Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

Fernando, B. P. W.; Sritharan, S. S.

doi:10.1080/07362994.2013.759482

Cited by 26 publications

(35 citation statements)

References 36 publications

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“…Remark 4.9. In contrast to the theory introduced in this section, one can also define a weak or a martingale solution to system (4.1), see [7,15,35]. For the case of a Wiener noise, we also refer to [5,9,16,17,33,41].…”

Section: The Stochastic Navier-stokes Equationsmentioning

confidence: 99%

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Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Benner

Trautwein

2019

Mathematische Nachrichten

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We consider the controlled stochastic Navier-Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier-Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists. K E Y W O R D SLévy process, local mild solution, stochastic Navier-Stokes equations, stochastic optimal control M S C ( 2 0 1 0 ) 93E20, 49J20, 35Q30 1444

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Section: The Stochastic Navier-stokes Equationsmentioning

confidence: 99%

“…In [33,41], weak solutions are considered with noise terms given by Wiener processes. Weak solutions with Lévy noise are considered in [7,15]. For three-dimensional domains, uniqueness is still an open problem and weak solutions are introduced as martingale solutions, see [5,9,16,17,35].…”

Section: Introductionmentioning

confidence: 99%

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Benner

Trautwein

2019

Mathematische Nachrichten

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“…Since C 1,1 (D( ); ℝ) 5 is a subspace of ℰ and is dense in UC (D(A); ℝ), the space of all uniformly continuous and bounded functions on D(A) (see [30]), we can extend ( P ) ≥0 to UC (D(A); ℝ). However, by the existence theorem due to Getoor (Proposition 4.1, [26] and also see Lemma 3.9, [21]), there exists a measure , ∈ D(A), ∈ [0, ] such that…”

Section: Lemma 42mentioning

confidence: 99%

“…By Theorem 2.1 [42], there exists a martingale solution of the problem (5.16) in [0, 1 ( )). We can recursively obtain a martingale solution ( Ω , ℙ , ℱ , X( , ) ) of the system (5.16) for the interval [0, ] (see [16,21]). Now, the proof of Proposition 5.5 is completed by proving lemmas given below.…”

Section: Invariant Measures and Ergodicit Ymentioning

confidence: 99%

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

Mohan

Sakthivel

Sritharan

2018

Mathematische Nachrichten

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In this work we construct a Markov family of martingale solutions for 3D stochastic Navier-Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing. K E Y W O R D S ergodicity, invariant measures, Lévy noise, Markov solutions, stochastic Navier-Stokes equations M S C ( 2 0 1 0 ) 35Q10, 37L40, 60H15, 60J75 1056

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“…Proof. We apply Theorem A.1 to get an H 1 2 2 (R)-valued solution, and then we show the existence of normalized conditional density by using the Getoor's lemma [22,Proposition 4.1] or [15,Lemma 3.9].…”

Section: )mentioning

confidence: 99%

Nonlinear filtering with correlated Lévy noise characterized by copulas

Fernando¹,

Hausenblas²

2018

Braz. J. Probab. Stat.

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The objective in stochastic filtering is to reconstruct the information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process.Usually X and Y are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Lévy) process. We are interested in the situation where both the state process X and the observation process Y are perturbed by coupled Lévy processes. More precisely, L = (L1, L2) is a 2-dimensional Lévy process in which the structure of dependence is described by a Lévy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and L for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like P(X(t) > a), where a is a threshold.Primary 60H15, 35R30, 60K35; secondary 46G10, 28B05

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Nonlinear Filtering of Stochastic Navier-Stokes Equation with Itô-Lévy Noise

Cited by 26 publications

References 36 publications

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise

Nonlinear filtering with correlated Lévy noise characterized by copulas

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