This chapter is devoted to substantiate the concept of locality (metastability, quasi-deterministic approximation) to be used. The framework here is set up by the SDE
in ℝd. Here, the coefficients are functions b ∈ C∞ (ℝd, ℝd) and σ ∈ C∞(ℝd, ℝd × d) and ε ≥ 0 parametrizes the intensity of (W t )t≥0 which denotes a Brownian motion in ℝd on1 a standard filtered probability space (ω, F,ℙ , (F t )t≥0) .Hence, for any ε > 0 and ξ ∈ ℝd, the stochastic process Xε,ξ solving (2.1) is a diffusion with drift b and covariance ε a, i.e. the generator of Xε is given by the differential operator
where the positive semi-definite matrix a is defined by
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© 2009 Springer-Verlag Berlin Heidelberg
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(2009). Locality and time scales of the underlying non-degenerate stochastic system: Freidlin-Wentzell theory. In: Local Lyapunov Exponents. Lecture Notes in Mathematics, vol 1963. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85964-2_2
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DOI: https://doi.org/10.1007/978-3-540-85964-2_2
Publisher Name: Springer, Berlin, Heidelberg
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