Stochastic Evolution Systems

Rozovsky, Boris L.; Lototsky, Sergey V.

doi:10.1007/978-3-319-94893-5

Cited by 59 publications

(49 citation statements)

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“…When α = 2, these SPDEs have been studied widely. W k,2 -theory was well established by Pardoux [15] and Rozovskii [16]. A more general W 2,q -theory was founded by Krylov [9][10][11] for 2 ≤ q < ∞.…”

Section: Introductionmentioning

confidence: 99%

Hölder estimates of mild solutions for nonlocal SPDEs

et al. 2019

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We consider nonlocal PDEs driven by additive white noises on R d . For L q integrable coefficients, we derive the existence and uniqueness, as well as Hölder continuity, of mild solutions. Precisely speaking, the unique mild solution is almost surely Hölder continuous with Hölder index 0 < θ < (1/2 -d/(qα))(1 ∧ α). Moreover, we show that any order γ (< q) moment of Hölder normal for u on every bounded domain of R + × R d is finite. MSC: 60H15; 35R60

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Section: Introductionmentioning

confidence: 99%

Hölder estimates of mild solutions for nonlocal SPDEs

et al. 2019

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“…( 13) for all t ∈ [0, T ] and for all ω in ⊂ with P( ) = 1. Furthermore, since the conditions are fulfilled for all k ∈ N, using Corollary 3 from p. 141 in [49], we obtain that ω ν n ,R,n…”

Section: Existence and Uniqueness Of The Approximating Sequencementioning

confidence: 82%

“…The equation above is a particular case of Eq. (1.1)-(1.2) in Chapter 4, Section 4.1, p. 129 in [49]. All assumptions required by Theorem 1 and Theorem 2 in [49], Chapter 4, are fulfilled.…”

Section: Existence and Uniqueness Of The Approximating Sequencementioning

confidence: 97%

“…(1.1)-(1.2) in Chapter 4, Section 4.1, p. 129 in [49]. All assumptions required by Theorem 1 and Theorem 2 in [49], Chapter 4, are fulfilled. Therefore there exists a unique solution ω ν n ,R,n t which belongs to the class L 2 (0, T ; W k,2 (T 2 )) ∩ C([0, T ], W k−1,2 (T 2 )) and satisfies Eq.…”

Section: Existence and Uniqueness Of The Approximating Sequencementioning

confidence: 98%

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Well-posedness for a stochastic 2D Euler equation with transport noise

Crisan

Lang

2022

Stoch PDE: Anal Comp

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We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.

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“…-912 -Such SPDEs were studied extensively in classical works [22,24,25]. It is a natural question, studied for instance in a series of papers by Gyöngy [17,18], to what extent such SPDEs are approximated by (random) PDEs, upon replacing the (Stratonovich) differential dW = dW (ω) byẆ ε (ω) dt, given a suitable family of smooth approximation (W ε ) to Brownian motion.…”

Section: Introductionmentioning

confidence: 99%

Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac

Diehl¹,

Friz²,

Stannat³

2017

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Stochastic Evolution Systems

Cited by 59 publications

References 0 publications

Hölder estimates of mild solutions for nonlocal SPDEs

Hölder estimates of mild solutions for nonlocal SPDEs

Well-posedness for a stochastic 2D Euler equation with transport noise

Stochastic partial differential equations: a rough paths view on weak solutions via Feynman–Kac

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