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.Comment: Published at http://dx.doi.org/10.1214/009117904000001071 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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 $H<1/2$.Comment: 31 page

The Stochastic Heat Equation with a Fractional-Colored Noise: Existence of the Solution

In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in (0,T) \times \bR^d, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$, and colored in space, with spatial covariance given by a function $f$. Our main result gives the necessary and sufficient condition on $H$ for the existence of the process solution. When $f$ is the Riesz kernel of order $\alpha \in (0,d)$ this condition is $H>(d-\alpha)/4$, which is a relaxation of the condition $H>d/4$ encountered when the noise $\dot B$ is white in space. When $f$ is the Bessel kernel or the heat kernel, the condition remains $H>d/4$