Raluca M. Balan

Intermittency for the wave and heat equations with fractional noise in time

In this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index H>1/2. The solutions of these equations are interpreted in the Skorohod sense. Using Malliavin calculus techniques, we obtain an upper bound for the moments of order p≥2 of the solution. In the case of the wave equation, we derive a Feynman–Kac-type formula for the second moment of the solution, based on the points of a planar Poisson process. This is an extension of the formula given by Dalang, Mueller and Tribe [Trans. Amer. Math. Soc. 360 (2008) 4681–4703], in the case H=1/2, and allows us to obtain a lower bound for the second moment of the solution. These results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time.

A Markov Property For Set-Indexed Processes

We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process.

A strong invariance principle for associated random fields

In this paper we generalize Yu’s strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n -> (infinity symbol). The main tools will be the Berkes-Morrow multi-parameter blocking technique, the Csorgo-Revesz quantile transform method and the Bulinski rate of convergence in the CLT for associated random fields.

A strong Markov property for set-indexed processes

We introduce adapted sets and optional sets and we study a type of strong Markov property for set-indexed processes that can be associated with the sharp Markov property defined by Ivanoff and Merzbach (Proceedings of the International Conference on Stochastic Models, June 1998, Carleton University, Can. Math. Soc. Conf. Proc. 26 (2000) 217).

Integration with respect to Lévy colored noise, with applications to SPDEs

In this article, we introduce a Lévy analogue of the spatially homogeneous Gaussian noise of [5], and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered measure μ on , whose density is given by for a symmetric complex-valued function h. Without assuming that the Fourier transform of μ is a non-negative function, we identify a large class of integrands with respect to this noise. As an application, we examine the linear stochastic heat and wave equations driven by this type of noise.

SPDEs with -Stable Lévy Noise: A Random Field Approach

This paper is dedicated to the study of a nonlinear SPDE on a bounded domain in , with zero initial conditions and Dirichlet boundary, driven by an -stable Levy noise with , , and possibly nonsymmetric tails. To give a meaning to the concept of solution, we develop a theory of stochastic integration with respect to this noise. The idea is to first solve the equation with “truncated” noise (obtained by removing from the jumps which exceed a fixed value ), yielding a solution , and then show that the solutions coincide on the event , for some stopping times converging to infinity. A similar idea was used in the setting of Hilbert-space valued processes. A major step is to show that the stochastic integral with respect to satisfies a th moment inequality. This inequality plays the same role as the Burkholder-Davis-Gundy inequality in the theory of integration with respect to continuous martingales.

Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise and Rough Initial Condition

In this article, we study the parabolic Anderson model driven by a space-time homogeneous Gaussian noise on

Recent Advances Related to SPDEs with Fractional Noise

\mathbb {R}_{+} \times \mathbb {R}^d