On the influence of reflective boundary conditions on the statistics of Poisson–Kac diffusion processes

Abstract

We analyze the influence of reflective boundary conditions on the statistics of Poisson-Kac diffusion processes, and specifically how they modify the Poissonian switching-time statistics. After addressing simple cases such as diffusion in a channel, and the switching statistics in the presence of a polarization potential, we thoroughly study Poisson-Kac diffusion in fractal domains. Diffusion in fractal spaces highlights neatly how the modification in the switching-time statistics associated with reflections against a complex and fractal boundary induces new emergent features of Poisson-Kac diffusion leading to a transition from a regular behavior at shorter timescales to emerging anomalous diffusion properties controlled by walk dimensionality of the fractal set.

Extended Poisson-Kac theory: A unifying framework for stochastic processes with finite propagation velocity

“…[100]. We remark that the occurrence of long-term subdiffusive scaling in stochastic processes possessing finite propagation velocity has also already been obtained for symmetric random walks on fractals [101] or generalised PK processes in pre-fractal media [102]. Motivated by these results, in this section we show that within our theoretical framework we can formulate an EPK process that can capture particularly subdiffusion solely as the result of microscopic correlations among its transition rates.…”

Markovian nature, completeness, regularity and correlation properties of generalized Poisson-Kac processes

“…Figure 4 depicts a portion of the orbit of a Poisson-Kac particle at a = 1. The orbit is almost everywhere smooth, and discontinuity in the velocity arise as a consequences either of the internal Poissonian switching or of the reflection at the boundary [52]. At longer time-scales, this regularity is broken as a consequence of the two mechanisms mentioned above, and the anomalous features of Brownian motion can be recovered as a long-term property.…”

“…Subsequently we analyze the implications of the emergent fractality considering a pattern formation problem arising from colloidal aggregation theory [71]. Poisson-Kac stochastic processes on fractals have been studied in [52]. The content of paragraph 4.1 is therefore a succinct review of the analysis developed in [52], albeit using different fractal models, to avoid repetition.…”

“…Poisson-Kac stochastic processes on fractals have been studied in [52]. The content of paragraph 4.1 is therefore a succinct review of the analysis developed in [52], albeit using different fractal models, to avoid repetition. Although, some degree of overlapping exists between the content of paragraph 4.1 and the analysis is [52], this review is nonetheless functional to the novel analysis developed in paragraph 4.2, and further in 5.…”

“…Finally, some observations on the role of correlation properties of noise sources and their influence on the dynamic properties of transport phenomena are addressed, using a Wiener model for comparison. implies in turn that the higher-dimensional Cattaneo equation does not preserve non-negativity [49], which is highly unpleasant when this equation is applied to describe the space-time evolution of molecular concentrations, absolute temperature fields, or probability density functions, which by definition attain strictly non-negative values.Recently, we have started a systematic study of Poisson-Kac processes and of their higher-dimensional extensions (GPK) with the fundamental goals just mentioned above [50,51,52,53,54]. In developing such a research program, one encounters some delicate issues related to the very basic properties of Poisson-Kac and GPK processes, that require a careful dissection in order to avoid misunderstanding and misinterpretations.One of this issues is the Markovian nature of these processes.…”