arXiv subjects

Alexandra Blessing

Publications and source records attributed to Alexandra Blessing.

13 recordsLinked to original sources

arXiv2026

Impact of spinning on the early-warning signs in non-Markovian stochastic systems

We construct early-warning signals for impending critical transitions in non-Markovian systems. We analyze stochastic forcings such as fractional Brownian motion, fractional Ornstein-Uhlenbeck processes and red noise in fast-slow systems exhibiting such transitions. We show that the effectiveness of indicators such as autocovariance, autocorrelation, and spectral density depends on several properties of the underlying system. In particular, we compare the influence of the Hurst index and the bifurcation type. We prove that the rotatory dynamics associated with a Hopf bifurcation substantially alters the scaling laws of these observables. Finally, we provide practical guidelines for implementing these signals and validate them on both theoretical and applied models.

math.PR↗

arXiv2026

Synchronization by noise for stochastic differential equations driven by fractional Brownian motion

We investigate synchronization by noise for stochastic differential equations (SDEs) driven by a fractional Brownian motion (fbm) with Hurst index $H\in(0,1)$. Provided that the SDE has a negative top Lyapunov exponent, we show that a weak form of synchronization occurs. To this aim we use tools from stochastic dynamical systems, random dynamical systems and characterize the support of an invariant measure of a random dynamical system in a non-Markovian setting.

math.PR↗

arXiv2026

Renormalization destroys a finite time bifurcation in the $\Phi^4_2$ equation

We study the singular $\Phi^4_2$ equation at a pitchfork bifurcation of the underlying deterministic dynamics. To this aim, we linearize the SPDE along its stationary solution and show that the support of its finite-time Lyapunov exponents (FTLEs) is the real line, regardless of the bifurcation parameter and in sharp contrast to the non-singular $\Phi^4_1$ equation. The proof relies on a support theorem for the stationary solution and its renormalized square.

math.PR↗

arXiv2025

On the approximation of finite-time Lyapunov exponents for the stochastic Burgers equation

We analyze stochastic partial differential equations (SPDEs) with quadratic nonlinearities close to a change of stability. To this aim we compute finite-time Lyapunov exponents (FTLEs), observing a change of sign based on the interplay between the distance towards the bifurcation and the noise intensity. A technical challenge is to provide a suitable control of the nonlinear terms coupling the dominant and stable modes of the SPDE and of the corresponding linearization. In order to illustrate our results we apply them to the stochastic Burgers equation.

math.PR↗

arXiv2025

Taylor-like approximations of center manifolds for rough differential equations

The dynamics of rough differential equations (RDEs) has recently received a lot of interest. For example, the existence of local random center manifolds for RDEs has been established. In this work, we present an approximation for local random center manifolds for RDEs driven by geometric rough paths. To this aim, we combine tools from rough path and deterministic center manifold theory to derive Taylor-like approximations of local random center manifolds. The coefficients of this approximation are stationary solutions of RDEs driven by the same geometric rough path as the original equation. We illustrate our approach for stochastic differential equations (SDEs) with linear and nonlinear multiplicative noise.

math.PR↗

arXiv2025

An integrable bound for semilinear rough partial differential equations with unbounded diffusion coefficients

This work develops moment bounds for the controlled rough path norm of the solution of semilinear rough partial differential equations.~The novel aspects are two-fold: first we consider rough paths of low time regularity $\gamma\in(1/4,1/2)$ and second treat unbounded diffusion coefficients. To this aim we introduce a suitable notion of a controlled rough path according to a monotone scale of Banach spaces and innovative control functions.

math.PR↗

arXiv2025

A mild rough Gronwall Lemma with applications to non-autonomous evolution equations

We derive a Gronwall type inequality for mild solutions of non-autonomous parabolic rough partial differential equations (RPDEs). This inequality together with an analysis of the Cameron-Martin space associated to the noise, allows us to obtain the existence of moments of all order for the solution of the corresponding RPDE and its Jacobian when the random input is given by a Gaussian Volterra process. Applying further the multiplicative ergodic theorem, these integrable bounds entail the existence of Lyapunov exponents for RPDEs. We illustrate these results for stochastic partial differential equations with multiplicative boundary noise.

