arXiv subjects

Raluca M. Balan

Publications and source records attributed to Raluca M. Balan.

44 records · Page 3Linked to original sources

arXiv2016

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 $p\geq2$ 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.

math.PR↗

arXiv2016

SPDEs with rough noise in space: Hölder continuity of the solution

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index $H \in (\frac14,\frac12)$. The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than $H$, and Hölder continuous in time of order smaller than $γ$, where $γ=H$ for the wave equation and $γ=H/2$ for the heat equation.

math.PR↗

arXiv2015

Itô formula for integral processes related to space-time Lévy white noise

In this article, we give a new proof of the Itô formula for some integral processes related to the space-time Lévy white noise introduced in Balan (2015) as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the $p$-th moment of the stochastic integral, and the Itô representation theorem leading to a chaos expansion similar to the Gaussian case.

math.PR↗

arXiv2015

Intermittency for the wave equation with Lévy white noise

In this article, we consider the stochastic wave equation in dimension 1 driven by the Lévy white noise introduced in Balan (2015). Using Rosenthal's inequality, we develop a maximal inequality for the moments of order $p \geq 2$ of the integral with respect to this noise. Based on this inequality, we show that this equation has a unique solution, which is weakly intermittent in the sense of Foondun and Khoshnevisan (2009) and Khoshnevisan (2014).

math.PR↗

arXiv2014

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients necessary and sufficient conditions for the finite dimensional convergence to an $α$-stable Lévy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $α\leq 1$. In the functional setting we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (1992) and improved by Louhichi and Rio (2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $S$ topology, introduced by Jakubowski (1997).

math.PR↗

arXiv2011

Linear SPDEs with harmonizable noise

Using tools from the theory of random fields with stationary increments, we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise (called harmonizable) is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (1999), and the fractional noise considered in Balan and Tudor (2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with harmonizable noise, under some mild conditions imposed on the Green function of the differential operator which appears in this equation. This methodology is applied to the study of the heat and wave equations (possibly replacing the Laplacian by one of its fractional powers), extending in this manner the results of Balan and Tudor (2010) to the case $H<1/2$.

math.PR↗

arXiv2010

The Stochastic Wave Equation with Multiplicative Fractional Noise: a Malliavin calculus approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index $H>1/2$, and has a homogeneous spatial covariance structure given by the Riesz kernel of order $α$. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is $α>d-2$, which coincides with the condition obtained in Dalang (1999), when the noise is white in time. Under this condition, we obtain estimates for the $p$-th moments of the solution, we deduce its Hölder continuity, and we show that the solution is Malliavin differentiable of any order. When $d \leq 2$, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.

math.PR↗

arXiv2005

A strong invariance principle for associated random fields

In this paper we generalize Yu's [Ann. Probab. 24 (1996) 2079-2097] strong invariance principle for associated sequences to the multi-parameter case, under the assumption that the covariance coefficient u(n) decays exponentially as n\to \infty. The main tools that we use are the following: the Berkes and Morrow [Z. Wahrsch. Verw. Gebiete 57 (1981) 15-37] multi-parameter blocking technique, the Csorgo and Revesz [Z. Wahrsch. Verw. Gebiete 31 (1975) 255-260] quantile transform method and the Bulinski [Theory Probab. Appl. 40 (1995) 136-144] rate of convergence in the CLT.

math.PR↗