Hh-transforms Of Positivity Preserving Semigroups, Perturbation Of Generalized Dirichlet Form And Related Topics

The well-known Doob- h-transform has been researched by many researchers (cf. [6,25,32,36,62,65,66,81] and the references therein) since the construction of conditional Brownian motion by Doob, L. J. in 1957 (cf. [30]). In this dissertation, given a-excessive h and co-a-excessive h we consider the hh-transforms of strongly continuous contraction dual semigroups (Tt)t>o and (Tt)t>o on L2(E;m), which are only positivity preserving semigroups, and get strongly continuous contraction semi-groups (Tth)t>0 and (Tth)t>0 with sub-Markovian property on L2(E:hh·m). More-over, we show that (Tth)t>0 and (Tth)t>0 are in duality in the space L2(E;hh·m). In particular, if one (say (Tt)t>0) of the dual semigroups (Tt)t>o and (Tt)t>0 is sub-Markovian, we can also get a pair of sub-Markovian dual semigroups (e-αtTt)t>0 and (Tth)t>0 on L2(E;h·m) (cf. corollary 2.2.4). Finally, under the framework of quasi-regular Dirichlet forms, we give a necessary and sufficient condition for (Tt)t>0 and (Tt)t>0 to be hh-associated with a pair of special standard processes which are in duality. Moreover, those right processes satisfy Hunt’s Hypothesis:if a set can not be reached immediately from almost every point, then it can not be reached forever, e.g. be aε-exceptional set). In particular, we obtain the similar result:for a pair of semigroups associated with a quasi-regular semi-Dirichlet form (cf. Theorem 2.2.7).The one-to-one correspondence between quasi-regular Dirichlet forms and right processes (cf. [14,52,53]) is a powerful tool in studying the classical potential theory and stochastic analysis. Although perturbation of Dirichlet forms has been studied intensively by many researchers, there are few researchers who consider the perturbation of generalized Dirichlet forms by signed smooth measures. For this kind of perturbation, we get several sufficient conditions for the perturbed form to be a generalized Dirichlet form (cf. Theorem 3.2.1, Theorem 3.2.2 and Theorem 3.2.3), and give one sufficient condition for the perturbed generalized Dirichlet form to be associated with a Markov process (cf. Theorem 3.2.5).Next we will focus on the potential terms of the perturbed Dirichlet forms, give one sufficient condition for them to be in the domain of the perturbed Dirichlet forms and are quasi-continuous. Finally, we prove two switching identities directly by perturbation of non-symmetric Dirichlet forms and the above results.In the last Chapter, by h-transform, perturbation of Dirichlet forms and Gir-sanov transform, we get some results about the asymptotic property of the additive functional of zero energy for Brownian motion (cf. Theorem 4.2.3).In details the contents of this dissertation are organized as follows.In the first section of Chapter I, we recall some background and give the main results of this dissertation. In the second section of this Chapter, we introduce some necessary concepts. In Chapter II, we start with h-transforms of positivity preserving forms and state some recent results about h-transforms in the first section. In the second section, we first define the hh-transforms of positivity preserving dual semigroups (Tt)t>0 and (Tt)t>0 on L2(E;m) (cf. Proposition 2.2.1), and show that (Tth)t>o and (Tth)t>0 are contraction operators on L1(E;hh·m) and L∞(E;hh m), respectively (cf. Proposition 2.2.1). Then by Riesz-Thorin theory we get that they are contraction operators on L2 (E:hh·m) and prove that they are strongly continuous contraction semigroups on L2(E;hh·m) (cf. Theorem 2.2.1, Theorem 2.2.2). In particular, if (Tt)t>0 is sub-Markovian, we can also get a pair of sub-Markovian dual semigroups (e-atTt)t>0 and (Tth)t>0 on L2(E;h·m) (cf. corollary 2.2.4). Finally, under the framework of quasi-regular Dirichlet form, we give a necessary and sufficient condition for (Tt)t>0 and (Tt)t>o to be associated with a pair of right processes in some sense (cf. Theorem 2.2.6). In particular, if the pair of semigroups is associated with a quasi-regular semi-Dirichlet form, we can show that they are also associated with a pair of dual Markov processes on L2(E;h·m) (cf. Theorem 2.2.7).In Chapter III, we mainly consider the perturbation of generalized Dirichlet forms and focus on the properties of certain potential items of the perturbed non-symmetric Dirichlet forms. In the first section, we consider the generalized Dirichlet form as follows: we show that ifμis a smooth measure in Hardy-class, then the domain and the norm of the perturbed formεμare the same as that of the formε(cf. Theorem 3.2.1) and are still generalized Dirichlet forms. Also, we give some conditions for the perturbed formεμto be generalized Dirichlet forms (cf. Theorem 3.2.1, Theorem 3.2.2), whenμis a signed smooth measure. Moreover, we give one sufficient condition for the perturbed generalized Dirichlet formεμto be associated with a right process. In the second section, given a quasi-regular Dirichlet form (ε,D(ε)) and the associated dual processes (X,X), a smooth measureμ, we consider the following potential terms, We get that if f∈L2(E;μ), a>0,p>0, then UAα+pμf, UAα+pμf∈D(εμ) and are quasi-continuous (cf. Theorem 3.3.1). Then by the theory of Dirichlet forms we give two switching identities directly (cf. Theorem 3.3.1), and finally we give an application of the switching identities (cf. Proposition 3.3.2):for a pair of dual processes (X, X) which are associated with a quasi-regular Dirichlet form, let At At be the positive continuous additive functional whose revuz measure isμ, (Y, Y) be the time-change processes of (X,X) by At and At, then Y and Y are a pair of dual right processes on L2(E;μ).In the last Chapter, we give some applications of h-transform and perturbation of Dirichlet forms in. Feynman-Kac functionals and the time-change processes of dual processes. The main results are as follows (cf. Theorem 4.2.3):If Lt-u is a martingale, u is bounded,▽u∈Kd-1 and ||E.[eMt-u]||9<∞, then for any x∈Rd, we have Here D(ε)b=D(ε) n L∞(Rd, dx).