Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

Röckner, Michael; Wu, Bo; Zhu, Rongchan; Zhu, Xiangchan

doi:10.1137/18m1211076

Cited by 17 publications

(16 citation statements)

References 33 publications

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“…The stochastic heat equation on Riemannian manifold had been studied detailed by [22](see also [17]). Here they introduced some notation.…”

Section: Stochastic Heat Equationmentioning

confidence: 99%

Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

Wu¹

2018

Preprint

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“…The stochastic heat equation on Riemannian manifold had been studied detailed by [22](see also [17]). Here they introduced some notation.…”

Section: Stochastic Heat Equationmentioning

confidence: 99%

Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

Wu¹

2018

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“…the law of Brownian motion or spatial white noise). • We mention that in [RWZZ20,CWZZ21] global martingale solutions were constructed for geometric stochastic heat equations by using a Dirichlet form approach. This relies on an integration by parts formula for the known invariant measure.…”

Section: Introductionmentioning

confidence: 99%

Global existence and non-uniqueness for 3D Navier--Stokes equations with space-time white noise

Hofmanová¹,

Zhu²,

Zhu³

2021

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We establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier-Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition inand implies also non-uniqueness in law. Contents 1. Introduction 2 1.1. Singular SPDEs 3 1.2. Convex integration 5 1.3. Decomposition 6 1.4. Final remarks 8 2. Preliminaries 9 2.1. Function spaces 9 2.2. Paraproducts, commutators and localizers 10 3. More regular stochastic perturbations 12 3.1. The case of γ ∈ (1/2, 3/2] 13 3.2. The case of γ ∈ (1/6, 1/2] 13 3.3. The case of γ ∈ (0, 1/6] 14 4. Paracontrolled solutions 15 4.1. Stochastic objects 15 Date: December 30, 2021. 2010 Mathematics Subject Classification. 60H15; 35R60; 35Q30. Key words and phrases. stochastic Navier-Stokes equations, probabilistically strong solutions, paracontrolled calculus, non-uniqueness in law, convex integration.This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 949981). The financial support by the DFG through the CRC 1283 "Taming uncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications" is greatly acknowledged. R.Z. is grateful to the financial supports of the NSFC (No. 11922103).

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“…This work is the continuity of [55], which is motivated by Tadahisa Funaki's pioneering work [33] and Martin Hairer's recent work [42]. Let M be a n-dimensional compact Riemannian manifold.…”

Section: Introductionmentioning

confidence: 99%

“…In [55], starting from the Wiener measure (or Brownian bridge measure) µ on C([0, 1], M) we use the theory of Dirichlet forms to construct a natural evolution which admits µ as an invariant measure. Moreover, the relation between the evolution constructed in [55] and (1.1) has also been discussed in [55] by using the Andersson-Driver approximation. It is conjectured in [55] that the Markov processes constructed by Dirichlet form in [55] have the same law as the solution to (1.1).…”

Section: Introductionmentioning

confidence: 99%

Stochastic Heat Equations for infinite strings with Values in a Manifold

Chen¹,

Wu²,

Zhu³

et al. 2018

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In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R + or R with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper [55] on finite volume case.Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution if the Ricci curvature is strictly positive and the nonergodicity of the process if the sectional curvature is negative.

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Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

Cited by 17 publications

References 33 publications

Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

Spectral gaps for the O-U/Stochastic heat processes on path space over a Riemannian manifold with boundary

Global existence and non-uniqueness for 3D Navier--Stokes equations with space-time white noise

Stochastic Heat Equations for infinite strings with Values in a Manifold

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