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Juraj Foldes

Publications and source records attributed to Juraj Foldes.

14 recordsLinked to original sources
arXiv2026

Stochastic Persistence in Infinite Dimensions

Motivated by infinite-dimensional ecological and biological models such as reaction-diffusion SPDEs and stochastic functional differential equations, we develop a general criteria for stochastic persistence (coexistence) in terms of an average lyapunov function, which was previously known only in finite dimensions. To apply our results to SPDEs we analyze the projective process, and we employ a combination of mild (stochastic convolution) and variational (lyapunov function) techniques. Our analysis also requires some nontrivial well-posedness and nonnegativity results for reaction-diffusion SPDEs, which we state and prove in great generality, extending the known results in the literature. Finally, we present several examples including ecological models (Lotka-Volterra), an epidemic model (SIR), and a model for turbulence. Notably we show that, as in the SDE case, coexistence in the Lotka-Volterra model is determined by the invasion rates.

math.PR↗
arXiv2024

Stochastic Extinction, An Average Lyapunov Function Approach

We study the stability of $\mathcal{M}_0$, an invariant subset of a Markov process $(X_t)_{t\geq 0}$ on a metric space $\mathcal{M}$. By building the theory of average Lyapunov functions, we formulate general criteria based on the signs of Lyapunov exponents that guarantee extinction ($X_t \to \mathcal{M}_0$ as $t \to \infty$). Additionally, we provide applications to a stochastic SIS epidemic model on a network with regime-switching, a stochastic differential equation version of the Lorenz system, a general class of discrete-time ecological models, and stochastic Kolmogorov systems. In many examples we improve existing results by removing unnecessary assumptions or providing sharper criteria for the extinction.

math.PR↗
arXiv2024

Higher order expansion for the probabilistic local well-posedness theory for a cubic nonlinear Schrödinger equation

In this paper, we study the probabilistic local well-posedness of the cubic Schrödinger equation (cubic NLS): \[ (i\partial_{t} + Δ) u = \pm |u|^{2} u \text{ on } [0,T) \times \mathbb{R}^{d}, \] with initial data being a Wiener randomization at unit scale of a given function $f$. We prove that a solution exists almost-surely locally in time provided \(f\in H^{S}_{x}(\mathbb{R}^{d})\) with \(S>\max\big(\frac{d-3}{4},\frac{d-4}{2}\big)\) for \(d\geq 3\). In particular, we establish that the local well-posedness holds for any \(S>0\) when \(d=3\). We also show that, under appropriate smallness conditions for the initial data, the solutions are global in time and scatter. The solutions are constructed as a sum of an explicit multilinear expansion of the flow in terms of the random initial data and of an additional smoother remainder term with deterministically subcritical regularity. This construction allows us to introduce a new and refined notion of graded scattering. We develop the framework of directional space-time norms to control the (probabilistic) multilinear expansion and the (deterministic) remainder term and to obtain improved bilinear probabilistic-deterministic Strichartz estimates.

math.AP↗
arXiv2023

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators

In this paper, we study the local well-posedness of the cubic Schrödinger equation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I \times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ being an operator of degree $σ\geq 2$. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation (i.e. $\mathscr{L} = Δ$) to any dimension and obtain results under natural assumptions for general $\mathscr{L}$.

math.AP↗
arXiv2021

The method of stochastic characteristics for linear second-order hypoelliptic equations

We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green's functions and to obtain a generalization of Bony's Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.

math.AP↗
arXiv2021

A functional Law of the Iterated Logarithm for weakly hypoelliptic diffusions at time zero

We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hormander condition but not the strong Hormander condition}. That is, SDEs in which the drift terms must be used in order to conclude hypoellipticity. As a corollary of this result, we obtain the almost sure behavior as time goes to zero of a given direction in the equation, even if noise is not present explicitly in that direction. The techniques used to prove the main results are based on large deviations applied to a non-trivial rescaling of the original system. In concrete examples, we show how to find the proper rescaling to obtain the functional LIL. Furthermore, we apply the main results to the problem of identifying regular points for hypoelliptic diffusions. Consequently, we obtain a control-theoretic criteria for a given point to be regular for the process.

math.PR↗
arXiv2021

Existence of traveling waves for a fourth order Schr\" odinger equation with mixed dispersion in the Helmholtz regime

In this paper, we study the existence of traveling waves for a fourth order Schr\" odinger equations with mixed dispersion, that is, solutions to $$Δ^2 u +βΔu +i V \nabla u +αu =|u|^{p-2} u,\ in\ \R^N ,\ N\geq 2.$$ We consider this equation in the Helmholtz regime, when the Fourier symbol $P$ of our operator is strictly negative at some point. Under suitable assumptions, we prove the existence of solution using the dual method of Evequoz and Weth provided that $p\in (p_1 , 2N/(N-4)_+)$. The real number $p_1$ depends on the number of principal curvature of $M$ staying bounded away from $0$, where $M$ is the hypersurface defined by the roots of $P$. We also obtained estimates on the Green function of our operator and a $L^p - L^q$ resolvent estimate which can be of independent interest and can be applied to other operators.

