Some remarkable martingales
Some remarkable pure martingales
Some remarks on boundary value problems for linear stochastic differential equations
Some remarks on Pitman's theorem
Some remarks on single jump processes
Some remarks on the heat flow for functions and forms.
Some remarks on the theory of stochastic integration
Some results on invariant measures in hydrodynamics
In questa nota, si presentano risultati di esistenza e di unicità di misure invarianti per l'equazione di Navier-Stokes che governa il moto di un fluido viscoso incomprimibile omogeneo in un dominio bidimensionale soggetto a una forzante che ha due componenti: una deterministica e una di tipo rumore bianco nella variabile temporale.
Some results on stochastic convolutions arising in Volterra equations perturbed by noise
Regularity of stochastic convolutions corresponding to a Volterra equation, perturbed by a white noise, is studied. Under suitable assumptions, hölderianity of the corresponding trajectories is proved.
Some theorems of random operator equations.
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.
Spectral approximation of infinite-dimensional Black-Scholes equations with memory.
Spectral statistics for random Schrödinger operators in the localized regime
We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy in the localized phase. Assume the density of states function is not too flat near . Restrict it to some large cube . Consider now , a small energy interval centered at that asymptotically contains infintely many eigenvalues when the volume of the cube grows to infinity. We prove that, with probability one in the large volume...
Speed of the Brownian loop on a manifold
We define the speed of the curved Brownian bridge as a white noise distribution operating on stochastic Chen integrals.