Some non-linear s. p. d. e's that are second order in time.

Dalang, Robert C.; Mueller, Carl

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The non-linear stochastic wave equation in high dimensions.

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Extending martingale measure stochastic integral with applications to spatially homogeneous S. P. D. E's.

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SPDEs with coloured noise: Analytic and stochastic approaches

Marco Ferrante, Marta Sanz-Solé (2006)

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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...

Transience and non-explosion of certain stochastic Newtonian systems.

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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 $u (t, \underset{̲}{x}) = (u (t, x_{1}), \dots, u (t, x_{d}))$ , where $t > 0, x_{1}, \dots, x_{d}$ 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 $u (t, \underset{̲}{x})$ . 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...

Competing super-Brownian motions as limits of interacting particle systems.

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Electronic Journal of Probability [electronic only]

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Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Éric Gautier (2005)

ESAIM: Probability and Statistics

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Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution...