Computing Optimal Trajectories of Noise-Driven Systems - Arizona Math
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Computing Optimal Trajectories of Noise-Driven Systems Robert S. Maier Daniel L. Stein Physics and Mathematics Departments University of Arizona APS DCOMP 2001 Conference June 25, 2001
Partially supported by NSF grant PHY-9800979.
Optimal Trajectories Occur in...
The large deviation theory of drift-diffusion processes. (Mathematical probability community, since 1960s.) But large deviation theory predicts only the leading-order behavior of mean escape times and related quantities, as the noise strength tends to zero.
Ray expansions applied to Fokker–Planck or Smoluchowski equations. (Matkowsky, Schuss et al., before 1990; Dykman et al., since 1980s or before; RSM et al., since 1990.) A much more powerful approach: can in principle yield arbitrarily fine asymptotics. – Recent work: RSM & Stein, PRL 2000, 2001; see also –
RSM & Stein, SIAM J. Appl. Math. 1997. http://uranium.math.arizona.edu/ rsm
1
Noise-Perturbed Bistable ODEs
A Fokker–Planck equation for
, and an Itoˆ Langevin equation for : #%$ (.+ " ! #'&)(#*&,+ - / ,0
2 3 $54 1 /6879: ;0< ! 494A@ (>+ Here = - ? , and 79: ; is a vector of independent unit-strength white noises. The factor is the applied noise strength.
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Suppose the drift field has two attactors B , BDC and a hyperbolic limit set E F between them. E.g., B , B ’ are point attractors and E F is a saddle; or B , B C are stable limit cycles I and E F is an unstable limit cycle. As , what !HG happens to JKMLNLPORQ , the mean time to escape the basin of B , and reach the vicinity of BSC ? 2
Ray Expansions; Stochastic Phenomenology If the quasistationary PDF NT
NTU / is
expanded in a ray expansion
T /WV XZY T /1
Y 2 [ \8^]`_badc X. e f g]
! !
and this is substituted into the Fokker–Planck equation, then the balance equations yield ODEs for Y T Y 2 h
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