Perturbation of Conservation Laws and Averaging on Manifolds

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

Slow-Fast Systems with Fractional Environment and Dynamics

“…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.…”

Rough Homogenisation with Fractional Dynamics

“…SDEs with symmetries are popular topics, see [DVMU16, AVMU19, ELJL10, Tak20, EL06, ELJL04]. See also [Li18b,Li12] for perturbation to symmetries.…”

“…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.…”

“…Theorem 3.3. [Li18b] Let Y be compact and Y i are in C 2 ∩ BC 1 . Suppose that σ0 and σ i are C 1 , where i = 1, .…”