2023/02/20 by Michael B. Marcus, Marcus, Michael B., Jay Rosen +1
Mathematics · #60E07 #60G15 #60G17 #60G99 #60J25 #60J55 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2302.10262
openalex publication_date 2023/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Y be a symmetric Borel right process with locally compact state space T⊆ R1 and potential densities u(x,y) with respect to some σ-finite measure on T. Let g and f be finite excessive functions for Y. Set ug, f(x,y)= u(x,y)+g(x)f(y), x,y∈ T. In this paper we take Y to be a symmetric Lévy process, or a diffusion, that is killed at the end of an independent exponential time or the first time it hits 0. Under general smoothness conditions on g, f, u and points d∈ T, laws of the iterated logarithm are found for Xk/2 =\Xk/2(t), t∈ T \, a k/2-permanental process with kernel \ug, f(x,y),x,y∈ T \, of the following form: For all integers k≥ 1, \limsupx → 0\frac| Xk/2( d+x)- Xk/2 (d)| ( 2 σ2(x)loglog 1/x)1/2= ( 2 X k/2 (d))1/2, a.s. , where, σ2(x)=u(d+x,d+x)+u(x,x)-2u(d+x,x). Using these limit theorems and the Eisenbaum Kaspi Isomorphism Theorem, laws of the iterated logarithm are found for the local times of certain Markov processes with potential densities that have the form of \ug, f(x,y),x,y∈ T \ or are slight modifications of it.