Perturbation Of Non-Symmetric Dirichlet Forms-the Corresponding Potential And The Infinitesimal Generator

The theory of Dirichlet form is a convenient and applicable mathematic instrument which makes us to cope with the operator semigroups and the related resolvents. There is an one to one correspondence between Dirichlet form, semigroups and resolvents. The intimate relationship among the semigroups the infinitesimal operator and the differential equation makes us recognize the stochastic processes more intuitively. Feynman-Kac semigroup has always been the research subject which interests both mathematicians and physics scientists. In this paper the main contents are the relations among the perturbed forms, the general Feynman-Kac semigroups, the related kernel (or say the corresponding potential) and the corresponding infinitesimal generator (refer to Picture 1-3). Given Dirichlet form(ε,10(ε)), we define the perturbed form (ε^μ, 10(ε^μ)) and the general Feynman-Kac semigroups perturbed by a signed smooth measureμas follows:Our motivation is to get a result similar to that for the Symmetric Dirichlet form, that is, the lower semi-boundedness of the perturbed form is equivalent to the strongly continuousness of the generalized Feynman-Kac semigroup. However, we find that the perturbation of Non-Symmetric Dirichlet form perturbed by a signed smooth measure is very complex. Saying precisely, we could not talk about the spectral decomposition of the nonsymmetric infinitesimal generator. However, we can still get the samiliar generalized resolvent equations under nonsymmetric situation. Applying these equations we could get the relationship (see Graph 1-3 or Theorem 3.2.2) between the perturbed form and the infinitesimal generator of the Feynman-Kac semigroup by comparing twoα-excessive functions.The rest of this paper is organized as follows. In chapter 1, we mainly discribe the basic notations and the definitions associated with the Dirichlet form. In chapter 2, we prove firstly that the form (ε^μ, D(ε^μ)) withμa smooth measure is still a Dirichlet form if (ε, D(ε)) is a Dirichlet form (see Theorem 2.2.1) and U_A^α+μ is the related kernel which is associated with the perturbed Dirichlet form (ε^μ, D(ε^μ)) (see Theorem 2.2.2), then we get a more general result.: for (?) p＞0, thatε_α^pμ act on U_ta^αp is the same as thatε_αact on the potential function U_A^α(refer to Theorem 2.3.1 and Remark 2.3.3); In chapter 3, we mainly discuss the perturbation of the Dirichlet form by a signed smooth measureμ=μ⁺－μ^- where bothμ⁺ andμ^- are smooth measures. Here we give a sufficient condition for U^α+μ(L²(E; m))(?)D(ε^μ) and we also get a sufficient condition for D(L^μ) to be dense in L²(E; m) (see Theorem 3.2.1). Whenμ∈S－S_K0, we get that the relation between L^μand (ε^μ, D(ε^μ)) is the same as that for the symmetric Dirichlet form (refer to Theorem 3.2.2), in the final section we discuss the analysis properties of the Kato-class measures.