Abstract. Existence and uniqueness for semilinear stochastic evolution equations with additive noise by means of finite dimensional Galerkin approximations is established and the convergence rate of the Galerkin approximations to the solution of the stochastic evolution equation is estimated.These abstract results are applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation and a stochastic Burgers equation. The estimated convergence rates are illustrated by numerical simulations.The main novelty in this article is to estimate the difference of the finite dimensional Galerkin approximations and of the solution of the infinite dimensional SPDE uniformly in space, i.e., in the L ∞ -topology, instead of the usual Hilbert space estimates in the L 2 -topology, that were shown before.Key words. Galerkin approximations, stochastic partial differential equation, stochastic heat equation, stochastic reaction diffusion equation, stochastic Burgers equation, strong error criteria. AMS subject classifications. 60H15, 35K901. Introduction. In this work we present a general abstract result for the spatial approximation of stochastic evolution equations with additive noise via Galerkin methods. This abstract result is applied to several examples of stochastic partial differential equations (SPDEs) of evolutionary type including a stochastic heat equation, a stochastic reaction diffusion equation and a stochastic Burgers equation. In all examples we need to verify the following conditions. First, we need the rate of approximation of the linear equation obtained by omitting the nonlinear term in the stochastic evolution equation. Then one needs a quite weak Lipschitz condition for the nonlinearity and finally a uniform bound on the sequence of approximations. These results are the key for the main theorem (see Theorem 3.1). The main novelty in this article is to estimate the difference of the finite dimensional Galerkin approximations and of the solution of the infinite dimensional SPDE uniformly in space, i.e., in the L ∞ -topology, instead of the usual Hilbert space estimates shown before in the L 2 -topology. Although there are several different methods using finite dimensional approximations like, for instance, spectral Galerkin, finite elements, or wavelets, we focus here on the spectral Galerkin method. Thus the finite dimensional approximations are given by an expansion in terms of the eigenfunctions of a dominant linear operator. This spectral Galerkin method is one of the key tools in the analysis of stochastic or deterministic PDEs. For SPDEs see, for example, [16,9,17,2], where the Galerkin method was used to establish the existence of solutions. Moreover, spectral methods are an effective tool for numerical simulations, especially on domains, like the interval,

The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates study of the problem of accuracy and stability of 3DVAR filters for the Navier-Stokes equation.We work in the limit of high frequency observations and derive continuous time filters. This leads to a stochastic partial differential equation (SPDE) for state estimation, in the form of a damped-driven Navier-Stokes equation, with mean-reversion to the signal, and spatially-correlated time-white noise. Both forward and pullback accuracy and stability results are proved for this SPDE, showing in particular that when enough low Fourier modes are observed, and when the model uncertainty is larger than the data uncertainty in these modes (variance inflation), then the filter can lock on to a small neighbourhood of the true signal, recovering from order one initial error, if the error in the observations modes is small. Numerical examples are given to illustrate the theory.

The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing in various areas of application. We present a complete analysis of the ensemble Kalman inversion with perturbed observations for a fixed ensemble size when applied to linear inverse problems. The well-posedness and convergence results are based on the continuous time scaling limits of the method. The resulting coupled system of stochastic differential equations allows to derive estimates on the long-time behaviour and provides insights into the convergence properties of the ensemble Kalman inversion. We view the method as a derivative free optimization method for the least-squares misfit functional, which opens up the perspective to use the method in various areas of applications such as imaging, groundwater flow problems, biological problems as well as in the context of the training of neural networks.AMS classification scheme numbers: 65N21, 62F15, 65N75, 65C30, 90C56 for Hilbert spaces (H 1 , ·, · H 1 ), (H 2 , ·, · H 2 ) and z 1 ∈ H 1 , z 2 ∈ H 2 . The empirical means are given bythe minimum of n and the first exit time of e s at radius n. Then, for all n ∈ N, from (14) (after rebasing the integration interval from [0, t] to [s, s + t]) we obtainAs τ n → ∞, applying Fatou's lemma on the left hand side and applying the monotone convergence theorem on the right hand side givesProof. By Lemma Appendix A.4 we can directly take expectations in (14) to obtains 2 ds.Note that by dropping the non-negative mixed terms j = k and by using Jensen's and Young's inequalityProof. The idea of this proof is based on Theorem 4.6.2 in [33]. We define the stochastic Lyapunov functionThe generator applied to V fulfillsis monotonically decreasing.Proof. The assertions follow by arguments similar to the proof of Proposition 4.11.Thus, for all t, s ≥ 0, it follows similarly to the proof of Lemma 4.1 that,...,J converges to zero almost surely as t → ∞. Proof. We define the Lyapunov function V (r, t) = t β 1 J J j=1 |r (j) | 2 and obtain LV (r, t) ≤ βt β−1 J J j=1 |r (j) | 2 − t β 1 J J j=1 r (j) , C(r) + 1 t α + R B r (j) .Thus, LV (r, t) ≤ 1 J J j=1 |r (j) | 2 β − λ min t t α + R t β−1 .

