2013
DOI: 10.1088/1367-2630/15/6/063034
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Area and perimeter covered by anomalous diffusion processes

Abstract: We investigate the geometric properties of two-dimensional continuous time random walks that are used extensively to model stochastic processes exhibiting anomalous diffusion in a variety of different fields. Using the concept of subordination, we determine exact analytical expressions for the average perimeter and area of the convex hulls for this class of non-Markovian processes. As the convex hull is a simple measure to estimate the home range of animals, our results give analytical estimates for the home r… Show more

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Cited by 32 publications
(24 citation statements)
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“…In recent years, convex hull calculations have been used to study diverse topics such as the size of spreading GPS-enabled drifters moving on the surfaces of lakes and rivers [21,22], star-forming clusters [23], forest fires [24], proteins [25,26], or clusters of contaminant particles [27]. Studies of the relationships between random walks, anomalous diffusion, extreme statistics and convex hulls have been motivated by animal home ranges [28][29][30][31][32][33][34]. Convex hulls have also been used to study analytical statistics of Burgers turbulence by analogy with Brownian motion [35][36][37].…”
Section: Introductionmentioning
confidence: 99%
“…In recent years, convex hull calculations have been used to study diverse topics such as the size of spreading GPS-enabled drifters moving on the surfaces of lakes and rivers [21,22], star-forming clusters [23], forest fires [24], proteins [25,26], or clusters of contaminant particles [27]. Studies of the relationships between random walks, anomalous diffusion, extreme statistics and convex hulls have been motivated by animal home ranges [28][29][30][31][32][33][34]. Convex hulls have also been used to study analytical statistics of Burgers turbulence by analogy with Brownian motion [35][36][37].…”
Section: Introductionmentioning
confidence: 99%
“…This procedure was successfully applied recently to compute the mean perimeter and the mean area of several two dimensional stochastic processes such as N independent Brownian motions in 2-d [17,24], random acceleration process in 2-d [25], 2-d branching Brownian motions with absorption with applications to edpidemic outbreak [26] and 2-d anomalous diffusion processes [27]. Very recently, this method was also successfully used to compute the exact mean perimeter of the convex hull of a planar Brownian motion confined to a half-space [28].…”
Section: Introductionmentioning
confidence: 99%
“…It was shown [14,18] recently that the problem of com-puting the mean perimeter and the mean area of the convex hull of an arbitrary two-dimensional stochastic process can be mapped to computing the extremal statistics of the one-dimensional component of the process. This procedure was successfully applied recently to compute the mean perimeter and the mean area of several twodimensional stochastic processes such as the random acceleration process in 2D [19], 2D branching Brownian motions with absorption and applications to edpidemic outbreak [20] and 2D anomalous diffusion processes [21]. Very recently, this method was also successfully used to compute the exact mean perimeter of the convex hull of a planar Brownian motion confined to a half-space [22].…”
Section: Introductionmentioning
confidence: 99%