Introduction to the Theory of (Non-Symmetric) Dirichlet Forms

Röckner, Michael; Ma, Zhiming

doi:10.1007/978-3-642-77739-4

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“…We get started by recalling some basic facts on bilinear forms. For a detailed study see for example [MR92] or [Fuk80].…”

Section: Dirichlet Formsmentioning

confidence: 99%

“…[FOT94], Chap. A.2, or [MR92], Chap. IV, obtained from the σ-algebras σ{ω(s) | 0 ≤ s ≤ t, ω ∈ Ω}, t ≥ 0.…”

Section: Locality Of the Quasi-regular Dirichlet Form (E Amentioning

confidence: 99%

“…Our approach is rather as in [AKR03], where Albeverio, Kondratiev and Röckner used Dirichlet form techniques, see [Fuk80], [MR92], to construct the diffusion corresponding to (E ,a , D(E ,a )) in the case Ω = R n and a the identity matrix. Our assumptions on for constructing the diffusion process corresponding to (E ,a , D(E ,a )) are still more general as in [AKR03] and rather as used by Fukushima in [Fuk85] where the author also considered the case Ω = R n .…”

Section: Introductionmentioning

confidence: 99%

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CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

Fattler

Grothaus

2008

Proceedings of the Edinburgh Mathematical Society

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Abstract. We give a Dirichlet form approach for the construction and analysis of elliptic diffusions in Ω ⊂ R n with reflecting boundary condition. The problem is formulated in an L 2 -setting w.r.t. a reference measure µ on Ω having an integrable, dx-a.e. positive, density w.r.t. the Lebesgue measure. The symmetric Dirichlet forms (E ,a , D(E ,a )) we consider are the closure of the symmetric bilinear forms, where a is a symmetric, elliptic, n × n-matrix-valued measurable function on Ω. Assuming that Ω is an open, relatively compact set with boundary ∂Ω of Lebesgue measure zero and that is satisfying the Hamza condition, we can show that (E ,a , D(E ,a )) is a local, quasi-regular Dirichlet form. Hence, it has an associated self-adjoint generator (L ,a , D(L ,a )) and diffusion process M ,a (i.e, an associated strong Markov process with continuous sample paths). Furthermore, since 1 ∈ D(E ,a ) (due to the Neumann boundary condition) and E ,a (1, 1) = 0, we obtain a conservative process M ,a (i.e., M ,a has infinite life time). Additionally assuming that √ ∈ W 1,2 (Ω) ∩ C(Ω) or that is bounded, Ω is convex and { = 0} has codimension ≥ 2 we can show that the set { = 0} has E ,a -capacity zero. Therefore, in this case we even can construct an associated conservative diffusion process in { > 0}. This is essential for our application to continuous N -particle systems with singular interactions. Note that for the construction of the self-adjoint generator (L ,a , D(L ,a )) and the Markov process M ,a we do not need to assume any differentiability condition on and a. The following explicit representation of the generator we obtain for √ ∈ W 1,2 (Ω) and a ∈ W 1,∞ (Ω):∂i(aij ∂j) + ∂i(log ) aij ∂j.Note that the drift term can be very singular, because we allow to be zero on a set of Lebesgue measure zero. Our assumptions in this paper even allow a drift which is not integrable w.r.t. the Lebesgue measure.

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“…We get started by recalling some basic facts on bilinear forms. For a detailed study see for example [MR92] or [Fuk80].…”

Section: Dirichlet Formsmentioning

confidence: 99%

“…[FOT94], Chap. A.2, or [MR92], Chap. IV, obtained from the σ-algebras σ{ω(s) | 0 ≤ s ≤ t, ω ∈ Ω}, t ≥ 0.…”

Section: Locality Of the Quasi-regular Dirichlet Form (E Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

Fattler

Grothaus

2008

Proceedings of the Edinburgh Mathematical Society

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“…), E-nest, martingale additive functional, continuous additive functionals, F loc , etc. we refer the reader to [10] and [15]. In particular, we recall that an increasing sequence of closed sets …”

Section: Introductionmentioning

confidence: 99%

Perturbation of symmetric Markov processes

Chen

Fitzsimmons

Kuwae

et al. 2007

Probab. Theory Relat. Fields

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We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower order perturbation of the L 2 -infinitesimal generator L of a general symmetric Markov process. An illuminating concrete example for Crucial to the development is to use the extension of Nakao's stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in [3].

