ISSN:
1572-9036
Keywords:
Primary 60H10
;
60J70
;
secondary 60F10
;
35P15
;
Diffusion process
;
large deviation
;
large time behavior
;
Fokker-Planck operator
;
simulated annealing
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We study the large-time behavior and rate of convergence to the invariant measures of the processes dX ε(t)=b(X) ε(t)) dt + εσ(X ε(t)) dB(t). A crucial constant Λ appears naturally in our study. Heuristically, when the time is of the order exp(Λ − α)/ε2 , the transition density has a good lower bound and when the process has run for about exp(Λ − α)/ε2, it is very close to the invariant measure. LetL ε=(ε2/2)Λ − ∇U · ∇ be a second-order differential operator on ℝd. Under suitable conditions,L z has the discrete spectrum $$\begin{gathered} 0 = \lambda _1^\varepsilon 〉 - \lambda _2^\varepsilon ...and lim \varepsilon ^2 log \lambda _2^\varepsilon = - \Lambda \hfill \\ \varepsilon \to 0 \hfill \\ \end{gathered} $$ LetU be a function from ℝd to [0,∞) with suitable conditions. A nonhomogeneous Markov processY(·) governed by $$dY(t) = - \nabla U(Y(t))dt + \sqrt {c/log(2 + t)} dB(t)$$ withY(0)=x is used to search for a global minimum ofU. LetS={x|U(x)=min y U(y)}. There exists a critical constantd * such that forc 〉 d *,Y(t) uniformly converges toS in probability overx in a compact set. The above statement fails forc 〈d *. Forc 〉 Λ,Y(t) converges weakly to a probability measure which does not depend on the starting pointx and concentrates onS. The techniques can also be used to study discrete simulated annealing.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01321859
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