Institute of Mathematics

Our in silico model was built to investigate the development process of the adaptive immune system. For simplicity, we concentrated on humoral immunity and its major components: T cells, B cells, antibodies, interleukins, non-immune self cells, and foreign antigens. Our model is a microscopic one, similar to the interacting particle models of statistical physics. Events are considered random and modelled by a continuous time, finite state Markov process, that is, they are controlled by independent exponential clocks. Our main purpose was to compare different theoretical models of the adaptive immune system and self--nonself discrimination: the ones that are described by well-known textbooks, and a novel one developed by our research group. Our theoretical model emphasizes the hypothesis that the immune system of a fetus can primarily learn what self is but unable to prepare itself for the huge, unknown variety of nonself. The simulation begins after conception, by developing the imm...

We give an extension of the Fekete's Subadditive Lemma for a set of submultiplicative functionals on countable product of compact spaces. Our method can be considered as an unfolding of he ideas [1]Theorem 3.1 and our main result is the extension of the symbolic dynamics results of .

The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, even beyond the appearence of shocks. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following , the stochastic version of Tartar-Murat theory of compensated compactness is extended to two-component stochastic models.

We study the asymptotic behavior of a self-interacting one-dimensional Brownian polymer first introduced by Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. The polymer describes a stochastic process with a drift which is a certain average of its local time. We show that a smeared out version of the local time function as viewed from the actual position of the process is a Markov process in a suitably chosen function space, and that this process has a Gaussian stationary measure. As a first consequence, this enables us to partially prove a conjecture about the law of large numbers for the end-to-end displacement of the polymer formulated in Durrett and Rogers [Probab. Theory Related Fields 92 (1992) 337--349]. Next we give upper and lower bounds for the variance of the process under the stationary measure, in terms of the qualitative infrared behavior of the interaction function. In particular, we show that in the locally self-repelling case (when the process is essentially pushed by the negative gradient of its own local time) the process is super-diffusive.

We present the derivation of the hydrodynamic limit under Eulerian scaling for a general class of one-dimensional interacting particle systems with two or more conservation laws. Following Yau's relative entropy method it turns out that in case of more than one conservation laws, in order that the system exhibit hydrodynamic behaviour, some particular identities reminiscent of Onsager's reciprocity relations must hold. We check validity of these identities whenever a stationary measure with product structure exists. It also follows that, as a general rule, the equilibrium thermodynamic entropy (as function of the densities of the conserved variables) is a globally convex Lax entropy of the hyperbolic systems of conservation laws arising as hydrodynamic limit. As concrete examples we also present a number of models modeling deposition (or domain growth) phenomena. The Onsager relations arising in the context of hydrodynamic limits under hyperbolic scaling seem to be novel. The fact that equilibrium thermodynamic entropy is Lax entropy for the arising Euler equations was noticed earlier in the context of Hamiltonian systems with weak noise, see ref. 7.

We investigate propagation of perturbations of equilibrium states for a wide class of 1D interacting particle systems. The class of systems considered incorporates zero range, K-exclusion, misanthropic, ''bricklayers'' models, and much more. We do not assume attractivity of the interactions. We apply Yau's relative entropy method rather than coupling arguments. The result is partial extension of T. Seppäläinen's recent paper. For 0 < b < 1/5 fixed, we prove that, rescaling microscopic space and time by N, respectively N 1+b , the macroscopic evolution of perturbations of microscopic order N −b of the equilibrium states is governed by Burgers' equation. The same statement should hold for 0 < b < 1/2 as in Seppäläinen's cited paper, but our method does not seem to work for b \ 1/5.

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie-Weiss Ising model and includes as well all ferromagnetic Curie-Weiss Potts and Curie-Weiss Heisenberg models. By de Finetti's theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that "ferromagnetism" is not however in itself sufficient and also study in some detail the Curie-Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie-Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a "formula" for the extension which is valid in many cases. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2007, Vol. 35, No. 3, 867-914. This reprint differs from the original in pagination and typographic detail. 1