Positivity of the density for the stochastic wave equation in two spatial dimensions

Chaleyat-Maurel, Mireille; Sanz–Solé, Marta

doi:10.1051/ps:2003002

Cited by 4 publications

(3 citation statements)

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“…Hence, their situation is a bit different as ours, as we deal with a system of SPDEs and we evaluate the solution at a single point (t, x) ∈]0, T ] × R d . In a similar context, Chaleyat and Sanz-Solé in [4] study the strict positivity of the density of the random vector (u(t, x 1 ), ...., u(t, x k )), 0 ≤ x 1 ≤ · · · ≤ x k ≤ 1, where u is the solution to the stochastic wave equation in two spatial dimensions, that is, take in equation (1.1) d = 2, k, q = 1, and L = ∂ 2 ∂t 2 − ∆. We will also apply the criterion of strict positivity to the case of a single equation, that is the solution to (1.1) with k, q = 1.…”

Section: Strict Positivity Of the Densitymentioning

confidence: 89%

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

Nualart

2012

Stochastics

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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.

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Section: Strict Positivity Of the Densitymentioning

confidence: 89%

Section: Strict Positivity Of the Densitymentioning

confidence: 99%

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

Nualart

2012

Stochastics

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“…The lower bound for E(J ε (z)) is a direct consequence of the results proved in [35] (see also [6]) on the positivity of the density for the solution of (1.4). To establish the upper bound, we apply (3.10) along with (3.13) and obtain…”

Section: Proof Of Theorem 31mentioning

confidence: 67%

Hitting probabilities for nonlinear systems of stochastic waves

Dalang

Sanz–Solé

2015

Memoirs of the AMS

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Abstract. We consider a d-dimensional random field u = {u(t, x)} that solves a non-linear system of stochastic wave equations in spatial dimensions k ∈ {1, 2, 3}, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent β. Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of R d , in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when d(2 − β) > 2(k + 1), points are polar for u. Conversely, in low dimensions d, points are not polar. There is however an interval in which the question of polarity of points remains open.

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Smoothness of the joint density for spatially homogeneous SPDEs

Huang

Nualart

et al. 2015

J. Math. Soc. Japan

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In this paper we consider a general class of second order stochastic partial differential equations on R d driven by a Gaussian noise which is white in time and it has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points (t, x 1 ), . . . , (t, x n ), t > 0, assuming some suitable regularity and non degeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.

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Positivity of the density for the stochastic wave equation in two spatial dimensions

Abstract: Abstract. We consider the random vector u(t, x) = (u(t, x1), . . . , u(t, x Mathematics Subject Classification. 60H15, 60H07.

Cited by 4 publications

References 14 publications

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

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

Hitting probabilities for nonlinear systems of stochastic waves

Smoothness of the joint density for spatially homogeneous SPDEs

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