On stochastic partial differential equations with spatially correlated noise: smoothness of the law

Márquez-Carreras, David; Mellouk, Mohamed; Sarrà, Mònica

doi:10.1016/s0304-4149(00)00099-5

Cited by 44 publications

(32 citation statements)

References 7 publications

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“…Remark 6.4 Theorem 6.2 provides a generalisation of Theorem 3 in [14] for the case of the three-dimensional wave equation (see also [15]). Moreover, it also generalises the results in [7] for the stochastic wave equation with space dimension d = 1, 2 and the stochastic heat equation in any space dimension.…”

Section: Remark 63supporting

confidence: 74%

“…We should mention that Theorem 6.2 in Section 6 provides an improvement of Theorem 3 in [14] for the case of the three-dimensional stochastic wave equation, since in this latter reference the integrability conditions concerning the Fourier transform of are much more involved (see also [15]). Moreover, it is worth mentioning that Theorem 6.2 also generalises known results on existence and smoothness of the density for the case of the stochastic heat and wave equation with dimensions d ≥ 1 and d = 1, 2, respectively (see [2,7,8,15]). …”

Section: Introductionsupporting

confidence: 55%

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Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

2007

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In this paper, we extend Walsh's stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang's one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d = 1, 2, 3. Moreover, for these particular examples, known results in the literature have been improved.

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Section: Remark 63supporting

confidence: 74%

Section: Introductionsupporting

confidence: 55%

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

2007

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“…In [13], the Malliavin differentiability and the smoothness of the density of u(t, x) was established when d = 1, and the extension to d > 1 can easily be done by working coordinate by coordinate. These results were extended in [18, Proposition 5.1].…”

Section: Then the Probability Law Of F Has An Infinitely Differentiabmentioning

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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“…Recently, in [6] a related problem has been studied. The results apply in particular to equations like (1.2) driven by a centered Gaussian noise with covariance given by…”

Section: S(t − S X − Y) B U(s Y) Ds Dy (13)mentioning

confidence: 99%