonomous Cauchy problem u(t) = A(t)u(t); u(s) = x s 2 X ; t s 2 IR ; (nCP ) on a Banach space X with possibly unbounded operators A(t), t 2 IR, on X, the solutions of (nCP) lead (under certain conditions) to a family (U(t; s)) ts in the space L(X) of bounded linear operators on X, satisfying the following properties: This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG. 1 (E1) the mapping (t; s) 7! U(t; s) from D := f(t; s) 2 IR 2 : t sg into L(X) is strongly continuous, (E2) U(s; s) = Id X , U(t; r)U(r; s) = U(t; s) for all t r s, (E3) there are constants M 1 and !...
First Order Perturbations Of Dirichlet Operators: Existence And Uniqueness
, 1996
"... We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embed ..."
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Cited by 5 (1 self)
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We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embedded in E. Assuming quasiregularity of (E 0 ; D(E 0 )) we show that there always exists a closed extension of Lu := L 0 u + hB; ruiH that generates a subMarkovian C 0 semigroup of contractions on L 2 (E; ¯) (resp. L 1 (E; ¯)), if B 2 L 2 (E; H;¯) and R hB; ruiH d¯ 0; u 0. If D is an appropriate core for (L 0 ; D(L 0 )) we show that there is only one closed extension of (L; D) in L 1 (E; ¯) generating a strongly continuous semigroup. In particular we apply our results to operators of type \Delta H +B \Delta r, where \Delta H denotes the GrossLaplacian on an abstract Wiener space (E; H; fl) and B = \Gammaid E + v, where v takes values in the CameronMartin s...
DISCRETE SPECTRUM AND ALMOST PERIODICITY
"... Abstract. The purpose of this note is to show that solutions of first and second order Cauchy problems with almost periodic inhomogeneity are almost periodic on the real line whenever the spectrum of the underlying operator is discrete. 1. ..."
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Cited by 4 (0 self)
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Abstract. The purpose of this note is to show that solutions of first and second order Cauchy problems with almost periodic inhomogeneity are almost periodic on the real line whenever the spectrum of the underlying operator is discrete. 1.
Existence of invariant manifolds for stochastic equations in infinite dimension
 J. Funct. Anal
, 2003
"... Abstract. We provide a Frobenius type existence result for finitedimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor [1 ..."
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Cited by 4 (1 self)
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Abstract. We provide a Frobenius type existence result for finitedimensional invariant submanifolds for stochastic equations in infinite dimension, in the spirit of Da Prato and Zabczyk [5]. We recapture and make use of the convenient calculus on Fréchet spaces, as developed by Kriegl and Michor [16]. Our main result is a weak version of the Frobenius theorem on Fréchet spaces. As an application we characterize all finitedimensional realizations for a stochastic equation which describes the evolution of the term structure of interest rates. 1.
Strong uniqueness for a class of infinite dimensional Dirichlet operators and applications to stochastic quantization
, 1997
"... Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L p setting is discussed. As a direct application essential self adjointness and strong uni ..."
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Cited by 3 (1 self)
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Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L p setting is discussed. As a direct application essential self adjointness and strong uniqueness in L p is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume ae R 2 . AMS Subject Classification Primary: 47 B 25, 81 S 20 Secondary: 31 C 25, 60 H 15, 81 Q 10 Key words and phrases: Dirichlet operators, essential selfadjointness, C 0  semigroups, generators, stochastic quantization, Markov uniqueness, apriori estimates Running head: Strong uniqueness for Dirichlet operators 1 Introduction The theory of Dirichlet forms is a rapidly developing field of modern analysis which has intimate relationships with potential theory, probability theory, diffe...
Towards a gauge theory for evolution equations on vectorvalued spaces
, 2009
"... Abstract. We investigate symmetry properties of vectorvalued diffusion and Schrödinger equations. For a separable Hilbert space H we characterize the subspaces of L 2 (R N, H) that are local (i.e., defined pointwise) and discuss the issue of their invariance under the time evolution of the differen ..."
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Cited by 3 (3 self)
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Abstract. We investigate symmetry properties of vectorvalued diffusion and Schrödinger equations. For a separable Hilbert space H we characterize the subspaces of L 2 (R N, H) that are local (i.e., defined pointwise) and discuss the issue of their invariance under the time evolution of the differential equation. In this context, the possibility of a connection between our results and the theory of gauge symmetries in mathematical physics is explored. 1. The abstract setting: global symmetries In mathematical physics, one is often interested in the formulation of gauge theories. These are field theories where solutions of the relevant equations are symmetric – i.e., invariant under some transformation group of the functional values. A prototypical example is given by quantum electrodynamics, which is a gauge theory with respect to the symmetry group U(1) (the unitary group) and leads to the introduction of the electromagnetic field. The usual framework to deal with gauge theories in a mathematically rigorous way is that of differential geometry. Aim of this note is to propose a possible approach based on operator theoretic methods, instead, borrowing some ideas from the theory of vector bundles. Let H be a separable complex Hilbert space and consider the Bochner space H: = L 2 (R N; H),
Characteristic Equations for the Spectrum of Generators
, 1997
"... We propose an abstract framework for the computation of the spectrum oe(A) of a linear operator A : D(A) ae X ! X on a Banach space X through a condition in a smaller Banach space X 1 . If this space is finite dimensional this yields a characteristic equation for oe(A). The method is tested for del ..."
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Cited by 1 (0 self)
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We propose an abstract framework for the computation of the spectrum oe(A) of a linear operator A : D(A) ae X ! X on a Banach space X through a condition in a smaller Banach space X 1 . If this space is finite dimensional this yields a characteristic equation for oe(A). The method is tested for delay, integrodifferential and population equations and is applicable to control and stability questions. 1. Motivation The spectrum oe(A) of the generator A of a strongly continuous semigroup on a Banach space X characterizes various qualitative and, in particular, asymptotic properties of the semigroup. We only mention that the negativity of the spectral bound of A, i.e., s(A) := sup fRe : 2 oe(A)g ! 0 implies uniform exponential stability for eventually norm continuous semigroups on Banach spaces or for positive semigroups on L p  spaces (see [Ne], Chapter 3.5). We refer to the monographs [Na1] and [Ne] where these and many more relations between spectrum and asymptotics are discussed...