Forward, backward, and weighted stochastic bridges

We define stochastic bridges as conditional distributions of stochastic paths that leave a specified point in phase-space in the past and arrive at another one in the future. These can be defined relative to either forward or backward stochastic differential equations and with the inclusion of arbitrary path-dependent weights. The underlying stochastic equations are not the same except in linear cases. Accordingly, we generalize the theory of stochastic bridges to include time-reversed and weighted stochastic processes. We show that the resulting stochastic bridges are identical, whether derived from a forward or a backward time stochastic process. A numerical algorithm is obtained to sample these distributions. This technique, which uses partial stochastic equations, is robust and easily implemented. Examples are given, and comparisons are made to previous work. In stochastic equations without a gradient drift, our results confirm an earlier conjecture, while generalizing this to cases with path-dependent weights. An example of a two-dimensional stochastic equation with no potential solution is analyzed and numerically solved. We show how this method can treat unexpectedly large excursions occurring during a tunneling or escape event, in which a system escapes from one quasistable point to arrive at another one at a later time.

Figure 1

Convergence of Brownian bridge variance (solid lines) to analytic result (dotted lines) with virtual time $\tau$, at fixed real time of $t=0$. Upper and lower lines indicate sampling errors of $\pm 1\sigma$.

Figure 2

Convergence of Brownian bridge variance (solid lines) to analytic result (dotted lines) with real time $t$, at a fixed virtual time $\tau =1$. Upper and lower lines indicate sampling errors of $\pm 1\sigma$.

Figure 3

Three-dimensional Brownian bridge variance surface evolving with real and virtual time.

Figure 4

Parametric plot of $x$ versus $y$ values for 10 random unconstrained stochastic trajectories for a linear stochastic differential equation with $\epsilon =0.01$, from time $t=0$ to $t=\pi$.

Figure 5

Parametric plot of final $x$ versus $y$ values for 10 random trajectories at $\tau =0.4$, for a linear stochastic bridge with $\epsilon =0.01$, from time $t=0$ to $t=\pi$.

Figure 6

Ten typical stochastic bridge trajectories, showing evolution of $x,y$ during the constrained transition from a stable to a metastable point for the Maier-Stein equations with $\alpha =1$, $\mu =1$, and $\epsilon =0.01$. Here, $\tau =1$ and $t=20$.

Figure 7

Parametric plots of $x$ versus $y$ for ten independent stochastic bridge trajectories, with $\alpha =-10$, $\mu =1$, and $\epsilon =0.01$. Initial condition is the stable point at $\mathit{x}=\left[1,0\right]$; total integration time is $\tau =2$ and $t=20$.

Figure 8

Ten typical stochastic bridge trajectories, showing evolution of $x,y$ during the constrained transition from a stable to a metastable point for the Maier-Stein equations with $\alpha =6.67$, $\mu =1$, and $\epsilon =0.01$. Here, $\tau =1$ and $t=20$.

Figure 9

Ten typical stochastic differential equation trajectories, showing evolution of $x,y$ for $t=20$, with the same parameters as in Fig. 8. Over this time scale, no escapes occur, and all trajectories are trapped near the stable region at $\mathit{x}=\left[1,0\right].$