arXiv subjects

Avi Mayorcas

Publications and source records attributed to Avi Mayorcas.

15 recordsLinked to original sources

arXiv2025

Ergodic Theory for Fractional SDE with Singular Coefficients

We show existence and uniqueness of invariant measures for SDE of the form \[ dX_t = g(X_t)dt + u(X_t)dt + dW^H_t \] where $W^H$ is a fractional Brownian motion (fBm) with Hurst parameter $H\in (0,\frac{1}{2})$, $u$ is a linearly dispersive term and $g$ is any $B^\alpha_{\infty,\infty}(\mathbb{R}^d)$ distribution in the class treated by Catellier--Gubinelli `16, i.e. $\alpha>1-\frac{1}{2H}$. The significant challenge is to combine the regularizing effect of the fBm with an ergodic theory suited to non-Markovian SDE. Concerning the latter our first main contribution is to construct a bona fide stochastic dynamical system (SDS) (Hairer `05 and Hairer--Ohashi `07) associated to the equation above. Since the solution map is only continuous in the support of the stationary noise process we weaken the definitions introduced by Hairer `05 and Hairer--Ohashi `07 but manage to retain the Doob--K'hashminksii provided by Hairer--Ohashi `07. Our second innovation is to introduce a family of flexible local-global stochastic sewing lemmas, in the vein of L\^e `20, which allows us to efficiently treat small and large scales simultaneously. By tuning the local scale as a function of $\|g\|_{B^\alpha_{\infty,\infty}}$ we are able to obtain the necessary continuity of the semi-group and stability estimates to show unique ergodicity for all $g\in B^{\alpha}_{\infty,\infty}(\mathbb{R}^d)$. We believe that these local-global sewing lemmas may be of independent interest.

math.PR↗

arXiv2025

Homogenisation of a Passive Scalar Transported by Locally Supported White Noise

Stochastic perturbations of transport type are a common and widely accepted way of representing turbulent effects in fluid dynamics models. In many known examples, it even leads to improved solution theory, a phenomenon known as \emph{regularization by noise}. A common thread in the recent literature on the topic is the so-called \emph{It\^o-Stratonovich diffusion limit}. By selecting Stratonovich transport noise with carefully arranged vector fields, one can show that the solution of certain SPDEs are close, in an appropriate topology, to an effective, deterministic, equation with a new effective second order elliptic operator, linked to the Ito-Stratonovich corrector. In this work, we deal with a passive scalar model with molecular diffusivity $\kappa$. Starting from the results in [Flandoli \emph{et al.}, 2022, \emph{Philos. Trans. Roy. Soc. A}, 380(2219)], we consider a transport noise made by a sum of independent and compactly supported vector fields. This setting is relevant for models of stratified turbulence which naturally occur in boundary layers and Boussinesq models. Due to the anisotropic nature of the noise, the identification of the limit equation is not straightforward as in all other examples known in literature, as the Ito-Stratonovich corrector is a generic second order elliptic operator with non-constant coefficients. Using tools from Homogenisation theory, we obtain a representation for the limiting effective diffusivity matrix. Exploiting this representation, we study asymptotics, in the $\kappa \rightarrow 0$ regime, of the effective diffusivity across a number of vector field regimes parametrised by the radius of their support. Finally, we provide a careful numerical analysis of the effective diffusivity, discovering a nonlinear behavior for $\kappa \rightarrow 0$, in some regimes.

math.AP↗

arXiv2025

Stochastic Analysis of Overlapping Generations Models Under Incomplete Markets

We provide a stochastic analysis of an overlapping-generations model under incomplete markets. By casting individual optimisation with idiosyncratic income risk into a forward-backward stochastic differential equation (FBSDE) system, we (i) establish existence and uniqueness of the dynamic general-equilibirum interest rate and (ii) derive semi-explicit formulas for both the equilibrium interest rate path and the natural borrowing limit - defined as the discounted expected shortfall of future income. Our FBSDE-based approach yields tractable policy functions and equilibrium mappings without relying on high-dimensional PDE methods, offering clear insights into how income dynamics and demographic structure drive intereate-rate fluctuations and credit constraints.

math.PR↗

arXiv2024

An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Small-Noise Results

We study an additive-noise approximation to Keller-Segel-Dean-Kawasaki dynamics, which is proposed as an approximate model to the fluctuating hydrodynamics of chemotactically interacting particles around their mean-field limit. As such, the interaction potential is given by the Green's function associated to Poisson's equation, which is singular around the origin. Two parameters play a key r\^{o}le in the approximation: the noise intensity $\varepsilon$ which captures the amplitude of fluctuations (tending to zero as the effective system size tends to infinity) and the correlation length $\delta$ which represents the effective scale under consideration. Let $\delta(\varepsilon)\to0$ as $\varepsilon\to0$. Under the relative scaling assumption $\lim_{\varepsilon\to0}\varepsilon\log(\delta(\varepsilon)^{-1})=0$ we obtain analogues of law of large numbers and large deviation principles in irregular spaces of distributions using methods of singular stochastic partial differential equations. The same techniques also yield a central limit theorem under the relative scaling $\lim_{\varepsilon\to0}\varepsilon^{1/2}\log(\delta(\varepsilon)^{-1})=0$. Assuming the more restrictive relative scaling $\lim_{\varepsilon\to0}\varepsilon^{1/2}\delta^{-\gamma-2}=0$ for some $\gamma\in(-1/2,0)$, we also obtain analogues of law of large numbers and large deviation principles in regular function spaces using a mixture of pathwise and probabilistic tools. We further describe consequences of these results relevant to applications of our approximation in studying continuum fluctuations of particle systems.

