arXiv subjects

Alexander Alecio

Publications and source records attributed to Alexander Alecio.

4 recordsLinked to original sources

arXiv2025

Noise induced Stability of a Mean-Field model of Systemic Risk with uncertain robustness

We consider a model for systemic risk comprising of a system of diffusion processes, interacting through their empirical mean. Each process is subject to a confining double-well potential with some uncertainty in the coefficients, corresponding to fluctuations in height of the potential barrier seperating the two wells. This is equivalent to studying a single McKean-Vlasov SDE with explicit dependence on its moments and, novelly, independently varying additive and multiplicative noise. Such non-linear SDEs are known to possess two phases: stable (ordered) and unstable (disordered). When the potential is purely bistable, the phase changes from stable to unstable when noise intensity is increased past a critical threshold. With the recent advances, it will be shown that the behaviour here is far richer: indeed, depending on the interpretation of the stochastic integral, the system exhibits phase changes that cannot occur in any regime where there is no uncertainty in the potential. Strikingly, this allows for the phenomenon of noise induced stability; situations where more noise can reduce the risk of system failure.

math.PR↗

arXiv2024

Stationary properties of the Gauss-Galerkin QMoM truncation of MV-SDEs

The Gauss Galerkin Method/Quadrature method of moments (GG-QMoM) closure scheme, introduced by Dawson, closes a truncated set of moment equations of an SDE by a Galerkin approximation of its law in the space of probability measures. Here, results are presented on stationary solutions of the closed equations, irrespective of the number of moments retained (hence not dependent on the convergence theorem). These are applied to polynomial MV-SDEs, with explicit dependence on its moments in the drift, which can possess multiple stationary solutions. Particularly, we show in the deterministic limit, there are as many stationary solutions as extrema of the potential (critically, not just the minima), preserving the bifurcation diagram of the full equations. Further, a scaling property of solutions is proven, allowing changes of stability to be directly probed. Finally, this is applied to the GG-QMoM truncation of Dawson-Shiino model, whose stationary measures and change in stability has been previously studied.

math.PR↗

arXiv2023

Phase transitions of McKean-Vlasov SDEs in Multi-well Landscapes

Phase transitions and critical behaviour of a class of MV-SDEs, whose concomitant non-local Fokker-Planck equation includes the Granular Media equation with quadratic interaction potential as a special case, is studied. By careful analysis of an implicit auxiliary integral equation, it is shown for a wide class of potentials that below a certain `critical threshold' there are exactly as many stationary measures as extrema of the potential, while above another the stationary measure is unique, and consequently phase transition(s) between. For symmetric bistable potentials, these critical thresholds are proven to be equal and a strictly increasing function of the aggregation parameter. Additionally, a simple condition is provided for symmetric multi-well potentials with an arbitrary number of extrema to demonstrate analogous behaviour. This answers, with considerably more generality, a conjecture of Tugaut [Stochastics, 86:2, 257-284]. To the best of our knowledge many of these results are novel. Others simplify the proofs of known results whilst greatly increasing their applicability.

math.PR↗

arXiv2023

Dimension reduction of noisy interacting systems

We consider a class of models describing an ensemble of identical interacting agents subject to multiplicative noise. In the thermodynamic limit, these systems exhibit continuous and discontinuous phase transitions in a, generally, nonequilibrium setting. We provide a systematic dimension reduction methodology for constructing low dimensional, reduced-order dynamics based on the cumulants of the probability distribution of the infinite system. We show that the low dimensional dynamics returns the correct diagnostic properties since it produces a quantitatively accurate representation of the stationary phase diagram of the system that we compare with exact analytical results and numerical simulations. Moreover, we prove that the reduced order dynamics yields also the prognostic, i.e., time dependent properties, as it provides the correct response of the system to external perturbations. On the one hand, this validates the use of our complexity reduction methodology since it retains information not only of the invariant measure of the system but also of the transition probabilities and time dependent correlation properties of the stochastic dynamics. On the other hand, the breakdown of linear response properties is a key signature of the occurrence of a phase transition. We show that the reduced response operators capture the correct diverging resonant behaviour by quantitatively assessing the singular nature of the susceptibility of the system and the appearance of a pole for real value of frequencies. Hence, this methodology can be interpreted as a low dimensional, reduced order approach to the investigation and detection of critical phenomena in high dimensional interacting systems in settings where order parameters are not known.

cond-mat.stat-mech↗