Non-Markovian diffusion equations and processes: analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.Comment: 43 pages, 19 figures, in press on Physica A (2008

Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel $\sim 1/t^{1+\theta}$ with $0 \leq \theta <1$. In the presence of an epidemic-like process of triggered shocks developing in a cascade of generations at or close to criticality, this "bare" kernel is renormalized into an even slower decaying response function $\sim 1/t^{1-\theta}$. Surprisingly, this means that the shorter the memory of the bare kernel (the larger $1+\theta$), the longer the memory of the response function (the smaller $1-\theta$). Here, we present a detailed investigation of this paradoxical behavior based on a generation-by-generation decomposition of the total response function, the use of Laplace transforms and of "anomalous" scaling arguments. The paradox is explained by the fact that the number of triggered generations grows anomalously with time at $\sim t^\theta$ so that the contributions of active generations up to time $t$ more than compensate the shorter memory associated with a larger exponent $\theta$. This anomalous scaling results fundamentally from the property that the expected waiting time is infinite for $0 \leq \theta \leq 1$. The techniques developed here are also applied to the case $\theta >1$ and we find in this case that the total renormalized response is a {\bf constant} for $t < 1/(1-n)$ followed by a cross-over to $\sim 1/t^{1+\theta}$ for $t \gg 1/(1-n)$.Comment: 27 pages, 4 figure

Glycoproteins in Claudin-Low Breast Cancer Cell Lines Have a Unique Expression Profile

Claudin proteins are components of epithelial tight junctions; a subtype of breast cancer has been defined by the reduced expression of mRNA for claudins and other genes. Here, we characterize the expression of glycoproteins in breast cell lines for the claudin-low subtype using liquid chromatography/tandem mass spectrometry. Unsupervised clustering techniques reveal a group of claudin-low cell lines that is distinct from nonmalignant, basal, and luminal lines. The claudin-low cell lines express F11R, EPCAM, and other proteins at very low levels, whereas CD44 is expressed at a high level. Comparison of mRNA expression to glycoprotein expression shows modest correlation; the best agreement occurs when the mRNA expression level is lowest and little or no protein is detected. These findings from cell lines are compared to those for tumor samples by the Clinical Proteomic Tumor Analysis Consortium (CPTAC). The CPTAC samples contain a group low in CLDN3. The samples low in CLDN3 proteins share many differentially expressed glycoproteins with the claudin-low cell lines. In contrast to the situation for cell lines or patient samples classified as claudin-low by RNA expression, however, most of the tumor samples low in CLDN3 protein express the estrogen receptor or HER2. These tumor samples express CD44 protein at low rather than high levels. There is no correlation between CLDN3 gene expression and protein expression in these CPTAC samples; hence, the claudin-low subtype defined by gene expression is not the same group of tumors as that defined by low expression of CLDN3 protein