Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by ${\psi }_p\left(x\right)={exp\left(\right|x|}^p)-1$ with 0 < p ≤ 2) of $ma{x}_{0\le t\le \tau }\left|B_t\right|$ or $|B_{\tau }|$ to be finite, where $B={\left(B_t\right)}_{t\ge 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}<\infty$ as soon as $E\left({\tau }^k\right)=O\left(C^k{k}^k\right)$ for some constant C > 0 as k → ∞ (or equivalently $\parallel \tau {\parallel }_{{\psi }_1}<\infty$). In particular, if τ ∼ Exp(λ) or $\left|N\right(0,{\sigma }^2\left)\right|$ then the last condition is satisfied, and we obtain $\parallel ma{x}_{0\le t\le \tau }\left|B_t\right|{\parallel }_{{\psi }_1}\le K\surd E\left(\tau \right)$ with some universal constant K > 0....

Consider a stochastic heat equation ∂tu=κ  ∂xx2u+σ(u)ẇ for a space–time white noise ẇ and a constant κ&gt;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...

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 $\begin{array}{ccc}\mathit{H}\mathrm{\left(}\mathrm{0}\mathit{,\omega }\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{;} \mathit{H}\mathrm{(}\mathit{s}\mathrm{+}\mathit{t,\omega }\mathrm{)}\mathrm{=}\mathit{H}\mathrm{\left(}\mathit{s,}\mathrm{\Phi }\mathrm{\right(}\mathit{t,\omega }\mathrm{\left)}\mathrm{\right)}\mathrm{+}\mathit{H}\mathrm{\left(}\mathit{t,\omega }\mathrm{\right)}& & \end{array}$ 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...

In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic gaussian regulator problem. For a completely observable controlled linear system driven by a fractional brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.

In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic Gaussian regulator problem. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.

The Kaczmarz algorithm of successive projections suggests the following concept. A sequence $\left(e_k\right)$ of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and $xₙ=x_{n-1}+\alpha ₙeₙ$, where $\alpha ₙ=⟨x-x_{n-1},eₙ⟩$. We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional Brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...