math.AP↗
arXiv2020

Sensitivity of steady states in a degenerately-damped stochastic Lorenz system

We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz '63 model. Specifically, we prove that when damping is absent in one of the temperature components, the system possesses a unique invariant probability measure if and only if noise acts on the convection variable. On the other hand, if there is a positive growth term on the vertical temperature profile, we prove that there is no normalizable invariant state. Our approach relies on the derivation and analysis of non-trivial Lyapunov functions which ensure positive recurrence or null-recurrence/transience of the dynamics.

math.PR↗
arXiv2020

Invariant measures and global well posedness for SQG equation

We construct an invariant measure $μ$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $μ$ are initial conditions of global, unique solutions of SQG, that depend continuously on the initial data. In addition, we show that the support of $μ$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measures $μ$ is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.

math.AP↗
arXiv2020

Singular radial solutions for the Lin-Ni-Takagi equation

We study singular radially symmetric solution to the Lin-Ni-Takagi equation for a supercritical power non-linearity in dimension $N\geq 3$. It is shown that for any ball and any $k \geq 0$, there is a singular solution that satisfies Neumann boundary condition and oscillates at least $k$ times around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph--Lundgren exponent.

math.AP↗
arXiv2018

Singular radial solutions for the Keller-Segel equation in high dimension

We study singular radially symmetric solution of the stationary Keller-Segel equation, that is, an elliptic equation with exponential nonlinearity, which is super-critical in dimension $N \geq 3$. The solutions are unbounded at the origin and we show that they describe the asymptotics of bifurcation branches of regular solutions. It is shown that for any ball and any $k \geq 0$, there is a singular solution that satisfies Neumann boundary condition and oscillates at least $k$ times around the constant equilibrium. Moreover, we prove that in dimension $3 \leq N \leq 9$ there are regular solutions satisfying Neumann boundary conditions that are close to singular ones. Hence, it follows that there exist regular solutions on any ball with arbitrarily fast oscillations. For generic radii, we show that the bifurcation branches of regular solutions oscillate in the bifurcation plane when $3\leq N\leq 9$ and approach to a singular solution. In dimension $N > 10$, we show that the Morse index of the singular solution is finite, and therefore the existence of regular solutions with fast oscillations is not expected.

math.AP↗
arXiv2018

Large Prandtl Number Asymptotics in Randomly Forced Turbulent Convection

We establish the convergence of statistically invariant states for the stochastic Boussinesq Equations in the infinite Prandtl number limit and in particular demonstrate the convergence of the Nusselt number (a measure of heat transport in the fluid). This is a singular parameter limit significant in mantle convection and for gasses under high pressure. The equations are subject to a both temperature gradient on the boundary and internal heating in the bulk driven by a stochastic, white in time, gaussian forcing. Here, the stochastic source terms have a strong physical motivation for example as a model of radiogenic heating. Our approach uses mixing properties of the formal limit system to reduce the convergence of invariant states to an analysis of the finite time asymptotics of solutions and parameter-uniform moment bounds. Here, it is notable that there is a phase space mismatch between the finite Prandtl system and the limit equation, and we implement methods to lift both finite and infinite time convergence results to an extended phase space which includes velocity fields. For the infinite Prandtl stochastic Boussinesq equations, we show that the associated invariant measure is unique and that the dual Markovian dynamics are contractive in an appropriate Kantorovich-Wasserstein metric. We then address the convergence of solutions on finite time intervals, which is still a singular perturbation. In the process we derive well-posed equations which accurately approximate the dynamics up to the initial time when the Prandtl number is large.

math.AP↗
arXiv2017

On the Born-Infeld equation for electrostatic fields with a superposition of point charges

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_kδ_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ where $N\ge3$, $a_k\in\mathbb R$ for all $k=1,\dots,n$, $x_k\in\mathbb R^N$ are the positions of the point charges, possibly non symmetrically distributed, and $δ_{x_k}$ is the Dirac delta distribution centered at $x_k$. For this problem, we give explicit quantitative sufficient conditions on $a_k$ and $x_k$ to guarantee that the minimizer of the energy functional associated to the problem solves the associated Euler-Lagrange equation. Furthermore, we provide a more rigorous proof of some previous results on the nature of the singularities of the minimizer at the points $x_k$'s depending on the sign of charges $a_k$'s. For every $m\in\mathbb N$, we also consider the approximated problem $$ -\sum_{h=1}^mα_hΔ_{2h}u=\sum_{k=1}^n a_kδ_{x_k}\quad\mbox{in }\mathbb R^N, \qquad\lim_{|x|\to\infty}u(x)=0 $$ where the differential operator is replaced by its Taylor expansion of order $2m$, see (2.1). It is known that each of these problems has a unique solution. We study the regularity of the approximating solution, the nature of its singularities, and the asymptotic behavior of the solution and of its gradient near the singularities.

math.AP↗