Abstract
In this chapter, we develop techniques for calculating the statistics of the time that it takes for a diffusion process in a bounded domain to reach the boundary of the domain. We then use this formalism to study the problem of Brownian motion in a bistable potential. Applications such as stochastic resonance and the modeling of Brownian motors are also presented. In Sect. 7.1, we motivate the techniques that we will develop in this chapter by looking at the problem of Brownian motion in bistable potentials. In Sect. 7.2, we obtain a boundary value problem for the mean exit time of a diffusion process from a domain. We then use this formalism in Sect. 7.3 to calculate the escape rate of a Brownian particle from a potential well. The phenomenon of stochastic resonance is investigated in Sect. 7.4. Brownian motors are studied in Sect. 7.5. Bibliographical remarks and exercises can be found in Sects. 7.6 and 7.7, respectively.
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Notes
- 1.
In other words, the integral \(\int _{-\infty }^{a}e^{-\beta V (y)}\,dy\) can be neglected.
- 2.
The boundary conditions are reflecting at \(x = -L\) and absorbing at x = 0, whence (7.16) is the correct formula to use.
- 3.
We assume, without loss of generality, that in the absence of the time-dependent forcing A 0 = 0, we have that X t is of mean zero, \(\int x\rho _{s}(x)\,dx = 0\). This assumption is satisfied for the symmetric bistable potential.
- 4.
Note that although in this figure, we have β −1 ∈ [0, 1], formula (7.45) is actually valid only for β ≫ 1.
- 5.
Equivalently, we can solve the stationary Fokker–Planck equation (6.165) as follows: We can consider the equation separately in the intervals [0, λ 1] and [λ 1, 1]. In these two intervals, the equation becomes a second-order differential equation with constant coefficients that we can solve. Using, then, the periodic boundary conditions, the normalization condition, and a continuity condition at x = λ 1, we can calculate the invariant distribution and then substitute it into (6.169) to calculate the effective drift.
- 6.
The stochastic process X t does not necessarily denote the position of a Brownian particle. We will, however, refer to X t as the particle position in the sequel.
- 7.
We use the notation \(X_{t}^{x,y,f}\) for the solution of (7.56) to emphasize its dependence on the initial conditions for X t , f(t), y(t). Also, in order to avoid problems with initial layers, we are assuming that ϕ depends explicitly only on X t ; the dependence on y(t), f(t) comes through (7.56).
- 8.
Compare this with the rescaling 6.162. In contrast to Sect. 6.6.2, here we will calculate the drift and diffusion coefficients in two steps.
- 9.
- 10.
In order to be more specific on the boundary conditions with respect to f and y, we need to specify their generator. For example, when y(t) is the Ornstein–Uhlenbeck process, the boundary conditions in y are that ρ should decay sufficiently fast as \(\vert y\vert \rightarrow +\infty\).
- 11.
More generally, the SNR is essentially the ratio between the strengths of the singular and continuous parts of the power spectrum:
$$\displaystyle{\mbox{ SNR} = 2\frac{\lim _{\varDelta \omega \rightarrow 0}\int _{\omega _{0}-\varDelta \omega }^{\omega _{0}+\varDelta \omega }S(\omega )\,d\omega } {S_{N}(\omega )}.}$$
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Pavliotis, G.A. (2014). Exit Problems for Diffusion Processes and Applications. In: Stochastic Processes and Applications. Texts in Applied Mathematics, vol 60. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1323-7_7
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