We consider particles obeying Langevin dynamics while being at known positions and having known velocities at the two endpoints of a given interval. Their motion in phase space can be modeled as an Ornstein–Uhlenbeck process conditioned at the two endpoints—a generalization of the Brownian bridge. Using standard ideas from stochastic optimal control we construct a stochastic differential equation (SDE) that generates such a bridge that agrees with the statistics of the conditioned process, as a degenerate diffusion. Higher order linear diffusions are also considered. In general, a timevarying drift is sufficient to modify the prior SDE and meet the endpoint conditions. When the drift is obtained by solving a suitable differential Lyapunov equation, the SDE models correctly the statistics of the bridge. These types of models are relevant in controlling and modeling distribution of particles and the interpolation of density functions.
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Brownian bridges for stochastic chemical processes—An approximation method based on the asymptotic behavior of the backward Fokker–Planck equation
A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science where one wants to control the endpoint of a stochastic process—e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift as it comes from a solution of a Backward Fokker–Planck (BFP) equation that is infeasible to compute for complex or highdimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, this paper uses the asymptotic properties of the BFP equation to generate an approximate drift and determine ways to correct (i.e., reweight) any errors incurred from this approximation. Because such a procedure avoids the solution of the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials.
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 Award ID(s):
 2126230
 NSFPAR ID:
 10366344
 Date Published:
 Journal Name:
 The Journal of Chemical Physics
 Volume:
 156
 Issue:
 18
 ISSN:
 00219606
 Page Range / eLocation ID:
 184108
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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