Displaying similar documents to “Some non-linear s. p. d. e's that are second order in time.”

SPDEs with coloured noise: Analytic and stochastic approaches

Marco Ferrante, Marta Sanz-Solé (2006)

ESAIM: Probability and Statistics

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We study strictly parabolic stochastic partial differential equations on ℝ d , ≥ 1, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are...

Regularity of solutions to stochastic Volterra equations

Anna Karczewska, Jerzy Zabczyk (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We study regularity of stochastic convolutions solving Volterra equations on R d driven by a spatially homogeneous Wiener process. General results are applied to stochastic parabolic equations with fractional powers of Laplacian.

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

Mireille Chaleyat-Maurel, Marta Sanz-Solé (2003)

ESAIM: Probability and Statistics

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We consider the random vector u ( t , x ̲ ) = ( u ( t , x 1 ) , ⋯ , u ( t , x d ) ) , where t > 0 , x 1 , ⋯ , x d are distinct points of ℝ 2 and u denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for u ( t , x ̲ ) . We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize...