关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stochastic equations in infinite dimensions

Part I. Deterministic Differential Equations: 1. Linear analysis 2. Galerkin approximation and finite elements 3. Time-dependent differential equations Part II. Stochastic Processes and Random Fields: 4. Probability theory 5. Stochastic processes 6. Stationary Gaussian processes 7. Random fields Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs) 9. Elliptic PDEs with random data 10. Semilinear stochastic PDEs.

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An Introduction to Computational Stochastic PDEs

Linear operators part ii (spectral theory)

https://repository.tudelft.nl/islandora/object/uuid%3Ad8941932-86a0-43aa-b0d7-130799bbcfc1/datastream/OBJ/download

Stochastic evolution equations in UMD Banach spaces

In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $p\in (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.

Stochastic integration in quasi-Banach spaces

Recently, Hairer et. al [14] showed that there exist SDEs with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L p -sense with respect to the initial value for every p ∈ [1,∞]. In this article we provide sufficient conditions on the coefficien t functions of the SDE and on p ∈ (0,∞] which ensure local Lipschitz continuity in the strong L p -sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear SDEs from the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.

/pdf/local-lipschitz-continuity-in-the-initial-value-and-strong-41pr863gac.pdf

Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Let X be a (quasi-)Banach space. Let d = (dn)n 1 be an X-valued sequence of random variables adapted to a ltration ( Fn)n 1 on a probability space ( ;A;P), dene F1 := (Fn : n 1) and let e = (en)n 1 be aF1-conditionally independent sequence on ( ;A;P) such that L(dnjFn 1) =L(enjF1) for all n 1 (F0 =f ;?g). If there exists a p2 (0;1) and a constant Dp independent of d and e such that one has, for all n 1, X k=1 dk p D pE n X k=1 ek p

/pdf/vector-valued-decoupling-and-the-burkholder-davis-gundy-4hv16tb239.pdf

Vector-valued decoupling and the Burkholder–Davis–Gundy inequality

In this article we prove pathwise Holder convergence with optimal rates of the implicit Euler scheme for the abstract stochastic Cauchy problem $$\begin{aligned} \left\{ \begin{aligned} dU(t)&= AU(t)\,dt + F(t,U(t))\,dt + G(t,U(t))\,dW_H(t);\quad t\in [0,T],\\ U(0)&=x_0. \end{aligned}\right. \end{aligned}$$ Here $$A$$ is the generator of an analytic $$C_0$$ -semigroup on a umd Banach space $$X,\,W_H$$ is a cylindrical Brownian motion in a Hilbert space $$H$$ , and the functions $$F:[0,T]\times X\rightarrow X_{\theta _F}$$ and $$G:[0,T]\times X\rightarrow {\fancyscript{L}}(H,X_{\theta _G})$$ satisfy appropriate (local) Lipschitz conditions. The results are applied to a class of second order parabolic SPDEs driven by multiplicative space-time white noise.

Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

We show that if a sequence of piecewise affine linear processes converges in the strong sense with a positive rate to a stochastic process that is strongly Holder continuous in time, then this sequence converges in the strong sense even with respect to much stronger Holder norms and the convergence rate is essentially reduced by the Holder exponent. Our first application hereof establishes pathwise convergence rates for spectral Galerkin approximations of stochastic partial differential equations. Our second application derives strong convergence rates of multilevel Monte Carlo approximations of expectations of Banach-space-valued stochastic processes.

/pdf/convergence-in-holder-norms-with-applications-to-monte-carlo-2lvfevyhbl.pdf

Sonja Cox

Papers

Stochastic integration in quasi-Banach spaces

Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Vector-valued decoupling and the Burkholder–Davis–Gundy inequality

Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions