Extended Poisson-Kac Theory: A Unifying Framework for Stochastic Processes

Stochastic processes play a key role for modeling a huge variety of transport problems out of equilibrium, with manifold applications throughout the natural and social sciences. To formulate models of stochastic dynamics the conventional approach consists in superimposing random fluctuations on a suitable deterministic evolution. These fluctuations are sampled from probability distributions that are prescribed a priori, most commonly as Gaussian or L\'evy. While these distributions are motivated by (generalised) central limit theorems they are nevertheless \textit{unbounded}, meaning that arbitrarily large fluctuations can be obtained with finite probability. This property implies the violation of fundamental physical principles such as special relativity and may yield divergencies for basic physical quantities like energy. Here we solve the fundamental problem of unbounded random fluctuations by constructing a comprehensive theoretical framework of stochastic processes possessing physically realistic finite propagation velocity. Our approach is motivated by the theory of L\'evy walks, which we embed into an extension of conventional Poisson-Kac processes. The resulting extended theory employs generalised transition rates to model subtle microscopic dynamics, which reproduces non-trivial spatio-temporal correlations on macroscopic scales. It thus enables the modelling of many different kinds of dynamical features, as we demonstrate by three physically and biologically motivated examples. The corresponding stochastic models capture the whole spectrum of diffusive dynamics from normal to anomalous diffusion, including the striking `Brownian yet non Gaussian' diffusion, and more sophisticated phenomena such as senescence. Extended Poisson-Kac theory can therefore be used to model a wide range of finite velocity dynamical phenomena that are observed experimentally.Comment: 26 pages, 5 figure

Weak Galilean invariance as a selection principle for coarse-grained diffusive models

Galilean invariance is a cornerstone of classical mechanics. It states that for closed systems the equations of motion of the microscopic degrees of freedom do not change under Galilean transformations to different inertial frames. However, the description of real world systems usually requires coarse-grained models integrating complex microscopic interactions indistinguishably as friction and stochastic forces, which intrinsically violate Galilean invariance. By studying the coarse-graining procedure in different frames, we show that alternative rules -- denoted as "weak Galilean invariance" -- need to be satisfied by these stochastic models. Our results highlight that diffusive models in general can not be chosen arbitrarily based on the agreement with data alone but have to satisfy weak Galilean invariance for physical consistency

Anomalous Processes with General Waiting Times: Functionals and Multipoint Structure

Many transport processes in nature exhibit anomalous diffusive properties with non-trivial scaling of the mean square displacement, e.g., diffusion of cells or of biomolecules inside the cell nucleus, where typically a crossover between different scaling regimes appears over time. Here, we investigate a class of anomalous diffusion processes that is able to capture such complex dynamics by virtue of a general waiting time distribution. We obtain a complete characterization of such generalized anomalous processes, including their functionals and multi-point structure, using a representation in terms of a normal diffusive process plus a stochastic time change. In particular, we derive analytical closed form expressions for the two-point correlation functions, which can be readily compared with experimental data.Comment: Accepted in Phys. Rev. Let

Spectral Properties of Stochastic Processes Possessing Finite Propagation Velocity.

This article investigates the spectral structure of the evolution operators associated with the statistical description of stochastic processes possessing finite propagation velocity. Generalized Poisson-Kac processes and Lévy walks are explicitly considered as paradigmatic examples of regular and anomalous dynamics. A generic spectral feature of these processes is the lower boundedness of the real part of the eigenvalue spectrum that corresponds to an upper limit of the spectral dispersion curve, physically expressing the relaxation rate of a disturbance as a function of the wave vector. We also analyze Generalized Poisson-Kac processes possessing a continuum of stochastic states parametrized with respect to the velocity. In this case, there is a critical value for the wave vector, above which the point spectrum ceases to exist, and the relaxation dynamics becomes controlled by the essential part of the spectrum. This model can be extended to the quantum case, and in fact, it represents a simple and clear example of a sub-quantum dynamics with hidden variables

Age representation of Levy walks: partial density waves, relaxation and first passage time statistics

