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Stochastic Analysis and Applications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsaa20 Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions Ummugul Bulut a & Edward J. Allen a a Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, USA Version of record first published: 18 Jun 2012.
To cite this article: Ummugul Bulut & Edward J. Allen (2012): Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions, Stochastic Analysis and Applications, 30:4, 553-567 To link to this article: http://dx.doi.org/10.1080/07362994.2012.649634
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Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions
UMMUGUL BULUT AND EDWARD J. ALLEN Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas, USA
Stochastic partial differential equations for the one-dimensional telegraph equation and the two-dimensional linear transport equation are derived from basic principles. The telegraph equation and the linear transport equation are well-known correlated random walk (CRW) models, that is, transport models characterized by correlated successive-step orientations. In the present investigation, these deterministic CRW equations are generalized to stochastic CRW equations. To derive the stochastic CRW equations, the possible changes in direction and particle movement for a small time interval are carefully determined. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the position intervals decrease, stochastic partial differential equations are derived for the telegraph and transport equations. Comparisons between numerical solutions of the stochastic partial differential equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.
Keywords Stochastic differential equation; Stochastic partial differential equation; Telegraph equation; Transport equation.
Mathematics Subject Classification 82C70; 60H15; 82C41; 60H10; 65C30.
1. Introduction Stochastic differential equations (SDEs) are providing additional insight into
Downloaded by [Texas Technology University] at 07:52 23 April 2013 randomly varying dynamical problems in mathematical biology, finance, engineering, chemistry, physics, and medicine [2, 3, 6, 9, 10, 13, 14]. Stochastic generalizations of deterministic models for certain dynamical systems are proving to be useful. In the present investigation, stochastic telegraph equations are derived in one-dimension and stochastic linear transport equations are derived in two-dimensions. Usually organisms and particles do not move in purely random directions. Often the current direction is correlated with the direction of prior movement.
Received April 13, 2011; Accepted April 25, 2011. This work was partially supported by NSF grant DMS-0718302. Address correspondence to Ummugul Bulut, Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409-1042, USA; E-mail: [email protected]
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This type of random walk is called a correlated random walk (CRW) [11]. In the present investigation, two well-known correlated random walk models are studied, specifically, the telegraph equation in one-dimension and the linear transport equation in two dimensions. These equations are useful, for example, in studying animal or particle movement. In the present investigation, stochastic telegraph and linear transport equations are derived from basic principles. To derive an SPDE for a correlated random walk, all independent variables are made discrete. A discrete stochastic model is then developed by carefully studying the changes that occur in a small time interval. Next, letting the time interval go to zero yields an SDE system. Finally, the Wiener processes in the SDE system are appropriately replaced by Brownian sheets and the intervals in the remaining independent variables are allowed to go to zero. The resulting stochastic partial differential equation is an SPDE model for a correlated random walk. Before deriving stochastic generalizations of correlated random walk equations in one- and two-dimensions, it is useful to review some properties of Brownian sheets. Then, in the following two sections, a stochastic version of the telegraph equation is derived and a stochastic version of the two-dimensional linear transport equation is derived. Computational results are next described that illustrate that the stochastic correlated random walk derivations are reasonable. The results are summarized in the final section.
2. Some Properties of Brownian Sheets Before deriving the stochastic partial differential equations, it is useful to consider several properties of Brownian sheets [4, 8, 18]. A Brownian sheet on 0 5 × 0 5 is illustrated in Figure 1. The Brownian sheet W x t satisfies:
t+ t x+ x 2 W x t ∼ dx dt N 0 x t t x t x Downloaded by [Texas Technology University] at 07:52 23 April 2013
Figure 1. A Brownian Sheet on 0 5 × 0 5 . (Figure available in color online.) SPDEs for Correlated Random Walks 555
That is, the Brownian sheet is independent and normally distributed over = + = = rectangular regions. In addition, if xj xmin j x for j 0 1 K, where x − = xmax xmin /K, then the Brownian sheet defines for j 1 2 K, the standard Wiener processes, Wj t , where
√ x 2 = j W x t x dWj t dx dt xj−1 t x = = Notice that if ti i t for t 0 1 M, then
t √ i = dWj t t i j ti−1 ∼ = = where i j N 0 1 for each j 1 2 K and i 1 2 M. In addition, from a two-dimensional Brownian sheet, an independent one-dimensional Wiener process in t can be defined for each x by x+ x t ∗ = √1 W x W t x lim dx x→0 x x x ∗ ∼ ≥ = In particular, W t x N 0 t for each t 0 and if x1 x2, then the Wiener ∗ ∗ process W t x1 is independent of the Wiener process W t x2 . (Notice that W ∗ t x is not a Brownian sheet but is a one-dimensional Wiener process for each value of x.) These definitions can be extended to higher dimensions. For = example, standard Wiener processes Wi j t can be defined for j 1 2 J and i = 1 2 I using a three dimensional Brownian sheet by
x + x y + y 3 = 1 i j W x y t dWi j t dy dx dt x y xi yj t y x
where W x y t is a three-dimensional Brownian sheet.
3. Derivation of Stochastic Correlated Random Walk Transport Models In this section, SPDEs are derived for the telegraph equation in one dimension and for the linear transport equation in two dimensions. In particular, the dynamical systems, with time discrete, are studied to determine the different independent Downloaded by [Texas Technology University] at 07:52 23 April 2013 random changes. As the time interval decreases, the discrete stochastic models lead to certain stochastic differential equation systems. Then, Brownian sheets are appropriately substituted for Wiener processes in SDE systems. When intervals in the secondary variables go to zero, the final SPDE models are derived.
3.1. Telegraph Equation for One-Dimensional Correlated Random Walk In this section, an SPDE is derived for a one-dimensional correlated random walk, specifically, the telegraph equation. The deterministic telegraph equation is first introduced and described before deriving a stochastic generalization. The deterministic telegraph equation is derived by considering right and left moving particles. Let x t and x t be the number densities of right and left moving particles at position x and time t. In particular, and have units of 556 Bulut and Allen
number of particles per unit distance. The rate of change for the right and left moving particles can be shown to satisfy the equations: