Stochastic equations in infinite dimensions

By Jin Feng, Thomas G. Kurtz. 2006

3,216 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/57851e9b37bb6fe9df2719c6085043613a795a6f

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Citations in the corpus

Explosive solutions of stochastic reaction-diffusion equations in mean Lp-norm (2011, neutral): “Stochastic reaction-diffusion equation, Parabolic Itô equations, Itô’s Formula, Positive Solutions, Blow-up of solutions, Explosions.”
Ito's formula in UMD Banach spaces and regularity of solutions of the Zakai equation (2008, background): “In the framework where E is the Hilbert space L2(Rd), the autonomous version of the problem (1.2) has been studied for instance by Da Prato, Iannelli and Tubaro [11] and Da Prato and Zabczyk [13], who proved the existence of strong solutions for this equation.”
The geometric minimum action method: A least action principle on the space of curves (2008, background): “3) below is replaced by a stochastic partial differential equation defining a stochastic process X with values in some suitable Hilbert space [2]; situations of this type are considered in Section 4.”
Cylindrical Lévy processes in Banach spaces (2009, background): “2 extends the concept of a solution of stochastic Cauchy problems on a Hilbert space or a Banach space driven by a Lévy process to the cylindrical situation, see [9] for the case of a Hilbert space and [12] for the case of a Banach space.”
The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion (2011, methodology): “First of all we recall the definition of weak solution according to Da Prato and Zabczyk [3].”
Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions (2011, background): “" Fernique’s theorem [12] states that for any exponent p ≥ 0, we have Eπ0[‖x‖s ] <∞.”
Brownian Representations of Cylindrical Local Martingales, Martingale Problem and Strong Markov Property of Weak Solutions of SPDEs in Banach Spaces (2005, background): “The contribution of Theorem 2 is that we do not consider only continuous local martingales in Hilbert or Banach spaces (as in [3] or [4]) but we represent cylindrical local martingales over a Banach space.”
Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces (2004, background): “1) on a bounded interval; see, for example, [7].”
The stochastic Fubini theorem revisited (2012, background): “There is already an extensive literature on such Fubini theorems (see [1,2,4,5,10,11]).”
Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations (2004, methodology): “For a comprehensive presentation of these equations see [8].”
Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups (2008, background): “Then we have (see [10]) that Z(t)+WA,σ(t), t ≥ 0, is a mild solution to (1.”
Stochastic optimal control of delay equations arising in advertising models (2004, background): “Unfortunately it is not possible, in general, to obtain regular solutions of the HJB equation (22) using the existing theory based on perturbations of Ornstein-Uhlenbeck semigroups (see e.g. Barbu and Da Prato [1], Da Prato and Zabczyk [5] and Gozzi [15], [16]).”
Limiting dynamics for stochastic wave equations (2008, background): “where | · |LQ2 is the Hilbert–Schmidt norm, see [17].”
Analysis of SPDEs arising in path sampling. Part I: The Gaussian case (2005, background): “The choice of this larger space does not affect any of the subsequent expressions (see also [4] for further details).”