5,463 research outputs found
Stochastic partial differential equations with singular terminal condition
In this paper, we first prove existence and uniqueness of the solution of a
backward doubly stochastic differential equation (BDSDE) and of the related
stochastic partial differential equation (SPDE) under monotonicity assumption
on the generator. Then we study the case where the terminal data is singular,
in the sense that it can be equal to + on a set of positive measure. In
this setting we show that there exists a minimal solution, both for the BDSDE
and for the SPDE. Note that solution of the SPDE means weak solution in the
Sobolev sense
Beyond Poisson-Boltzmann: Numerical sampling of charge density fluctuations
We present a method aimed at sampling charge density fluctuations in Coulomb
systems. The derivation follows from a functional integral representation of
the partition function in terms of charge density fluctuations. Starting from
the mean-field solution given by the Poisson-Boltzmann equation, an original
approach is proposed to numerically sample fluctuations around it, through the
propagation of a Langevin like stochastic partial differential equation (SPDE).
The diffusion tensor of the SPDE can be chosen so as to avoid the numerical
complexity linked to long-range Coulomb interactions, effectively rendering the
theory completely local. A finite-volume implementation of the SPDE is
described, and the approach is illustrated with preliminary results on the
study of a system made of two like-charge ions immersed in a bath of
counter-ions
Random attractors for degenerate stochastic partial differential equations
We prove the existence of random attractors for a large class of degenerate
stochastic partial differential equations (SPDE) perturbed by joint additive
Wiener noise and real, linear multiplicative Brownian noise, assuming only the
standard assumptions of the variational approach to SPDE with compact
embeddings in the associated Gelfand triple. This allows spatially much rougher
noise than in known results. The approach is based on a construction of
strictly stationary solutions to related strongly monotone SPDE. Applications
include stochastic generalized porous media equations, stochastic generalized
degenerate p-Laplace equations and stochastic reaction diffusion equations. For
perturbed, degenerate p-Laplace equations we prove that the deterministic,
infinite dimensional attractor collapses to a single random point if enough
noise is added.Comment: 34 pages; The final publication is available at
http://link.springer.com/article/10.1007%2Fs10884-013-9294-
The rational SPDE approach for Gaussian random fields with general smoothness
A popular approach for modeling and inference in spatial statistics is to
represent Gaussian random fields as solutions to stochastic partial
differential equations (SPDEs) of the form , where
is Gaussian white noise, is a second-order differential
operator, and is a parameter that determines the smoothness of .
However, this approach has been limited to the case ,
which excludes several important models and makes it necessary to keep
fixed during inference.
We propose a new method, the rational SPDE approach, which in spatial
dimension is applicable for any , and thus remedies
the mentioned limitation. The presented scheme combines a finite element
discretization with a rational approximation of the function to
approximate . For the resulting approximation, an explicit rate of
convergence to in mean-square sense is derived. Furthermore, we show that
our method has the same computational benefits as in the restricted case
. Several numerical experiments and a statistical
application are used to illustrate the accuracy of the method, and to show that
it facilitates likelihood-based inference for all model parameters including
.Comment: 28 pages, 4 figure
Super-Brownian motion as the unique strong solution to an SPDE
A stochastic partial differential equation (SPDE) is derived for
super-Brownian motion regarded as a distribution function valued process. The
strong uniqueness for the solution to this SPDE is obtained by an extended
Yamada-Watanabe argument. Similar results are also proved for the Fleming-Viot
process.Comment: Published in at http://dx.doi.org/10.1214/12-AOP789 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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