Hybrid colored noise process with space-dependent switching rates

Paul C. Bressloff and Sean D. Lawley

Phys. Rev. E 96, 012129 – Published 13 July 2017

Abstract

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the Itô-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an Itô SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the Itô form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit $(κ \to 0)$ and fast switching limit $(ε \to 0)$ depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain Itô, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit $ε, κ \to 0$ with $ε / κ^{2}$ fixed.

Received 23 May 2017

DOI:https://doi.org/10.1103/PhysRevE.96.012129

Physics Subject Headings (PhySH)

Brownian motion Diffusion Intracellular transport Stochastic processes

Statistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Paul C. Bressloff and Sean D. Lawley

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA

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Vol. 96, Iss. 1 — July 2017

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Images

Figure 1
Convergence of hybrid colored noise process to various SDEs depending on ratio of correlation time to mean switching time for spatially constant switching rates. The red, blue, and black curves are the respective distributions of (2.6), (2.12), and (4.9), which are Gaussians with zero mean and variance $2 t \bar{D}$ , $2 t \hat{D}$ , and $2 t [ν_{η} \hat{D} + (1 - ν_{η}) \bar{D}]$ . The squares, circles, and plusses are the empirical distribution of the hybrid colored noise process (2.4) with mean switching time $ε = 10^{- 4}$ and respective correlation times $κ = ε^{2} (η = 10^{- 12})$ , $κ = ε^{1 / 4} (η = 10^{2})$ , or $κ = ε^{1 / 2} (η = 1)$ for $10^{6}$ trials. Simulations of the hybrid colored noise process were performed according to the statistically exact algorithm described in Sec. 5. The distributions are at time $t = 1$ with $D_{0} = 10^{- 2}$ , $D_{1} = 1$ , $α = 1 - β = 0.9091$ , and thus $\bar{D} = . 1$ , $\hat{D} = 0.3306 \bar{D}$ , and $ν = 1 / 2$ .
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Figure 2
Effective diffusivity of hybrid colored noise process as a function of ratio of correlation time to mean switching time, $η = κ^{2} / ε$ , for spatially constant switching rates. The red dashed line is $\bar{D}$ in Eq. (2.7), the blue dotted line is $\hat{D}$ in Eq. (2.11), and the black solid line is the effective diffusivity, $ν_{η} \hat{D} + (1 - ν_{η}) \bar{D}$ , with $ν_{η}$ in Eq. (4.8). The black diamonds are one half of the empirical variance of $10^{6}$ simulations of the hybrid colored noise process until time $t = 1$ . The parameters, $ε, α, β, D_{0}, D_{1}$ , are given in the caption of Fig. 1.
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Figure 3
Convergence of hybrid colored noise process to various SDEs depending on the ratio of correlation time to mean switching time for space-dependent switching rates, $α (x) = β (- x) = tanh (10 x) + 1$ . The red, blue, and black curves are the respective empirical distributions of (2.6), (2.12), and (4.9), for $10^{6}$ trials. The squares, circles, and plusses are the empirical distribution of the hybrid colored noise process (2.4) with mean switching time $ε = 10^{- 3}$ and respective correlation times $κ = ε^{2} (η = 10^{- 9})$ , $κ = ε^{1 / 3} (η = 10)$ , or $κ = ε^{1 / 2} (η = 1)$ for $10^{6}$ trials. Simulations of (2.12), (2.6), and (4.9) were performed using the Euler-Maruyama method with a discrete time step of size $10^{- 3}$ . Simulations of the hybrid colored noise process were performed according to the statistically exact algorithm described in Sec. 5. The distributions are at time $t = 1$ with $D_{0} = 10^{- 1}$ and $D_{1} = 1$ .
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Figure 4
Top: Space-dependent switching rates, $α (x) = β (- x) = tanh (x) + 1$ . Bottom: Effective diffusivity $\bar{D} (x)$ in Eq. (2.7) and effective diffusivity $\hat{D} (x)$ in Eq. (2.11) for $D_{0} = 10^{- 1}$ , $D_{1} = 1$ , and $α (x) = β (- x) = tanh (x) + 1$ .
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