Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Brzeźniak, Zdzisław; Neerven, J. M. A. M. van; Veraar, Mark; Weis, Lutz

doi:10.1016/j.jde.2008.03.026

Cited by 77 publications

(99 citation statements)

References 25 publications

(50 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In our approach, we need a smallness condition on L f and L b and are not able to take the above-mentioned linear part into account yet. There is a possibility that the operator-theoretic approach of [11] works in such a setting. We also refer the reader to subsection 5.3 for a discussion on the smallness condition.…”

Section: Main Resultmentioning

confidence: 99%

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Neerven¹,

Veraar²,

Weis³

2012

SIAM J. Math. Anal.

115

View full text Add to dashboard Cite

Abstract. We prove maximal L p -regularity for the stochastic evolution equation dU (t) + AU (t) dt = F (t, U (t)) dt + B(t, U (t)) dW H (t), t ∈ [0, T ], U (0) = u 0 , under the assumption that A is a sectorial operator with a bounded H ∞ -calculus of angle less than 1. Introduction. Maximal L p -regularity techniques have been pivotal in much of the recent progress in the theory of parabolic evolution equations (see [2,22,25,54,76,86] and the references therein). Among other things, such techniques provide a systematic and powerful tool to study nonlinear and time-dependent parabolic problems.For stochastic parabolic evolution equations, maximal L p -regularity results have been obtained previously by Krylov for second order problems on R d [44,46,47,48,49], by Kim for second order problems on bounded domains in R d [43], and by Mikulevicius and Rozovskii for Navier-Stokes equations [63]. A systematic theory of maximal L p -regularity for stochastic evolution equations, however, based on abstract operator-theoretic properties of the operators governing the equation, has yet to be developed. A first step towards such a theory has been taken in our recent paper [68], where it was shown that if A is a sectorial operator with a bounded H ∞ -calculus of angle <

show abstract

Section: Main Resultmentioning

confidence: 99%

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Neerven¹,

Veraar²,

Weis³

2012

SIAM J. Math. Anal.

115

View full text Add to dashboard Cite

show abstract

“…[14,Theorems 5,6,8]). Sinceφ 1 is subhomogeneous and 0 ∈ H n , there is a constant C > 0 (independent of n) such that…”

Section: Additive Noisementioning

confidence: 99%

Strong solutions for stochastic partial differential equations of gradient type

Gess

2012

Journal of Functional Analysis

View full text Add to dashboard Cite

Unique existence of analytically strong solutions to stochastic partial differential equations (SPDE) with drift given by the subdifferential of a quasi-convex function and with general multiplicative noise is proven. The proof applies a genuinely new method of weighted Galerkin approximations based on the "distance" defined by the quasi-convex function. Spatial regularization of the initial condition analogous to the deterministic case is obtained. The results yield a unified framework which is applied to stochastic generalized porous media equations, stochastic generalized reaction-diffusion equations and stochastic generalized degenerated p-Laplace equations. In particular, higher regularity for solutions of such SPDE is obtained.

show abstract

“…(ii) Once the well posedness of the equation (3) generated by L is obtained, it implies various extensions of the results on the corresponding boundary value problems, problems with unbounded coefficients, fractional dynamics, Malliavin calculus, Banach space valued SDE, stochastic monotonicity (see [5,6,16,18,20,26]) obtained earlier for particular cases.…”

mentioning

confidence: 86%

The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Kolokoltsov

2010

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Ito's construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy-Khintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein-Kantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.

show abstract

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Cited by 77 publications

References 25 publications

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Maximal $L^p$-Regularity for Stochastic Evolution Equations

Strong solutions for stochastic partial differential equations of gradient type

The Lévy–Khintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Contact Info

Product

Resources

About