A Khasminskii type averaging principle for stochastic reaction–diffusion equations

By Sandra Cerrai. 2008

189 citations recorded by the source corpus.

= g(Xf (t), Y8(t)), Y8(O) y e I~k for some parameter 0 0. But in applications that is more interesting is the behavior of X, (t) for t in intervals of order e-1 or even larger. Actually, it is indeed on those time scales that the most significant changes happen, such as exit from the neighborhood of an equilibrium point or of a periodic trajectory. With the natural time scaling t F t/E, if we set XE(t) := Xe(t/s) and ', (t) := Y(t/s), (1.1) can be rewritten as dX, (t) = b(X,. (t), Y(t)), Xe(O) = x E Rn, (1.2) dd.t. 1 dtye YY I Y g Oro (t), Y,, (0), F (0)= y E R'k

Subfields: Computational Theory and Mathematics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/28b1767c8f4b668a4f21e9ac6eca220764cacc85

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Citations in the corpus

Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations☆ (2011, background): “It was followed by [5] and [6] that the first author extended the results in the [4] to the general case, i.”
Option pricing under hybrid stochastic and local volatility (2013, background): “…averaging principle and ergodic theorem or, more directly, asymptotic diffusion limit theory of stochastic differential equations with a small parameter which was initiated by Khasminskii (1966) and developed by Papanicolaou (1978), Asch et al. (1991), Kim (2004), and Cerrai (2009), among others.”
Strong and weak orders in averaging for SPDEs (2012, methodology): “As far as stochastic partial differential equations are concerned, in [4] both the Khasminskii technique and a martingale problem approach are used; in [5] a modified Poisson equation is the essential tool.”
Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component (2018, methodology): “The averaging principle for such SPDE systems has been proved in a very general framework in [8], [10] for globally Lipschitz coefficients, and later in [9] in the case of non-globally Lipschitz continuous coefficients.”
Two-time-scales hyperbolic–parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles (2017, methodology): “Moreover, we will work in the framework of Green functions, which is a little different from the previous works [6, 7, 16, 18] investigating the coupled hyperbolic-parabolic equations.”
Diffusion approximation for fully coupled stochastic differential equations (2020, result): “[2, 3, 13, 22, 29, 34, 46, 52], see also [10, 11, 12, 50] for similar results for SPDEs.”