On the density of systems of non-linear spatially homogeneous SPDEs

Abstract

In this paper, we consider a system of k second order non-linear stochastic partial differential equations with spatial dimension d ≥ 1, driven by a q-dimensional Gaussian noise, which is white in time and with some spatially homogeneous covariance. The case of a single equation and a one-dimensional noise, has largely been studied in the literature. The first aim of this paper is to give a survey of some of the existing results. We will start with the existence, uniqueness and Hölder's continuity of the solution. For this, the extension of Walsh's stochastic integral to cover some measure-valued integrands will be recalled. We will then recall the results concerning the existence and smoothness of the density, as well as its strict positivity, which are obtained using techniques of Malliavin calculus. The second aim of this paper is to show how these results extend to our system of SPDEs. In particular, we give sufficient conditions in order to have existence and smoothness of the density on the set where the columns of the diffusion matrix span R k . We then prove that the density is strictly positive in a point if the connected component of the set where the columns of the diffusion matrix span R k which contains this point has a non void intersection with the support of the law of the solution. We will finally check how all these results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in dimension d ∈ {1, 2, 3}.AMS 2000 subject classifications: 60H15, 60H07.Key words and phrases. Spatially homogeneous Gaussian noise, Malliavin calculus, nonlinear stochastic partial differential equations, strict positivity of the density.

Hitting probabilities for nonlinear systems of stochastic waves

“…(b) strict positivity of the density p t,x on R d . Property (b) has been recently established in [35]. As for the upper bound in (a), we show here that this hypothesis can be significantly weakened.…”

Smoothness of the joint density for spatially homogeneous SPDEs

“…By Section 3 in [11], (H2) holds when d = 1 with κ 1 , κ 2 ∈ (0, H 1 ) and when d = 2, it is satisfied for κ 1 , κ 2 ∈ (0, H 1 + H 2 − 1). By Theorem 6.1 in [7], when d = 3 (H2) is satisfied with κ 1 , κ 2 ∈ (0, min(…”

“…In [11] it is proved that condition (H3) is satisfied for η = 3 − β and (2.8) holds with η 1 = κ 1 + 3 − β > η. Again, when ε is small enough, to make x and w + y in the support of the measure Γ 3 (r), we must have |x 1 − y 1 | > C for some positive constant C. So and (2.9) is satisfied with η 2 = min(2(H 2 + H 3 ) − 1, 2(H 1 + H 3 ) − 1, 2(H 1 + H 2 ) − 1), which is strictly greater than η.…”

“…The basic ingredient is to impose that leading terms as ε → 0 in the matrix are the diagonal ones given by (1.2) (see the hypotheses (H3) and (H4) below). These hypotheses are related, although different, to the ones imposed by Nualart in [11] to establish the smoothness of the density for the solution of a system of SDPEs. The paper is organized as follows.…”

Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $$k \ge 1$$

“…One further fact about p t,x (·) that we will need is provided by the following recent result of Nualart [19]. We first define precisely the driving noise that appears in (1.1).…”