Skip to main content

Locality and time scales of the underlying non-degenerate stochastic system: Freidlin-Wentzell theory

  • Chapter
  • 1050 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1963)

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

$$dX_t^{\varepsilon,x} = b(X_t^{\varepsilon,x})\;dt + \sqrt{\varepsilon} \;\sigma \;(X_t^{\varepsilon,x})\; dW_t \;\;\;\; (t\geq 0), \; X_0^{\varepsilon,x} = x \in {\rm R}^d$$
((2.1))

in ℝd. Here, the coefficients are functions bC (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

$$G^{\varepsilon} f := \sum_{_{_{_{i=1}}}}^{^{d}} b_i {\partial f\over{\partial x_i}} + {\varepsilon \over 2} \sum_{_{_{_{i,j=1}}}}^{^{d}} a_{ij} {{\partial^2f} \over {\partial x_i \partial x_j}} \;\;\; (f \in C^2 ({\rm R}^d, {\rm R}))$$

where the positive semi-definite matrix a is defined by

$$a := \sigma \; \sigma^*$$

Keywords

  • Potential Function
  • Exit Time
  • Deterministic System
  • Large Deviation Principle
  • Exit Rate

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(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

Download citation