Lévy walks (LWs) define a fundamental class of finite velocity stochastic processes that can be introduced as a special case of continuous time random walks. Alternatively, there is a hyperbolic representation of them in terms of partial probability density waves. Using the latter framework we explore the impact of aging on LWs, which can be viewed as a specific initial preparation of the particle ensemble with respect to an age distribution. We show that the hyperbolic age formulation is suitable for a simple integral representation in terms of linear Volterra equations for any initial preparation. On this basis relaxation properties, i.e. the convergence towards equilibrium of a generic thermodynamic function dependent on the spatial particle distribution, and first passage time statistics in bounded domains are studied by connecting the latter problem with solute release kinetics. We find that even normal diffusive LWs, where the long-term mean square displacement increases linearly with time, may display anomalous relaxation properties such as stretched exponential decay. We then discuss the impact of aging on the first passage time statistics of LWs by developing the corresponding Volterra integral representation. As a further natural generalization the concept of LWs with wearing is introduced to account for mobility losses

PTLD Burkitt Lymphoma in a Patient with Remote Lymphomatoid Granulomatosis.

Posttransplant lymphoproliferative disorder (PTLD) is a potentially fatal complication of solid organ transplantation. The majority of PTLD is of B-cell origin, and 90% are associated with the Epstein-Barr virus (EBV). Lymphomatoid granulomatosis (LG) is a rare, EBV-associated systemic angiodestructive lymphoproliferative disorder, which has rarely been described in patients with renal transplantation. We report the case of a patient with renal transplantation for SLE, who presented, 9 months after renal transplantation, an EBV-associated LG limited to the intracranial structures that recovered completely after adjustment of her immunosuppressive treatment. Nine years later, she developed a second PTLD disorder with central nervous system initial manifestation. Workup revealed an EBV-positive PTLD Burkitt lymphoma, widely disseminated in most organs. In summary, the reported patient presented two lymphoproliferative disorders (LG and Burkitt's lymphoma), both with initial neurological manifestation, at 9 years interval. With careful reduction of the immunosuppression after the first manifestation and with the use of chemotherapy combined with radiotherapy after the second manifestation, our patient showed complete disappearance of neurologic symptoms and she is clinically well with good kidney function. No recurrence has been observed by radiological imaging until now

On a stochastic partial differential equation with non-local diffusion

In this paper, we prove existence, uniqueness and regularity for a class of stochastic partial differential equations with a fractional Laplacian driven by a space-time white noise in dimension one. The equation we consider may also include a reaction term

Meningeal Relapse of Nodular Lymphocyte Predominant Hodgkin Lymphoma Transformed to T-Cell/Histiocyte-Rich Large B-Cell Lymphoma: A Case Report.

Central nervous system involvement in Hodgkin lymphoma is extremely rare, especially in nodular lymphocyte predominant Hodgkin lymphoma (NLPHL), which usually carries a favorable prognosis. Here we report a case of a young patient with NLPHL, who developed a progressive and fatal neurological deterioration requiring a very extensive work-up including two biopsies to obtain the diagnosis of T-cell/histiocyte-rich large B-cell lymphoma like transformation. This report, which includes post-mortem analysis, highlights the correlations between clinical, radiological, and biological data but also the difficulties encountered in reaching the correct diagnosis

Chronic myelogenous leukemia with acquired c-kit activating mutation and transient bone marrow mastocytosis

Mutations of the c-kit gene have been reported in myeloproliferative disorders. We describe here a case of Ph + (b2a2) chronic myelogenous leukemia that, during the course of disease, showed an unusual bone marrow mast-cell infiltration. A mutational screening for the c-kit gene, performed on DNA routinely cryopreserved during the follow-up, evidenced the D816Y-activating mutation as an additional genetic abnormality. Treatment with imatinib mesylate resulted in a substantial decrease of the BCR-ABL/ABL ratio and in the absence of c-kit mutation. It is likely that the superimposed c-kit mutation, in this case, may account for the transient bone marrow mastocytosis