arXiv subjects

Andreas Søjmark

Publications and source records attributed to Andreas Søjmark.

5 recordsLinked to original sources

arXiv2026

A stochastic Fokker--Planck equation for the mean-field limit of a population of noisy integrate-and-fire neurons

We study a densely connected system of excitatory integrate-and-fire neurons which are subject to common noise. The system incorporates a gradual transmission of action potentials and captures the re- and hyperpolarization phases through a random refractory period followed by a reset to a randomized level below the rest potential. As the number of neurons tends to infinity, we show that there is weak convergence to a unique membrane potential density governed by a stochastic Fokker--Planck equation with a well-defined spike transmission rate. The latter is driven by the mean cumulative spike count, which is non-differentiable but shown to satisfy a generalized flux condition. We obtain the uniqueness of the Fokker--Planck equation from energy estimates in the dual of the first Sobolev space. Finally, we give a conditional McKean--Vlasov representation of the membrane potential density as the law of a representative neuron given the common noise.

math.AP↗

arXiv2026

Probabilistic estimates for a system of noisy integrate-and-fire neurons

In this note, we establish various probabilistic estimates for an interacting particle system that describes the evolution of the membrane potentials in a network of excitatory integrate-and-fire neurons, which are subject to both idiosyncratic and common noise. These estimates serve to support a separate work that studies the large population limit and the corresponding stochastic Fokker--Planck equation for the membrane potential density.

math.PR↗

arXiv2025

Functional weak convergence of stochastic integrals for moving averages and continuous-time random walks

There is an extensive theory of weak convergence for moving averages and continuous-time random walks (CTRWs) with respect to Skorokhod's M1 and J1 topologies. Here we address the fundamental question of how this translates into functional limit theorems in the M1 or J1 topology for stochastic integrals driven by these processes. As an important application, we provide weak approximation results for general SDEs driven by time-changed Lévy processes. Such SDEs and their associated fractional Fokker--Planck--Kolmogorov equations are central to models of anomalous diffusion in statistical physics. Our results yield a rigorous functional characterisation of these as continuum limits of the underlying models driven by CTRWs. With regard to strictly M1 convergent moving averages and correlated CTRWs, it turns out that the convergence of stochastic integrals can fail decidedly and fundamental new challenges arise compared to the J1 setting. Nevertheless, we identify natural classes of integrand processes for which there is M1 convergence of the stochastic integrals. We also show that these results are flexible enough to yield functional limit theorems in the M1 topology for certain stochastic delay differential equations driven by moving averages.

math.PR↗

arXiv2025

Contagious McKean--Vlasov problems with common noise: from smooth to singular feedback through hitting times

We consider a family of McKean--Vlasov equations arising as the large particle limit of a system of interacting particles on the positive half-line with common noise and feedback. Such systems are motivated by structural models for systemic risk with contagion. This contagious interaction is such that when a particle hits zero, the impact is to move all the others toward the origin through a kernel which smooths the impact over time. We study a rescaling of the impact kernel under which it converges to the Dirac delta function so that the interaction happens instantaneously and the limiting singular McKean--Vlasov equation can exhibit jumps. Our approach provides a novel method to construct solutions to such singular problems that allows for more general drift and diffusion coefficients and we establish weak convergence to relaxed solutions in this setting. With more restrictions on the coefficients we can establish an almost sure version showing convergence to strong solutions. Under some regularity conditions on the contagion, we also show a rate of convergence up to the time the regularity of the contagion breaks down. Lastly, we perform some numerical experiments to investigate the sharpness of our bounds for the rate of convergence.

math.PR↗

arXiv2024

Functional CLTs for subordinated Lévy models in physics, finance, and econometrics

We present a simple unifying treatment of a broad class of applications from statistical mechanics, econometrics, mathematical finance, and insurance mathematics, where (possibly subordinated) Lévy noise arises as a scaling limit of some form of continuous-time random walk (CTRW). For each application, it is natural to rely on weak convergence results for stochastic integrals on Skorokhod space in Skorokhod's J1 or M1 topologies. As compared to earlier and entirely separate works, we are able to give a more streamlined account while also allowing for greater generality and providing important new insights. For each application, we first elucidate how the fundamental conclusions for J1 convergent CTRWs emerge as special cases of the same general principles, and we then illustrate how the specific settings give rise to different results for strictly M1 convergent CTRWs.

math.PR↗