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Mazyar Ghani Varzaneh

Publications and source records attributed to Mazyar Ghani Varzaneh.

13 recordsLinked to original sources
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

On the negativity of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion

We provide sign information for the top Lyapunov exponent for a stochastic differential equation driven by fractional Brownian motion. To this aim we analyze the stochastic dynamical system generated by such an equation, obtain a random dynamical system and construct an appropriate invariant measure. Suitable estimates for its density together with Birkhoff's ergodic theorem imply the negativity of the top Lyapunov exponent by increasing the noise intensity.

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

Invariant manifolds and stability for rough differential equations

We prove the existence of local stable, unstable, and center manifolds for stochastic semiflows induced by rough differential equations driven by rough paths valued stochastic processes around random fixed points of the equation. Examples include stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $H > \frac{1}{4}$. In case the top Lyapunov exponent is negative, we derive almost sure exponential stability of the solution.

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 $γ\in(1/4,1/2)$ 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↗
arXiv2024

Singular stochastic delay equations driven by fractional Brownian motion: Dynamics, longtime behaviour, and pathwise stability

We study differential equations with a linear, path dependent drift and discrete delay in the diffusion term driven by a $γ$-Hölder rough path for $γ> \frac{1}{3}$. We prove well-posedness of these systems and establish a priori bounds for their solutions. Applying these results to an equation driven by a multidimensional fractional Brownian motion with Hurst paramter $H \in \big(\frac{1}{3},1\big)$, we can prove the existence of a Lyapunov spectrum for the linearized system that describes its long-time behaviour. Furthermore, we can deduce the existence of local stable, unstable, and center manifolds for the nonlinear equation. As an application, we can prove that under suitable conditions, the solution exhibits pathwise local exponential stability. En passant, we present new, concise and relatively short proofs for some classical results for $\mathcal{C}_0$-semigroups that build on the Multiplicative Ergodic Theorem formulated on Banach spaces.

math.PR↗
arXiv2023

An integrable bound for rough stochastic partial differential equations with applications to invariant manifolds and stability

We study semilinear rough stochastic partial differential equations as introduced in [Gerasimovi{č}s, Hairer; EJP 2019]. We provide $\mathcal{L}^p(Ω)$-integrable a priori bounds for the solution and its linearization in case the equation is driven by a suitable Gaussian process. Using the Multiplicative Ergodic Theorem for Banach spaces, we can deduce the existence of a Lyapunov spectrum for the linearized equation around stationary points. The existence of local stable, unstable, and center manifolds around stationary points is also provided. In the case where all Lyapunov exponents are negative, local exponential stability can be deduced. We illustrate our findings with several examples.

math.PR↗
arXiv2023

Introduction to rough paths theory

These notes are an extended version of the course "Introduction to rough paths theory" given at the XXV Brazilian School of Probability in Campinas in August 2022. Their aim is to give a consise overview to Lyon's theory of rough paths with a special focus on applications to stochastic differential equations.

math.PR↗
arXiv2022

The geometry of controlled rough paths

We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô-Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.

math.PR↗
arXiv2019

Oseledets splitting and invariant manifolds on fields of Banach spaces

We prove a semi-invertible Oseledets theorem for cocycles acting on measurable fields of Banach spaces, i.e. we only assume invertibility of the base, not of the operator. As an application, we prove an invariant manifold theorem for nonlinear cocycles acting on measurable fields of Banach spaces.

math.PR↗
arXiv2019

A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces

We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of this article, we use our results to prove a stable manifold theorem for non-linear singular SDDEs.

math.PR↗