On generalized feynman-kac transformation for markov processes associated with semi-dirichlet forms

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• 作者：Xinfang HAN ^{xfanghan@163.com" class="auth_mail" title="E-mail the corresponding author} ; Li MA ; ^{malihnsd@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：semi-Dirichlet form ; generalized Feynman-Kac semigroup ; strong continuity ; lower semi-bounded ; representation of local continuous additive functional with zero quadratic variation
• 刊名：Acta Mathematica Scientia
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：36
• 期：6
• 页码：1683-1698
• 全文大小：281 K

文摘

Suppose that X is a right process which is associated with a semi-Dirichlet form (ɛ, D(ɛ)) on L²(E; m). Let J be the jumping measure of (ɛ, D(ɛ)) satisfying J(E × E − d) < ro. Let u G D(ɛ)_b: = D(ɛ) ∩ L^∞ (E; m), we have the following Fukushima's decomposition ũ (X_t) − ũ (X_o) = M^u t + N^u_t. Define P^u_t f (x) = Ex[e^Nu_tf (X_t)]. Let Q^u(f, g) = ɛ (f, g) + ɛ (u, fg) for f, g ∈ D(ɛ)_b. In the first part, under some assumptions we show that (Q^u, D(ɛ)_b) is lower semi-bounded if and only if there exists a constant α _o ≥ 0 such that ||P_t^u ||2 ≤ e^{α ot} for every t > 0. If one of these assertions holds, then (P_t^u)_{t≥ o} is strongly continuous on L²(E; m). If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J(E × E − d) < ∞. Some examples are also given in this part. Let A_t be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of A_tand give two sufficient conditions for P_t^Af (x) = E_x [e^A_tf (Xt)] to be strongly continuous.