947 research outputs found
Existence of weak solutions for a class of semilinear stochastic wave equations
We prove existence of weak solutions (in the probabilistic sense) for a
general class of stochastic semilinear wave equations on bounded domains of
driven by a possibly discontinuous square integrable martingale.Comment: 21 pages, final versio
Well-posedness of semilinear stochastic wave equations with H\"{o}lder continuous coefficients
We prove that semilinear stochastic abstract wave equations, including wave
and plate equations, are well-posed in the strong sense with an
-H\"{o}lder continuous drift coefficient, if . The
uniqueness may fail for the corresponding deterministic PDE and well-posedness
is restored by adding an external random forcing of white noise type. This
shows a kind of regularization by noise for the semilinear wave equation. To
prove the result we introduce an approach based on backward stochastic
differential equations. We also establish regularizing properties of the
transition semigroup associated to the stochastic wave equation by using
control theoretic results
Weak convergence for a spatial approximation of the nonlinear stochastic heat equation
We find the weak rate of convergence of the spatially semidiscrete finite
element approximation of the nonlinear stochastic heat equation. Both
multiplicative and additive noise is considered under different assumptions.
This extends an earlier result of Debussche in which time discretization is
considered for the stochastic heat equation perturbed by white noise. It is
known that this equation has a solution only in one space dimension. In order
to obtain results for higher dimensions, colored noise is considered here,
besides white noise in one dimension. Integration by parts in the Malliavin
sense is used in the proof. The rate of weak convergence is, as expected,
essentially twice the rate of strong convergence.Comment: 19 page
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