In this article we derive rigorously amplitude equations for stochastic PDEs with quadratic nonlinearities, under the assumption that the noise acts only on the stable modes and for an appropriate scaling between the distance from bifurcation and the strength of the noise. We show that, due to the presence of two distinct timescales in our system, the noise (which acts only on the fast modes) gets transmitted to the slow modes and, as a result, the amplitude equation contains both additive and multiplicative noise. As an application we study the case of the one dimensional Burgers equation forced by additive noise in the orthogonal subspace to its dominant modes. The theory developed in the present article thus allows to explain theoretically some recent numerical observations from [Rob03].

This paper gives theoretical results on spinodal decomposition for the stochastic Cahn-Hilliard-Cook equation, which is a Cahn-Hilliard equation perturbed by additive stochastic noise. We prove that most realizations of the solution which start at a homogeneous state in the spinodal interval exhibit phase separation, leading to the formation of complex patterns of a characteristic size.In more detail, our results can be summarized as follows. The Cahn-Hilliard-Cook equation depends on a small positive parameter ε which models atomic scale interaction length. We quantify the behavior of solutions as ε → 0. Specifically, we show that for the solution starting at a homogeneous state the probability of staying near a finitedimensional subspace Y ε is high as long as the solution stays within distance r ε = O(ε R ) of the homogeneous state. The subspace Y ε is an affine space corresponding to the highly unstable directions for the linearized deterministic equation. The exponent R depends on both the strength and the regularity of the noise.

The paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach. We prove existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under nondegeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure

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The Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion.The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar stochastic differential equation. We focus here on a toy model with similar properties than the one arising in the context of Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with non-globally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler-Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming.We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cut-off or localisation, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence. 1 stay near the origin and have p-moments at least up to p < 3. Then, any moment of the difference between u andũ n will explode for h → 0.There has been significant progress in the field of strongly convergent numerical schemes for SDEs with non-global Lipschitz-continuous nonlinearities. Standard references for numerical methods for SDEs, like [17] and [25,22], show strong convergence of the Euler-Maruyama method only for globally Lipschitz-continuous drift and diffusion terms. Higham, Mao and Stuart [10] proved a conditional result about strong convergence of the Euler-Maruyama discretization for non-globally Lipschitz SDEs given that moments of both solutions and the discretization stay bounded. This means that the question of strong convergence was replaced by the question whether moments of the numerical scheme stay bounded, but Hutzenthaler, Jentzen and Kloeden [12] answered the latter to the negative, even proving that moments of the Euler-Maruyama scheme always explode in finite time if either drift or diffusion term are not globally Lipschitz. Instead, they proposed a slight modification of the EM method, the so-called "tamed" EM method (and implicit equivalents) in [13,11].The numerical scheme (2) arising in the EnKF analysis bears resemblance to the "tamed" methods used throughout the literature. In case only the drift is non-globally Lipschitz a similar idea to (2) was used already in [13], but there the drift-tamed nonlinearity is strictly bounded by one, while in EnKF it is still allowed to grow linearly. In increment-ta...