Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

Es-Sarhir, Abdelhadi; Stannat, Wilhelm

doi:10.1016/j.jfa.2010.02.017

Cited by 14 publications

(14 citation statements)

References 12 publications

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“…In the next lemma, Lemma 2.2, we present a slightly different formulation of Lemma 2.2, that is, we add and substract in (19) the generator in (14) and the noise operator in (15). Lemma 2.2 is thus an immediate consequence of Lemma 2.1 and its proof is therefore omitted.…”

Section: Notationmentioning

confidence: 99%

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Hutzenthaler¹,

Jentzen²

2020

Ann. Probab.

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We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the L p -distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the L q -distances of the differences of the local characteristics for suitable p, q > 0. As application of our perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with non-globally monotone coefficients. As another application of our perturbation theory, we prove strong convergence rates for spectral Galerkin approximations of solutions of semilinear SPDEs with non-globally monotone nonlinearities including Cahn-Hilliard-Cook type equations and stochastic Burgers equations. Further applications of the perturbation theory include the regularity of solutions of SDEs with respect to the initial values and small-noise analysis for ordinary and partial differential equations.

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Section: Notationmentioning

confidence: 99%

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Hutzenthaler¹,

Jentzen²

2020

Ann. Probab.

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“…Our objective in this paper is to study the stability and accuracy properties of this stochastic model. Here stability refers to the contraction of two different trajectories of the filter (14), started at two different points, but driven by the same observed data; and accuracy refers to estimating the difference between the true trajectory of (1) which underlies the data, and the output of the filter (14). Similar questions are studied in finite dimensions in [32].…”

Section: Derivation Of the Continuous-time Limitmentioning

confidence: 99%

“…Whilst it should be possible to establish such bounds, using the techniques in [17] or [14], doing so is technically involved, as one needs to use random φ's in the definition of Z φ . In order to keep the presentation at a reasonable level, we refrain from giving details on this point.…”

Section: Pullback Stabilitymentioning

confidence: 99%

Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation

et al. 2013

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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.

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“…Let us remark that for the stochastic case it is even possible to bound exponential moments of the L 2 -norm. See [38,106].…”

Section: A-priori Estimates (1d-deterministic)mentioning

confidence: 99%

Stochastic PDEs and Lack of Regularity

Blömker

Romito

2015

Jahresber. Dtsch. Math. Ver.

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We review results on the existence and uniqueness for a surface growth model with or without space--time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open.\ud In the physically relevant dimension d=2 and with the physically relevant space--time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems.\ud Using spectral Galerkin method or any other type of regularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution. \ud We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up

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Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

Cited by 14 publications

References 12 publications

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation

Stochastic PDEs and Lack of Regularity

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