Eva Löcherbach

Reinhard Höpfner; Eva Löcherbach

Introduction Harris recurrence Stable increasing processes and Mittag Leffler processes The main theorem Proofs for subsection 3.1 - sufficient condition Proofs for subsection 3.1 - necessary condition Nummelin splitting in discrete time Nummelin-like splitting for general continuous time Harris processes and proofs for subsection 3.3 Overview: assumptions (H1) - (H6) References.

A. De Masi; Antonio Galves; Eva Löcherbach; Errico Presutti

This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses. This system can be informally described as follows. Each neuron spikes randomly following a point process with rate depending on its membrane potential. At its spiking time, the membrane potential of the spiking neuron is reset to the value 0 and, simultaneously, the membrane potentials of the other neurons are increased by an amount of potential

Antonio Galves; Eva Löcherbach

Eva Löcherbach

\frac{1}{N}

Eva Löcherbach; Dasha Loukianova; Oleg Loukianov

R. Höpfner; M. Hoffmann; Eva Löcherbach

1N. This mimics the effect of chemical synapses. Additionally, the effect of electrical synapses is represented by a deterministic drift of all the membrane potentials towards the average value of the system. We show that, as the system size N diverges, the distribution of membrane potentials becomes deterministic and is described by a limit density which obeys a non linear PDE which is a conservation law of hyperbolic type.

Eva Löcherbach; Enza Orlandi

We consider a new class of non Markovian processes with a countable number of interacting components. At each time unit, each component can take two values, indicating if it has a spike or not at this precise moment. The system evolves as follows. For each component, the probability of having a spike at the next time unit depends on the entire time evolution of the system after the last spike time of the component. This class of systems extends in a non trivial way both the interacting particle systems, which are Markovian (Spitzer in Adv. Math. 5:246–290, 1970) and the stochastic chains with memory of variable length which have finite state space (Rissanen in IEEE Trans. Inf. Theory 29(5):656–664, 1983). These features make it suitable to describe the time evolution of biological neural systems. We construct a stationary version of the process by using a probabilistic tool which is a Kalikow-type decomposition either in random environment or in space-time. This construction implies uniqueness of the stationary process. Finally we consider the case where the interactions between components are given by a critical directed Erdös-Rényi-type random graph with a large but finite number of components. In this framework we obtain an explicit upper-bound for the correlation between successive inter-spike intervals which is compatible with previous empirical findings.

Eva Löcherbach

We consider Markov processes built from pasting together pieces of strong Markov processes which are killed at a position dependent rate and connected via a transition kernel. We give necessary and sufficient conditions for local absolute continuity of probability laws for such processes on a suitable path space and derive an explicit formula for the corresponding likelihood ratio process. The main tool is the consideration of the process between successive jumps – what we call ‘elementary experiments’ – and criteria for absolute continuity of laws of the process there. We apply our results to systems of branching diffusions with interactions and immigrations.

Pierre Hodara; Eva Löcherbach

Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p > 1, ∃a ∈ R such that ∀x ∈ R, ExT p a < ∞ and EνT p/2 a < ∞, where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form Pν ∣∣∣1t ∫ t 0 f (Xs)ds − μ(f ) ∣∣∣ ≥ ε ) ≤ K(p) 1 tp/2 1 εp A(f ). Here f bounded or bounded and compactly supported and A(f ) = ‖f ‖∞ when f is bounded and A(f ) = μ(|f |) when f is bounded and compactly supported. We also give, under some conditions on the coefficients of X, a polynomial control of ExT p a from above and below. This control is based on a generalized Kac’s formula (see Theorem 4.1) for the moments Exf (Ta) of a differentiable function f . Résumé. Considérons une diffusion récurrente positive avec loi initiale ν et probabilité invariante μ. Pour tout a ∈ R, soit Ta le temps d’atteinte du point a. Supposons qu’il existe p > 1 et un point a ∈ R tels que pour tout x ∈ R, ExT p a < ∞ et EνT p/2 a < ∞. Alors nous obtenons l’inégalité de déviation non-asymptotique suivante: Pν ∣∣∣1t ∫ t 0 f (Xs)ds − μ(f ) ∣∣∣ ≥ ε ) ≤ K(p) 1 tp/2 1 εp A(f ), où f est une fonction bornée ou une fonction bornée à support compact. Ici, A(f ) = ‖f ‖∞ dans le cas d’une fonction bornée et A(f ) = μ(|f |) dans le cas d’une fonction bornée à support compact. De plus, sous certaines conditions sur les coefficients de la diffusion, nous obtenons une minoration et majoration, polynomiale en x, de ExT p a . Ce résultat est basé sur une formule de Kac généralisée (voir théoréme 4.1) pour les moments Exf (Ta) où f est une fonction dérivable. MSC: 60F99; 60J55; 60J60

R. Höpfner; Eva Löcherbach

We consider finite systems of diffusing particles in R with branching and immigration. Branching of particles occurs at position dependent rate. Under ergodicity assumptions, we estimate the position-dependent branching rate based on the observation of the particle process over a time interval [0, t]. Asymptotics are taken as t → ∞. We introduce a kernel-type procedure and discuss its asymptotic properties with the help of the local time for the particle configuration. We compute the minimax rate of convergence in squared-error loss over a range of Holder classes and show that our estimator is asymptotically optimal.