Eva Löcherbach

Limit theorems for null recurrent Markov processes

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.

Hydrodynamic Limit for Interacting Neurons

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

Likelihood Ratio Processes for Markovian Particle Systems with Killing and Jumps

\frac{1}{N}

Non-parametric Estimation of the Death Rate in Branching Diffusions

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.

Neighborhood radius estimation for Variable-neighborhood Random Fields

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.

LAN and LAMN for systems of interacting diffusions with branching and immigration

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.

Hawkes processes with variable length memory and an infinite number of components

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

On local asymptotic normality for birth and death on a flow

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.