A ${W}^{n}_{2}$ -Theory of Stochastic Parabolic Partial Differential Systems on C 1-domains

Kim, Kyeong Hun; Lee, Kijung

doi:10.1007/s11118-012-9302-0

Cited by 16 publications

(27 citation statements)

References 23 publications

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“…The necessity of such theory came from stochastic partial differential equations (SPDEs) and is well explained in [19]. For SPDEs in weighted Sobolev spaces, we refer the reader to [27,13,12,25,16].…”

Section: Introductionmentioning

confidence: 99%

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

Dong

Kim

2015

Advances in Mathematics

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We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assumed to be only measurable either in the time or the spatial variable. As functions of the other variables the coefficients have small bounded mean oscillation (BMO) semi-norms. The lower-order coefficients are allowed to blow up near the boundary with a certain optimal growth condition. As a corollary, we also obtain the corresponding results for elliptic equations.2010 Mathematics Subject Classification. 35J25, 35K20, 35R05.

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Section: Introductionmentioning

confidence: 99%

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

Dong

Kim

2015

Advances in Mathematics

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“…For p = 2 and X a Hilbert space, this discussion applies to the present setting as well. See also [32] for a related result for systems.…”

Section: 2mentioning

confidence: 96%

Maximal $${\gamma}$$ γ -regularity

2015

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Abstract. In this paper, we prove maximal regularity estimates in "square function spaces" which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results for both deterministic and stochastic equations in L p -spaces with 1 < p < ∞. For stochastic equations, the case 1 < p < 2 was not covered in the literature so far. Moreover, the "square function spaces" allow initial values with the same roughness as in the L 2 -setting.

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“…We note that under non-degeneracy conditions SIDEs have been investigated with various generalities in the literature, and very nice results on their solvability in L p -spaces have recently been obtained. In particular, L p -theories for such equations have been developed in [22], [23], [29], [30] and [31], which extend some results of the L p theory of Krylov [24] to certain classes of equations with non local operators. See also [7], [11] and [35] in the case of deterministic equations.…”

Section: Introductionmentioning

confidence: 99%

On $$L_p$$-solvability of stochastic integro-differential equations

Gyöngy

2020

Stoch PDE: Anal Comp

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A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.2010 Mathematics Subject Classification. Primary 45K05, 60H15, 60H20, 60J75; Secondary 35B65.Its derivatives in x ∈ R d up to order max{ m , 3} exist and are continuous in x ∈ R d such that |D k ξ| ≤ξ k = 0, 1, 2, ..., max{ m , 3} :=m for all (ω, t, x, z) ∈ Ω × H T × Z. Moreover, K −1 ≤ | det(I + θDξ t,z (x))| for all (ω, t, x, z, θ) ∈ Ω × H T × Z × [0, 1], where I is the d × d identity matrix, and Dξ denotes the Jacobian matrix of ξ in x ∈ R d . Assumption 2.3. The function η = (η i ) maps Ω × [0, T ] × R d × Z into R d such that Assumption 2.2 holds with η andη in place of ξ andξ, respectively.

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A ${W}^{n}_{2}$ -Theory of Stochastic Parabolic Partial Differential Systems on C 1-domains

Cited by 16 publications

References 23 publications

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

Elliptic and parabolic equations with measurable coefficients in weighted Sobolev spaces

Maximal $${\gamma}$$ γ -regularity

On $$L_p$$-solvability of stochastic integro-differential equations

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