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In this paper, we consider a quasi-linear stochastic heat equation on [0, 1], with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the realvalued mild solution of the SPDE perturbed by this family of noises converges in law, in the space ([0, T ] × [0, 1]) of continuous functions, to the solution of the white noise driven SPDE. Making use of a suitable continuous functional of the stochastic convolution term, we show that it suffices to tackle the linear problem. For this, we prove that the corresponding family of laws is tight and we identify the limit law by showing the convergence of the finite dimensional distributions. We have also considered two particular families of noises to that our result applies. The first one involves a Poisson process in the plane and has been motivated by a one-dimensional result of Stroock, which states that the family of processes n t 0 (−1) N (n 2 s) ds, where N is a standard Poisson process, converges in law to a Brownian motion. The second one is constructed in terms of the kernels associated to the extension of Donsker's theorem to the plane.

In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion B H,K with parameters H ∈ (0, 1) and K ∈ (1, 2) such that HK ∈ (0, 1). A remarkable difference between the case K ∈ (0, 1) and our situation is that this process is a semimartingale when 2HK = 1.2000 Mathematics Subject Classification. Primary 60G15; Secondary 60G18.

Abstract. We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes.As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and sub-fractional Brownian motion.