Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation

Quer-Sardanyons, Lluís; Sanz–Solé, Marta

doi:10.1016/s0022-1236(03)00065-x

Cited by 38 publications

(50 citation statements)

References 13 publications

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“…This result provides a generalisation of Theorem 3 in [13], where the authors deal with the three-dimensional stochastic wave equation and a slightly stronger condition than Eq. 1.3 is assumed.…”

Section: Introductionsupporting

confidence: 66%

“…As a consequence of the proof of Proposition 3 from [13], it is readily checked that Thus, we get that (t, x) = σ (u(t − ·, x − * )) (t − ·, x − * ).…”

Section: Proof Of Proposition 51mentioning

confidence: 72%

“…can be extended to the space H T ; for details, we refer to [8], p. 805, or [13], p. 3. We will also denote by W(g) the Gaussian random variable associated with an element g ∈ H T .…”

Section: Extension Of Walsh's Stochastic Integralmentioning

confidence: 99%

“…As it is pointed out in [13], Appendix A (see also [6]), when is the fundamental solution of the wave equation in R d , with d = 1, 2, 3, then condition (6.31) is satisfied with γ = 3.…”

Section: Remark 63mentioning

confidence: 99%

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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: Introductionsupporting

confidence: 66%

“…As a consequence of the proof of Proposition 3 from [13], it is readily checked that Thus, we get that (t, x) = σ (u(t − ·, x − * )) (t − ·, x − * ).…”

Section: Proof Of Proposition 51mentioning

confidence: 72%

“…can be extended to the space H T ; for details, we refer to [8], p. 805, or [13], p. 3. We will also denote by W(g) the Gaussian random variable associated with an element g ∈ H T .…”

Section: Extension Of Walsh's Stochastic Integralmentioning

confidence: 99%

“…As it is pointed out in [13], Appendix A (see also [6]), when is the fundamental solution of the wave equation in R d , with d = 1, 2, 3, then condition (6.31) is satisfied with γ = 3.…”

Section: Remark 63mentioning

confidence: 99%

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Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

2007

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“…The solution constructed in [1] had moments of all orders but no spatial sample path regularity was established. Absolute continuity and smoothness of the probability law was studied in [16] and [17] (see also the recent paper [13]). Hölder continuity of the solution was only recently established in [6], and sharp exponents were also obtained.…”

Section: Introductionmentioning

confidence: 99%

The Non-Linear Stochastic Wave Equation in High Dimensions

Conus

Dalang

2008

Electron. J. Probab.

120

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We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on p-th moments of the solution (p 1), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal .

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Gaussian Upper Density Estimates for Spatially Homogeneous SPDEs

Quer-Sardanyons

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. We consider a general class of SPDEs in R d driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring that the density of the corresponding mild solution admits an upper estimate of Gaussian type. The proof is based on the formula for the density arising from the integration-by-parts formula of the Malliavin calculus. Our result applies to the stochastic heat equation with any space dimension and the stochastic wave equation with d ∈ {1, 2, 3}. In these particular cases, the condition on the spectral measure turns out to be optimal.

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Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation

Cited by 38 publications

References 13 publications

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

The Non-Linear Stochastic Wave Equation in High Dimensions

Gaussian Upper Density Estimates for Spatially Homogeneous SPDEs

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