Smoothness with Respect to Noise Parameters for Parabolic/hyperbolic Anderson Model with Regular or Rough Noise in Space

Abstract

In this work, we study the continuity in law of the solutions of two linear multiplicative SPDEs, namely, the parabolic Anderson model (PAM) and the hyperbolic Anderson model (HAM). The forcing term under investigation is examined in two cases: (i) the regular noise, with the spatial covariance given by the Riesz kernel of order α ∈ (0, d) in spatial dimension d ≥ 1; (ii) the rough noise, which is a fractional noise in space with Hurst index H < 1/2 in spatial dimension d = 1. In both cases, the noise is assumed to be colored in time. The similar problem for the case of the white noise in time was considered in [13, 23]. For the initial condition, we consider two scenarios: (a) constant initial condition; (b) initial condition given by a signed Borel measure on Rd. In the case of constant initial condition, we prove that the solution is continuous in law in the space C([0,T]×Rd) of continuous functions, with respect to the spatial parameter (α or H) of the noise. In the case of general initial condition (given by a measure), the weak convergence of the solution with respect to spatial parameter of the noise is obtained in the space C([t0, T ] × Rd). The solution is understood in Skorohod sense, using Malliavin Calculus, a stochastic calculus of variations theory that proves beneficial in exploring various aspects in the theory of SPDEs. The results corresponding to cases (a) and (b) above are contained in the recent article [7], and respectively, in the preprint [30].