Stochastic partial differential equations : analysis and computations最新文献

The Metropolis-Adjusted Langevin Algorithm (MALA) is a Markov Chain Monte Carlo method which creates a Markov chain reversible with respect to a given target distribution, ${\pi }^N$ , with Lebesgue density on $R^N$ ; it can hence be used to approximately sample the target distribution. When the dimension N is large a key question is to determine the computational cost of the algorithm as a function of N. The measure of efficiency that we consider in this paper is the expected squared jumping distance (ESJD), introduced in Roberts et al. (Ann Appl Probab 7(1):110-120, 1997). To determine how the cost of the algorithm (in terms of ESJD) increases with dimension N, we adopt the widely used approach of deriving a diffusion limit for the Markov chain produced by the MALA algorithm. We study this problem for a class of target measures which is not in product form and we address the situation of practical relevance in which the algorithm is started out of stationarity. We thereby significantly extend previous works which consider either measures of product form, when the Markov chain is started out of stationarity, or non-product measures (defined via a density with respect to a Gaussian), when the Markov chain is started in stationarity. In order to work in this non-stationary and non-product setting, significant new analysis is required. In particular, our diffusion limit comprises a stochastic PDE coupled to a scalar ordinary differential equation which gives a measure of how far from stationarity the process is. The family of non-product target measures that we consider in this paper are found from discretization of a measure on an infinite dimensional Hilbert space; the discretised measure is defined by its density with respect to a Gaussian random field. The results of this paper demonstrate that, in the non-stationary regime, the cost of the algorithm is of $O\left(N^{1/2}\right)$ in contrast to the stationary regime, where it is of $O\left(N^{1/3}\right)$ .

We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $R^2$ . Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.

In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of $R^d$ . The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green's function is not required.