THE DIFFUSION LIMIT OF TRANSPORT EQUATIONS DERIVED FROM VELOCITY-JUMP PROCESSES ∗ THOMAS HILLEN † AND HANS G. OTHMER ‡ SIAM J. APPL. MATH. c  2000 Society for Industrial and Applied Mathematics Vol. 61, No. 3, pp. 751–775 Abstract. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approxi- mations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak–Keller–Segel–Alt model for chemotaxis. Key words. aggregation, chemotaxis equations, diffusion approximation, velocity-jump pro- cesses, transport equations AMS subject classifications. 35Q80, 60J15, 65U05, 92B05, 35B05, 35K50, 35K55 PII. S0036139999358167 1. Introduction. There are two major approaches used to describe the mo- tion of biological organisms: (i) a space-jump process in which the individual jumps between sites on a lattice, and (ii) a velocity-jump process in which discontinuous changes in the speed or direction of an individual are generated by a Poisson process [36]. The former leads to a renewal equation in which the kernel governs the waiting time between jumps and the redistribution after a jump and determines the type of partial differential equation that describes the asymptotic behavior of the evolution [36]. In this paper we analyze the diffusion approximation to the transport equation ∂ ∂t p(x,v,t)+ v ·∇p(x,v,t)= -λp(x,v,t)+ λ  V T (v,v ′ )p(x,v ′ ,t)dv ′ (1.1) describing a velocity-jump process. Here p(x,v,t) denotes the density of particles at spatial position x ∈ Ω ⊂ R n , moving with velocity v ∈ V ⊂ R n at time t ≥ 0 [36]. Here λ is the (constant) turning rate and 1/λ is a measure of the mean run length between velocity jumps. In general λ may be space dependent and depend on internal and external variables as well. The turning kernel T (v,v ′ ) gives the probability of a velocity jump from v ′ to v if a jump occurs, and implicit in the above formulation is the assumption that the choice of a new velocity is independent of the run length. The turning kernel may also be space dependent. When applied to the bacterium E. coli, the kernel T includes a bias, as described later, and the turning frequency must * Received by the editors June 23, 1999; accepted for publication (in revised form) March 29, 2000; published electronically August 29, 2000. http://www.siam.org/journals/siap/61-3/35816.html † Department of Mathematics, University of Utah, Salt Lake City, UT 84112 (hillen@math.utah. edu). This research of this author was supported by the Deutsche Forschungsgemeinschaft. ‡ Department of Mathematics, University of Minnesota, Minneapolis, MN 55455 (ohmer@math. umn.edu). The research of this author was supported in part by NIH grant GM 29123 and by NSF grant DMS 9805494. 751