• 대한수학회지
• /
• 제35권3호
• /
• pp.583-611
• /
• 1998

We consider a general class of real-valued Levy processes {X(t), $t\geq0$}, and obtain suitable large deviation results for the empiricals L(t, A) defined by $t^{-1}{\int^t}_01_A$(X(s)ds for t > 0 and a Borel subset A of R. These results are used to obtain the asymptotic behavior of P{Z(t) < a}, where Z(t) = $sup_{u\leqt}\midx(u)\mid$ as $t\longrightarrow\infty$, in terms of the rate function in the large deviation principle. A subclass of these processes is the Feller class: there exist nonrandom functions b(t) and a(t) > 0 such that {(X(t) - b(t))/a(t) : t > 0} is stochastically compact, i.e., each sequence has a weakly convergent subsequence with a nondegenerate limit. The stable processes are in this class, but it is much larger. We consider processes in this class for which b(t) may be taken to be zero. For any t > 0, we consider the renormalized process ${X(u\psi(t))/a(\psi(t)),u\geq0}$, where $\psi$(t) = $t(log log t)^{-1}$, and obtain large deviation probability estimates for $L_{t}(A)$ := $(log log t)^{-1}$${\int_{0}}^{loglogt}1_A$$(X(u\psi(t))/a(\psi(t)))dv$. It turns out that the upper and lower bounds are sharp and depend on the entire compact set of limit laws of {X(t)/a(t)}. The results extend to random walks in the Feller class as well. Earlier results of this nature were obtained by Donsker and Varadhan for symmetric stable processes and by Jain for random walks in the domain of attraction of a stable law.

• 대한수학회지
• /
• 제38권2호
• /
• pp.227-274
• /
• 2001

In this paper we intended to discuss the following topics: (1) Notation, generalities, Markov processes. The close relationship between (generators of) Markov processes and the martingale problem is exhibited. A link between the Korovkin property and generators of Feller semigroups is established. (2) Feynman-Kac semigroups: 0-order regular perturbations, pinned Markov measures. A basic representation via distributions of Markov processes is depicted. (3) Dirichlet semigroups: 0-order singular perturbations, harmonic functions, multiplicative functionals. Here a representation theorem of solutions to the heat equation is depicted in terms of the distributions of the underlying Markov process and a suitable stopping time. (4) Sets of finite capacity, wave operators, and related results. In this section a number of results are presented concerning the completeness of scattering systems (and its spectral consequences). (5) Some (abstract) problems related to Neumann semigroups: 1st order perturbations. In this section some rather abstract problems are presented, which lie on the borderline between first order perturbations together with their boundary limits (Neumann type boundary conditions and) and reflected Markov processes.