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“…MaOverbeck-Röckner [8]). For a positivity preserving form with a lower bound −γ on L 2 (X; m) is said to be local if (E γ , F) is local in the sense of Chapter V. Definition 1.1 in [9]. Theorem 1.1.…”

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confidence: 99%

Invariant sets and ergodic decomposition of local semi-Dirichlet forms

Kuwae

2011

Forum Mathematicum

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Weakly invariant set and strongly invariant setLet X be a separable metric space and m a σ-finite Borel measure on X. Let (T t ) t≥0 be a C 0 -semigroup on L 2 (X; m) and (T t ) t≥0 the dual C 0 -semigroup of (T t ) t≥0 on L 2 (X; m). An m-measurable subset B of X is said to be weakly invariant with respect to (T t ) t≥0 if I B c T t I B u = 0 for any t > 0 and u ∈ L 2 (X; m), equivalently B c is weakly invariant with respect to (T t ) t≥0 . An m-measurable subset B of X is said to be (strongly) invariant with respect to (T t ) t≥0 if T t I B u = I B T t u for any t > 0 and u ∈ L 2 (X; m). Clearly, the strong invariance implies the weak one and B is strongly invariant if and only if both B and B c are weakly invariant. So if (T t ) t≥0 is symmetric, then the weak invariance is equivalent to the strong one. Fix γ ≥ 0. A bilinear form (E, F) is said to be a positivity preserving form with a lower bound F) is a coercive closed form having the property that for u ∈ F, u + ∧ 1 ∈ F and2 . Any semi-Dirichlet form with a lower bound −γ on L 2 (X; m) is automatically a positivitiy preserving form with a lower bound −γ on L 2 (X; m).Let (E, F) be a semi-Dirichlet form or positivity preserving form with a lower bound −γ on L 2 (X; m). Then there exists a C 0 -semigroup (T t ) t≥0 on L 2 (X; m) such that (e −γt T t ) t≥0 is contractive on L 2 (X; m) and its resolvent (G α ) α>γ defined byFor a positivity preserving (resp. semi-Dirichlet) form with a lower bound −γ on L 2 (X; m) is said to be quasi-regular if (E γ , F) is a quasi-regular positivity preserving (resp. semi-Dirichlet) form on L 2 (X; m) in the sense of Ma-Röckner [10] (resp. MaOverbeck-Röckner [8]). For a positivity preserving form with a lower bound −γ on L 2 (X; m) is said to be local if (E γ , F) is local in the sense of Chapter V. Definition 1.1 in [9]. Theorem 1.1. Suppose that (T t ) t≥0 is associated with a quasi-regular local positivity preserving form (E, F) with a lower bound −γ on L 2 (X; m). Then the weak invariance with respect to (T t ) t≥0 is equivalent to the strong invariance.

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Introduction to the Theory of (Non-Symmetric) Dirichlet Forms

Abstract: Typesetting: Camera ready by author 41/3140 -5 432 10-Printed on acid-free paper Z.-M. M. dedicates this book to his wife. M. R. dedicates this book to his father on the occasion of his 65th birthday.

Cited by 808 publications

References 0 publications

CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

CONSTRUCTION OF ELLIPTIC DIFFUSIONS WITH REFLECTING BOUNDARY CONDITION AND AN APPLICATION TO CONTINUOUS N-PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS

Perturbation of symmetric Markov processes

Invariant sets and ergodic decomposition of local semi-Dirichlet forms

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