Figure 8 - uploaded by Kaitlin Hill

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# Most probable paths overlaid on the probability density generated by Monte-Carlo simulations of Equation (28) in a parameter regime in which (t max , 0) / ∈ B + . The solid black line corresponds to the most probable path numerically computed as a stationary curve of Equation (37) and the dashed magenta curve is the mean of the probability density. All other curves follow the same convention as in Figure 7. In (a) the parameters are r + = 2, r − = 3, T = 0.25, A + = 3, A − = 1, a = −0.5, σ = 0.22, and ε = σ 4 . In (b) the parameters are r + = 2, r − = 3, T = 0.25, A + = 4, A − = 3, a = −0.5, and σ = 0.07, and ε = σ 4 .

Source publication

We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider...

## Contexts in source publication

**Context 1**

... since we used a Gaussian kernel for the mollification, the functional forms of f ε and its various derivatives are in terms of Gaussian and error functions and thus Equation (37) can be numerically solved using a Forward Time Centered Space finite difference scheme. In Figure 8 we plot the numerically computed most probable path overlaid on the density of tipping events for two sets of parameters in which (t max , 0) / ∈ B + . In Figure 8(a) the n + nullcline intersects Σ and thus B − contains regions in which x > 0. ...

**Context 2**

... Figure 8 we plot the numerically computed most probable path overlaid on the density of tipping events for two sets of parameters in which (t max , 0) / ∈ B + . In Figure 8(a) the n + nullcline intersects Σ and thus B − contains regions in which x > 0. Moreover, (t max , 0) lies in a crossing region. ...

**Context 3**

... in this case, the most probable path intersects Σ slightly to the right of (t max , 0) / ∈ B + while the mean of the density appears to pass precisely through (t max , 0). In Figure 8(b) the parameters are selected so that both nullclines n + and n − intersect Σ and thus, in addition to B − having nontrivial intersection with x > 0, B + contains regions in which x < 0. Again, (t max , 0) lies in a crossing region but in this case the most probable path and the mean of the tipping events are both to the left of (t max , 0). However, in this case there is remarkable agreement between the most probable path and the mean of the tipping events. ...

**Context 4**

... since we used a Gaussian kernel for the mollification, the functional forms of f ε and its various derivatives are in terms of Gaussian and error functions and thus Equation (37) can be numerically solved using a Forward Time Centered Space finite difference scheme. In Figure 8 we plot the numerically computed most probable path overlaid on the density of tipping events for two sets of parameters in which (t max , 0) / ∈ B + . In Figure 8(a) the n + nullcline intersects Σ and thus B − contains regions in which x > 0. ...

**Context 5**

... Figure 8 we plot the numerically computed most probable path overlaid on the density of tipping events for two sets of parameters in which (t max , 0) / ∈ B + . In Figure 8(a) the n + nullcline intersects Σ and thus B − contains regions in which x > 0. Moreover, (t max , 0) lies in a crossing region. ...

**Context 6**

... in this case, the most probable path intersects Σ slightly to the right of (t max , 0) / ∈ B + while the mean of the density appears to pass precisely through (t max , 0). In Figure 8(b) the parameters are selected so that both nullclines n + and n − intersect Σ and thus, in addition to B − having nontrivial intersection with x > 0, B + contains regions in which x < 0. Again, (t max , 0) lies in a crossing region but in this case the most probable path and the mean of the tipping events are both to the left of (t max , 0). However, in this case there is remarkable agreement between the most probable path and the mean of the tipping events. ...