## 1 Introduction
Previous numerical solutions of various FP equations in both one-dimensional and two-dimensional have been implemented based on the CN approach [21; 22; 23; 24]. [PERSON], [PERSON], and [PERSON] [21] investigated the effect of different orders of nonlinear interaction on the geometric structure by considering the case when the equilibrium is a stable point. [PERSON], [PERSON], and [PERSON] [22] presented the time-evolution of probability density functions (PDFs) in a toy model of self-organized shear flows. Using theoretical analyses of limiting cases, as well as numerical solutions of the full FP equation, a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing was presented. [PERSON], [PERSON], and [PERSON] [23] reported the PDFs using direct numerical solutions of the FP equation for a classical double-well stochastic resonance, showing clear signature in information length. [PERSON], [PERSON], and [PERSON] [24] investigated the case where a stationary PDF that is unimodal for \(\delta\)-correlated stochastic noise becomes bimodal when the noise has a finite correlation time. As for the periodic-in-space systems [25; 26; 27], the Fourier series approach might be potentially better than finite-differencing in space with the CN method. For example, in the work by [PERSON] and [PERSON] [25], the effect of different spatially periodic, deterministic forces on the information geometry of stochastic processes was explored. [PERSON] and [PERSON] [26] presented time-dependent PDFs for a nonlinear stochastic process with a cubic force using analytical and computational studies. Numerical simulation of the FP equation revealed detailed evolution of the time-dependent PDF. [PERSON] and [PERSON] [27] studied the time-dependent dispersion properties of overdamped tracer particles diffusing in a near critically tilted one-dimensional periodic potential under the influence of an additional constant tilting force. Stochastic noise is ubiquitous and plays a crucial role in the evolution of many different systems such as the PLL. Given uncertainty inherent in the systems, a probabilistic description is essential for understanding the dynamics of stochastic systems. A direct numerical solution to the transient behavior of the PLL phase-error process was carried out by [PERSON] and [PERSON] [28] by comparing the FP equation to the one-dimensional heat-flow equation. The numerical method for solving the modified FP equation employed in references [29; 30; 31] is the [PERSON]'s forward time/central space (FTCS) explicit method. To elucidate the link between various parameters and the information geometry of the phase-error statistics, in the presence of additive white Gaussian noise, it is useful to quantify the difference among different probability density functions (PDFs) and compare their evolution. Based on the preliminary study presented in reference [29], extension of the work for more comprehensive treatment and profound insights on the behavior of PLL statistics will be presented in this paper. The remainder of this paper is organized as follows. In Section 2, the phase-locked loop and normalized FP equation are reviewed; applications of the FP equation are also addressed. In Section 3, the numerical methods using finite difference technique for discretization of the model equation, known as the transport (advection-diffusion) PDE are discussed; numerical formulations are then developed for the FP equation. Numerical implementation and analysis on the PLL statistics are given in Section 4. Illustration and discussion for both the transient and stationary statistics, including the nonmodulo-2\(\pi\) probability density function (PDF), modulo-2\(\pi\) PDF, and cycle slipping density function, with various parameters, such as the detuning factor and signal-to-noise ratio (SNR), are presented. Section 5 summarizes the main results of this paper.