math.PR↗

arXiv2025

Pathwise mild solutions for superlinear stochastic evolution equations and their attractors

We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck type process, which is shown to be tempered in suitable function spaces. This property, together with a bootstrapping argument based on the regularizing effect of parabolic evolution families, is then applied to prove the global well-posedness and the existence of a random attractor for reaction-diffusion equations with random non-autonomous generators and nonlinearities satisfying certain growth and dissipativity assumptions.

math.PR↗

arXiv2024

Detecting random bifurcations via rigorous enclosures of large deviations rate functions

The main goal of this work is to provide a description of transitions from uniform to non-uniform snychronization in diffusions based on large deviation estimates for finite time Lyapunov exponents. These can be characterized in terms of moment Lyapunov exponents which are principal eigenvalues of the generator of the tilted (Feynman-Kac) semigroup. Using a computer assisted proof, we demonstrate how to determine these eigenvalues and investigate the rate function which is the Legendre-Fenichel transform of the moment Lyapunov function. We apply our results to two case studies: the pitchfork bifurcation and a two-dimensional toy model, also considering the transition to a positive asymptotic Lyapunov exponent.

math.DS↗

arXiv2024

Global well-posedness of 2D Navier-Stokes with Dirichlet boundary fractional noise

In this paper, we prove the global well-posedness and interior regularity for the 2D Navier-Stokes equations driven by a fractional noise acting as an inhomogeneous Dirichlet-type boundary condition. The model describes a vertical slice of the ocean with a relative motion between the two surfaces and can be thought of as a stochastic variant of the Couette flow. The relative motion of the surfaces is modeled by a Gaussian noise which is coloured in space and fractional in time with Hurst parameter greater than 3/4.

math.AP↗

arXiv2024

Quantitative instability for stochastic scalar reaction-diffusion equations

This work studies the instability of stochastic scalar reaction diffusion equations, driven by a multiplicative noise that is white in time and smooth in space, near to zero, which is assumed to be a fixed point for the equation. We prove that if the Lyapunov exponent at zero is positive, then the flow of non-zero solutions admits uniform bounds on small negative moments. The proof builds on ideas from stochastic homogenisation. We require suitable corrector estimates for the solution to a Poisson problem involving an infinite-dimensional projective process, together with a linearisation step that hinges on quantitative parametrix-like arguments. Overall, we are able to construct an appropriate Lyapunov functional for the nonlinear dynamics and address some problems left open in the literature.

math.PR↗

arXiv2024

Concentration estimates for SPDEs driven by fractional Brownian motion

The main goal of this work is to provide sample-path estimates for the solution of slowly time-dependent SPDEs perturbed by a cylindrical fractional Brownian motion. Our strategy is similar to the approach by Berglund and Nader for space-time white noise. However, the setting of fractional Brownian motion does not allow us to use any martingale methods. Using instead optimal estimates for the probability that the supremum of a Gaussian process exceeds a certain level, we derive concentration estimates for the solution of the SPDE, provided that the Hurst index $H$ of the fractional Brownian motion satisfies $H>\frac14$. As a by-product, we also obtain concentration estimates for one-dimensional fractional SDEs valid for any $H\in(0,1)$.

math.PR↗

arXiv2023

Finite-time Lyapunov exponents for SPDEs with fractional noise

We estimate the finite-time Lyapunov exponents for a stochastic partial differential equation driven by a fractional Brownian motion (fbm) with Hurst index $H\in(0,1)$ close to a bifurcation of pitchfork type. We characterize regions depending on the distance from bifurcation, the Hurst parameter of the fbm and the noise strength where finite-time Lyapunov exponents are positive and thus indicate a change of stability. The results on finite-time Lyapunov exponents are novel also for SDEs perturbed by fractional noise.

math.PR↗