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On the global maximum of the solution to a stochastic heat equation with compact-support initial data

Mohammud Foondun, Davar Khoshnevisan (2010)

Annales de l'I.H.P. Probabilités et statistiques

Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ>0. Under some suitable conditions on the initial function u0 and σ, we show that the quantities lim sup t→∞t−1sup x∈Rln El(|ut(x)|2) and lim sup t→∞t−1ln E(sup x∈R|ut(x)|2) are equal, as well as bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x↦ut(x) are highly concentrated...

On the helix equation

Mohamed Hmissi, Imene Ben Salah, Hajer Taouil (2012)

ESAIM: Proceedings

This paper is devoted to the helices processes, i.e. the solutions H : ℝ × Ω → ℝd, (t, ω) ↦ H(t, ω) of the helix equation H ( 0 ,ω ) = 0 ;   H ( s + t,ω ) = H ( s, Φ ( t,ω ) ) + H ( t,ω ) where Φ : ℝ × Ω → Ω, (t, ω) ↦ Φ(t, ω) is a dynamical system on a measurable space (Ω, ℱ).More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined...

Peano type theorem for random fuzzy initial value problem

Marek T. Malinowski (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider the random fuzzy differential equations and show their application by an example. Under suitable conditions the Peano type theorem on existence of solutions is proved. For our purposes, a notion of ε-solution is exploited.

Probabilistic well-posedness for the cubic wave equation

Nicolas Burq, Nikolay Tzvetkov (2014)

Journal of the European Mathematical Society

The purpose of this article is to introduce for dispersive partial differential equations with random initial data, the notion of well-posedness (in the Hadamard-probabilistic sense). We restrict the study to one of the simplest examples of such equations: the periodic cubic semi-linear wave equation. Our contributions in this work are twofold: first we break the algebraic rigidity involved in our previous works and allow much more general randomizations (general infinite product measures v.s. Gibbs...

Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

In this continuation of the preceding paper (Part I), we consider a sequence ( F ₙ ) n ≥ 0 of i.i.d. random Lipschitz mappings → , where is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) X ₙ x = F ₙ ∘ . . . ∘ F ₁ ( x ) starting at x ∈ . The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I, the Chacon-Ornstein...

Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity

Marc Peigné, Wolfgang Woess (2011)

Colloquium Mathematicae

Consider a proper metric space and a sequence ( F ₙ ) n ≥ 0 of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) X ₙ x = F ₙ ∘ . . . ∘ F ₁ ( x ) starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic...

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