通过贝塞尔函数的占据时间
Occupation times via Bessel functions
摘要：
连续时间马尔可夫过程的占据时间密度的这项研究受到了E.Nir等人（2006）在单分子FRET光谱领域的工作的启发。在那里，单个分子在两个或多个状态之间波动，实验观测值取决于状态的占据时间分布。为了在数学上描述可观测的，需要计算单个状态的占用时间分布。本文考虑具有可数多个状态的马尔可夫过程。为了找到一个stete占领时间密度，我们使用傅立叶变换和拉普拉斯变换的组合的方式，允许逆的傅里叶变换。在Z上简单连续时间随机游动的情况下，我们得到了一个明确的占据时间密度的表达式。此外，我们还研究了谱测度在Karlin-McGregor对角化，试图通过修改贝塞尔函数表示占领时间密度。
英文摘要：
This study of occupation time densities for continuous-time Markov processes was inspired by the work of E.Nir et al (2006) in the field of Single Molecule FRET spectroscopy. There, a single molecule fluctuates between two or more states, and the experimental observable depends on the state's occupation time distribution. To mathematically describe the observable there was a need to calculate a single state occupation time distribution. In this paper, we consider a Markov process with countably many states. In order to find a one-stete occupation time density, we use a combination of Fourier and Laplace transforms in the way that allows for inversion of the Fourier transform. We derive an explicit expression for an occupation time density in the case of a simple continuous time random walk on Z. Also we examine the spectral measures in Karlin-McGregor diagonalization in an attempt to represent occupation time densities via modified Bessel functions.
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