664 research outputs found
Perturbation of strong Feller semigroups and well-posedness of semilinear stochastic equations on Banach spaces
We prove a Miyadera-Voigt type perturbation theorem for strong Feller
semigroups. Using this result, we prove well-posedness of the semilinear
stochastic equation dX(t) = [AX(t) + F(X(t))]dt + GdW_H(t) on a separable
Banach space E, assuming that F is bounded and measurable and that the
associated linear equation, i.e. the equation with F = 0, is well-posed and its
transition semigroup is strongly Feller and satisfies an appropriate gradient
estimate. We also study existence and uniqueness of invariant measures for the
associated transition semigroup.Comment: Revision based on the referee's comment
A variational approach to dissipative SPDEs with singular drift
We prove global well-posedness for a class of dissipative semilinear
stochastic evolution equations with singular drift and multiplicative Wiener
noise. In particular, the nonlinear term in the drift is the superposition
operator associated to a maximal monotone graph everywhere defined on the real
line, on which no continuity nor growth assumptions are imposed. The hypotheses
on the diffusion coefficient are also very general, in the sense that the noise
does not need to take values in spaces of continuous, or bounded, functions in
space and time. Our approach combines variational techniques with a priori
estimates, both pathwise and in expectation, on solutions to regularized
equations.Comment: 35 page
Well-posedness and asymptotic behavior for stochastic reaction-diffusion equations with multiplicative Poisson noise
We establish well-posedness in the mild sense for a class of stochastic
semilinear evolution equations with a polynomially growing quasi-monotone
nonlinearity and multiplicative Poisson noise. We also study existence and
uniqueness of invariant measures for the associated semigroup in the Markovian
case. A key role is played by a new maximal inequality for stochastic
convolutions in spaces.Comment: Final versio
Parameter estimation for semilinear SPDEs from local measurements
This work contributes to the limited literature on estimating the diffusivity
or drift coefficient of nonlinear SPDEs driven by additive noise. Assuming that
the solution is measured locally in space and over a finite time interval, we
show that the augmented maximum likelihood estimator introduced in Altmeyer,
Reiss (2020) retains its asymptotic properties when used for semilinear SPDEs
that satisfy some abstract, and verifiable, conditions. The proofs of
asymptotic results are based on splitting the solution in linear and nonlinear
parts and fine regularity properties in -spaces. The obtained general
results are applied to particular classes of equations, including stochastic
reaction-diffusion equations. The stochastic Burgers equation, as an example
with first order nonlinearity, is an interesting borderline case of the general
results, and is treated by a Wiener chaos expansion. We conclude with numerical
examples that validate the theoretical results.Comment: corrected versio
Approximation and convergence of solutions to semilinear stochastic evolution equations with jumps
We prove that the mild solution to a semilinear stochastic evolution equation
on a Hilbert space, driven by either a square integrable martingale or a
Poisson random measure, is (jointly) continuous, in a suitable topology, with
respect to the initial datum and all coefficients. In particular, if the
leading linear operators are maximal (quasi-)monotone and converge in the
strong resolvent sense, the drift and diffusion coefficients are uniformly
Lipschitz continuous and converge pointwise, and the initial data converge,
then the solutions converge.Comment: 28 pages, no figure
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