math.PR↗

arXiv2024

Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion

We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles $N\to\infty$, with quantitative sharp rates of order $N^{-1/2}$. Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter $H\in (0,+\infty)\setminus \mathbb{N}$ of the driving noises. In particular, as $H$ decreases to $0$, interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.

math.PR↗

arXiv2024

Large Deviations of the $\Phi^4_3$ Measure via Stochastic Quantisation

The $\Phi^4_3$ measure is one of the easiest non-trivial examples of a Euclidean quantum field theory (EQFT) whose rigorous construction in the 1970's has been one of the celebrated achievements of constructive quantum field theory. In recent years, progress in the field of singular stochastic PDEs, initiated by the theory of regularity structures, has allowed for a new construction of the $\Phi^4_3$ EQFT as the invariant measure of a previously ill-posed Langevin dynamics, a strategy originally proposed by Parisi and Wu ('81) under the name stochastic quantisation. We apply the same methodology to obtain a large deviation principle (LDP) for the family of periodic $\Phi^4_3$ measures at varying temperature. In addition, we show that the rate functional of the LDP and the $\Phi^4_3$ action functional coincide up to a constant. We wish to highlight that while our main result had previously been obtained by Barashkov (2022), the main focus of this work is on the approach through the stochastic quantisation equation.

math.PR↗

arXiv2023

Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion

We show pathwise uniqueness of multiplicative SDEs, in arbitrary dimensions, driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ with volatility coefficient $\sigma$ that is at least $\gamma$-H\"older continuous for $\gamma > \frac{1}{2H} \vee \frac{1-H}{H}$. This improves upon the long-standing results of [Lyo94 , Lyo98 , Dav08] which cover the same regime but require $\sigma$ to be at least $\frac{1}{H}$-H\"older continuous. Our central innovation is to combine stochastic averaging estimates with refined versions of the stochastic sewing lemma, due to [L\^e20, Ger22, MP22].

math.PR↗

arXiv2023

Blow-up for a Stochastic Model of Chemotaxis Driven by Conservative Noise on $\mathbb{R}^2$

We establish criteria on the chemotactic sensitivity $χ$ for the non-existence of global weak solutions (i.e. \textit{blow-up} in finite time) to a stochastic Keller--Segel model with spatially inhomogeneous, conservative noise on $\mathbb{R}^2$. We show that if $χ$ is sufficiently large then \emph{blow-up} occurs with probability $1$. In this regime our criterion agrees with that of a deterministic Keller--Segel model with increased viscosity. However, for $χ$ in an intermediate regime, determined by the variance of the initial data and the spatial correlation of the noise, we show that \emph{blow-up} occurs with positive probability.

math.AP↗

arXiv2023

Pathwise Regularisation of Singular Interacting Particle Systems and their Mean Field Limits

We investigate the regularizing effect of certain perturbations by noise in singular interacting particle systems under the mean field scaling. In particular, we show that the addition of a suitably irregular path can regularise these dynamics and we recover the McKean--Vlasov limit under very broad assumptions on the interaction kernel; only requiring it to be controlled in a possibly distributional Besov space. In the particle system we include two sources of randomness, a common noise path $Z$ which regularises the dynamics and a family of idiosyncratic noises, which we only assume to converge in mean field scaling to a representative noise in the McKean--Vlasov equation.

math.PR↗

arXiv2023

Singular SPDEs on Homogeneous Lie Groups

The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $$\partial_t u = \mathfrak{L} u+ F(u, \xi)\ ,$$ where the differential operator $\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $$\partial_t u = \sum_{i} X^2_i u + u (\xi-c)$$ on the compact quotient of an arbitrary Carnot group.

math.PR↗

arXiv2022

An Additive-Noise Approximation to Keller-Segel-Dean-Kawasaki Dynamics: Local Well-Posedness of Paracontrolled Solutions

Using the method of paracontrolled distributions, we show the local well-posedness of an additive noise approximation to the fluctuating hydrodynamics of the Keller-Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green's function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

math.PR↗

arXiv2022

A Stochastic Model of Chemorepulsion with Additive Noise and Nonlinear Sensitivity

We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. We show that for any suitable initial data there exists a pathwise unique, global solution to the SPDE. Furthermore we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder-Besov space of positive regularity, which the solution law converges to exponentially fast. We also establish tail bounds on the invariant measure that are heavier than Gaussian when measured using any $L^p$ norm.

math.PR↗

arXiv2021

Introduction to SPDEs from Probability and PDE

Lecture notes accompanying an 8hr hour mini-course on SPDE given at Bo\u{g}azi\c{c}i University, Istanbul in June/July 2025. They are based on earlier notes of a shorter mini-course given at the University of Oxford in 2021. The main focus of these notes is on an exposition of the variational method for monotone and coercive SPDE. A recap of the necessary functional analysis, operator theory and stochastic analysis on Hilbert spaces is included, with additional references. The final chapter contains a discussion of the pathwise approach.

math.PR↗

arXiv2021

Distribution dependent SDEs driven by additive fractional Brownian motion

We study distribution dependent stochastic differential equations with irregular, possibly distributional drift, driven by an additive fractional Brownian motion of Hurst parameter $H\in (0,1)$. We establish strong well-posedness under a variety of assumptions on the drift; these include the choice $$B(\cdot,μ) = f\astμ(\cdot) + g(\cdot),\quad f,g\in B^α_{\infty,\infty}, \quad α>1-1/2H,$$ thus extending the results by Catellier and Gubinelli [9] to the distribution dependent case. The proofs rely on some novel stability estimates for singular SDEs driven by fractional Brownian motion and the use of Wasserstein distances.

math.PR↗

arXiv2021

Distribution dependent SDEs driven by additive continuous noise

We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [16]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

math.PR↗