SDDEs limits solutions to sublinear reaction-diffusion SPDEs.
Self-intersection local time of -superprocess
Small random perturbations of infinite dimensional dynamical systems and nucleation theory
Small time asymptotics of Ornstein-Uhlenbeck densities in Hilbert spaces.
Sobolev solution for semilinear PDE with obstacle under monotonicity condition.
Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise.
Solutions statistiques homogènes des systèmes différentiels paraboliques et du système de Navier-Stokes
Some Dirichlet spaces obtained by subordinate reflected diffusions.
In this paper we want to show how well-known results from the theory of (regular) elliptic boundary value problems, function spaces and interpolation, subordination in the sense of Bochner and Dirichlet forms can be combined and how one can thus get some new aspects in each of these fields.
Some non-linear s. p. d. e's that are second order in time.
Some properties of superprocesses under a stochastic flow
For a superprocess under a stochastic flow in one dimension, we prove that it has a density with respect to the Lebesgue measure. A stochastic partial differential equation is derived for the density. The regularity of the solution is then proved by using Krylov’s Lp-theory for linear SPDE.
Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process
Stochastic partial differential equations on are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted -space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations...
Sparse finite element methods for operator equations with stochastic data
Let be a strongly elliptic operator on a -dimensional manifold (polyhedra or boundaries of polyhedra are also allowed). An operator equation with stochastic data is considered. The goal of the computation is the mean field and higher moments , , , of the solution. We discretize the mean field problem using a FEM with hierarchical basis and degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment for . The key tool...
Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains
We use the scale of Besov spaces , 1/τ = α/d + 1/p, α > 0, p fixed, to study the spatial regularity of solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains ⊂ ℝ. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.
SPDEs in spaces.
SPDEs with coloured noise: Analytic and stochastic approaches
We study strictly parabolic stochastic partial differential equations on , 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 related, deriving...
SPDEs with pseudodifferential generators: the existence of a density
We consider the equation du(t,x)=Lu(t,x)+b(u(t,x))dtdx+σ(u(t,x))dW(t,x) where t belongs to a real interval [0,T], x belongs to an open (not necessarily bounded) domain , and L is a pseudodifferential operator. We show that under sufficient smoothness and nondegeneracy conditions on L, the law of the solution u(t,x) at a fixed point is absolutely continuous with respect to the Lebesgue measure.
Stability of semilinear equations with boundary and pointwise noise
Stability of stationary and periodic solutions equations in Banach space.
Stationary solutions for heat equation perturbed by general additive noise.