Abstract:We prove a stochastic averaging theorem for stochastic differential equations in which the slow and the fast variables interact. The approximate Markov fast motion is a family of Markov process with generator L x for which we obtain a locally uniform law of large numbers and obtain the continuous dependence of their invariant measures on the parameter x. These results are obtained under the assumption that L x satisfies Hörmander's bracket conditions, or more generally L x is
“…In one dimension, the diffusion coefficient would be Y f 2 (x, y)π(dy) 1 2 . Similar results are known to hold on manifolds, see [Li18] and references therein. In contrast, here however both f and ḡ are just naïvely averaged.…”
We prove an averaging principle for interacting slow-fast systems driven by independent fractional Brownian motions. The mode of convergence is in Hölder norm in probability. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, partially improving a recent result of Panloup and Richard.
“…In one dimension, the diffusion coefficient would be Y f 2 (x, y)π(dy) 1 2 . Similar results are known to hold on manifolds, see [Li18] and references therein. In contrast, here however both f and ḡ are just naïvely averaged.…”
We prove an averaging principle for interacting slow-fast systems driven by independent fractional Brownian motions. The mode of convergence is in Hölder norm in probability. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, partially improving a recent result of Panloup and Richard.
“…SDEs with symmetries are popular topics, see [DVMU16, AVMU19, ELJL10, Tak20, EL06, ELJL04]. See also [Li18b,Li12] for perturbation to symmetries.…”
Section: Recent Progress On Slow/fast Markovian Dynamicsmentioning
confidence: 99%
“…If the Strong Hörmander's condition holds and Y is compact, then the Markov process with generator m k=1 (X i ) 2 + X 0 has a unique invariant probability measure. We state a theorem for Y compact, a version with Y not compact can also be found in [Li18b].…”
“…Theorem 3.2. [Li18b] Let Y be a compact manifold. Suppose that Y i (x, •) are bounded vector fields on Y with bounded derivatives and C ∞ in both variables.…”
We review recent developments of slow/fast stochastic differential equations, and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noise and providing new examples. The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and 'Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.
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