Dissertation > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Theory of probability ( probability theory, probability theory ) > Random process > Markov process

hh-transforms of Positivity Preserving Semigroups, Perturbation of Generalized Dirichlet form and Related Topics

Author HanXinFang
Tutor MaZhiMing;HouZhenTing
School Central South University
Course Probability Theory and Mathematical Statistics
Keywords Yasumasa semigroup h (?) - transform Generalized Dirichlet forms Disturbance Right process
CLC O211.62
Type PhD thesis
Year 2011
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Doob, L. J 1957 to consider and structural conditions Brownian motion (see [30]), Doob 's-h-transform has been the concern of many scholars (see [6,25,32,36,62,65,66 , 81] and references therein). Either to one pair L2 of (E; m) on the strong Yasumasa resistance of the continuous compression Semigroups (Tt) T GT; 0 and (Tt) T GT; 0, by their one pair of alpha-excessive function h and total hh-transform them from the yoke-α-excessive function h one pair of L2 (E; strongly continuous compression hh · m) on the Markov semigroup (Tt / h) t gt; 0 and (Tt / h) t gt; 0 (see Theorem 2.2.1, Theorem 2.2.2), and to get them in new space L2 (E; hh.m), dual (see Theorem 2.2.3) also analyzed their infinitesimal generator (see Proposition 2.2.1 and Proposition 2.2.2). In particular, if (TT) T GT; o and (TT) T GT; in 0 (GT Suppose (TT) T; 0) Markov, then we get a pair of L2 (E; HM) on the dual strongly continuous compression Markov semi-group (e-atTt) t gt; 0 and (Tt / h) t gt; 0 (see Theorem 2.2.4). Finally, the intended regular Dirichlet forms framework, this paper gives (Tt) t gt; 0 and (Tt) t gt; 0 in a certain sense, combined with the right process, necessary and sufficient conditions for its contact Yasumasa (£ D (ε)) is intended to regularization of, then get the right process under the new measure hh.m weak duality, and meet the general assumptions Hunt-: from almost all points of departure are not immediately reach the collection forever can not reach (ε-exceptional set) (see Theorem 2.2.6). In particular, when (Tt) t gt; 0 and (Tt) t gt; 0 associated with a pseudo-regular semi-Dirichlet type, similar results (see Theorem 2.2.7). Quasiregular Dirichlet forms one-to-one relationship with the right process (see [14,52,53]), addressed random analysis provides a powerful tool to study the potential of the classic. The Dirichlet disturbance and close contact with the operator disturbances and generalized Feynman-Kac semigroup been a hot research topic in international Dirichlet forms and its related fields. Generalized Dirichlet-symbolic measure smooth disturbance has been the number of people did not discuss this issue in this article do some research into the perturbed quadratic form is still several sufficient conditions of generalized Dirichlet forms (see Theorem 3.2.1, Theorem 3.2.2, Theorem 3.2.3), and the perturbed generalized Dirichlet type sufficient condition combined with Markov process (see Theorem 3.2.5). In particular, the article also problem of asymmetric dieldrin disturbance. Any to a non-symmetric quasi-regular Dirichlet forms, we study a special class of potential items, get them quasicontinuous and characterize such special geopotential perturbed Dirichlet forms defined domain sufficient condition (see Theorem 3.3.1); then take advantage of this result is direct proof that two commonly used conversion equation (see Proposition 3.3.1). In the final chapter of this article comprehensive utilization of the h-transform, dieldrin type disturbances and the Girsanov transform three methods portray the Brownian motion zero-energy additive functionals asymptotic behavior (see Theorem 4.2.3). Let the sequence of chapters briefly describe the main content of this article. Chapter 1 describes the research background and the main findings; Section II describes some of the basic concepts and some known results. The second Section given Yasumasa type H-transform some of the latest results and introduces the problem to be solved in this article. Section II first define L2 (E; m) space Yasumasa semigroup (Tt) t gt; 0 and (Tt) t gt; hh-transform: easy to verify that the transformation semigroup has Markov property Then we prove gt; (Tth) t 0 and (Tth) t gt; 0 are L1 (E; poerator hh-m) and L ∞ (E; hh · m) (see Proposition 2.2. 1), then the use of the Riesz-Thorin interpolation theorem to obtain them are L2 (E; hh · m) of compression on the operator (see Proposition 2.2.1), and thus prove that they are in L2 (E; hh · m) on strongly continuous compression dual semigroup (see Theorem 2.2.1, Theorem 2.2.2, Theorem 2.2.3). Finally, the intended regular Dirichlet forms framework, this paper gives (Tt) t gt; 0 and (Tt) t gt; necessary and sufficient condition for 0 in a certain sense, combined with a pair of dual right process (see Theorem 2.2.6) . In particular, in conjunction with the proposed regular semi-Dirichlet forms semigroup, we get a similar result. Chapter generalized Dirichlet disturbance, and to explore the nature of a class of potential items asymmetric dieldrin disturbance. The first section, we briefly describe the background of generalized Dirichlet forms and asymmetric dieldrin disturbance. Section II study follows the form of the generalized Dirichlet type of disturbance: when a smooth measure μ belongs to the Hardy-class, the perturbed quadratic εμ definition domain, as well as the norm (see Lemma 3.2.1 remains unchanged ) and εμ still generalized Dirichlet type (see Theorem 3.2.1); given μ symbol smooth measure, εμ the generalized Dirichlet sufficient condition (see Theorem 3.2.3), and the generalized Dirichlet forms combined with the right process, a sufficient condition (see Theorem 3.2.5). Let (X, X) in the third quarter, with the proposed regular (non-symmetric) Dirichlet forms (ε, D (ε)) Contact a pair of dual Markov processes. μ is a smooth measure any f ∈ L2 (E; μ), any real number α, p-gt; 0, defined as follows bits potential items: we get UAα pμf well UAα pμf is intended to be continuous and in the after disturbance Di's type domain D (εμ) (see Theorem 3.3.1); thus the use of the theory of Dirichlet forms the two conversion equation (see Proposition 3.3.1), and given the results of an application (see Proposition 3.3 .2): When a pair of dual process (X, X) associated with a Dirichlet forms, for any smooth measure μ and its only the corresponding positive continuous additive functional At, and At (Y, Y) were induced by the AT and At the (X, X) of the time conversion process, then Y and Y is L2 (E; μ) on a pair of dual right process. Chapter h-transform and dieldrin type of disturbance is given in the asymptotic behavior of the generalized Feynman-Kac functionals and dual process time transform, the following results were obtained (see Theorem 4.2.3): If Lt / -u \

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