Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

Nualart, David; Quer-Sardanyons, Lluís

doi:10.1007/s11118-007-9055-3

Cited by 68 publications

(107 citation statements)

References 14 publications

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“…For T 2 , using Cauchy-Schwartz inequality, our assumption on b ′ , Minkowski's inequality and the estimate (5.26) in [13], we obtain the bound…”

Section: Existence and Smoothness Of The Densitymentioning

confidence: 95%

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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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“…For T 2 , using Cauchy-Schwartz inequality, our assumption on b ′ , Minkowski's inequality and the estimate (5.26) in [13], we obtain the bound…”

Section: Existence and Smoothness Of The Densitymentioning

confidence: 95%

“…We shall use the stochastic integral defined in [4, Section 2.3] (see also [13,Section 3]). We briefly review the construction and properties of this integral.…”

Section: Preliminariesmentioning

confidence: 99%

Smoothness of the joint density for spatially homogeneous SPDEs

Huang

Nualart

et al. 2015

J. Math. Soc. Japan

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“…where we have used elementary properties of the Fourier transform (see also Dalang [3], Nualart and Quer-Sardanyons [18], and Dalang and Quer-Sardanyons [8] for properties of the stochastic integral). This last formula is convenient since…”

Section: Remark 11mentioning

confidence: 99%

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

Dalang

Khoshnevisan

Nualart

2013

Stoch PDE: Anal Comp

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We consider a system of d non-linear stochastic heat equations in spatial dimension k ≥ 1, whose solution is an R d -valued random field u = {u(t , x), (t, x) ∈ R + × R k }. The d-dimensional driving noise is white in time and with a spatially homogeneous covariance defined as a Riesz kernel with exponent β, where 0 < β < (2 ∧ k). The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish an upper bound on the two-point density, with respect to Lebesgue measure, of the R 2d -valued random vector (u(s, y), u(t, x)), that, in particular, quantifies how this density degenerates as (s, y) → (t, x). From this result, we deduce a lower bound on hitting probabilities of the process u, in terms of Newtonian capacity. We also establish an upper bound on hitting probabilities of the process in terms of Hausdorff measure. These estimates make it possible to show that points are polar when d >

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“…the regularity of the probability measure induced by the solution [26,24,31], large deviation principles [25,19], Varadhan estimates [20,30], support theorems [21,16], path properties such as Hölder continuity [29,12] and much more. See also the references in these works for a more detailed account.…”

Section: Introductionmentioning

confidence: 99%

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

Ascanelli

Süß²

2018

Stochastic Processes and their Applications

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In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.

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Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

Cited by 68 publications

References 14 publications

Smoothness of the joint density for spatially homogeneous SPDEs

Smoothness of the joint density for spatially homogeneous SPDEs

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

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

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