Diffusion in an annihilating environment

In this paper we study the following system of reaction-diffusion equations: ∂ /∂t = ∆ − V + λδ0, (0, x) ≡ 0, ∂V/∂t = − V, V (0, x) ≡ 1. Here (t, x) and V (t, x) are functions of time t ∈ [0,∞) and space x ∈ R. This system describes a continuum version of a model in which particles are injected at the origin at rate λ, perform independent simple symmetric random walks on Z, and are annihilated at rate 1 by traps located at the sites of Z in such a way that the trap disappears with the particle. This lattice model was studied by a number of authors, who obtained the asymptotic size and shape of the front separating the zone of particles from the zone of traps as well as the asymptotic particle density profile to leading order, in the limit of large time. The continuum model has similar behavior but allows for a more detailed study. As t increases, the particle density (t, · ) inflates and the trap density V (t, · ) deflates on a growing ball with radius R∗(t) centered at the origin. We derive the sharp asymptotics of the front position R∗(t), identify the shape of V (t, · ) near the surface of the ball, and obtain the limiting profile of (t, · ) inside the ball after appropriate scaling. We also identify the analogues of the total number and the age distribution of particles that are alive. It turns out that the cases d ≥ 3, d = 2, and d = 1 exhibit different behavior. Acknowledgment. J.G. and S.A.M. gratefully acknowledge the hospitality of EURANDOM, where a part of this research was carried out. Mathematics Subject Classification (2000): 60J70, 60J65 ∗ Partially supported by the DFG-Schwerpunkt 1033 2 J. Gärtner, F. den Hollander and S.A. Molchanov 1. Basic equations and elementary observations 1.1. Motivation We begin by describing the microscopic model that motivates our investigation. Initially, let each site of the lattice Z be occupied by a ‘trap’. Suppose that the origin acts as a source that produces ‘particles’ according to a Poisson stream with rate λ > 0. Each particle performs a simple symmetric random walk at rate 1, independently of the other particles. If a particle meets a trap, then at rate 1 both the particle and the trap are annihilated. Thus, particles interact with each other indirectly, via the annihilation of traps. In this model, it is of interest to locate the ‘front’ that separates the zone of particles from the zone of traps, to describe the evolution of the densities of particles and traps both near and away from this front, to derive a macroscopic scaling limit for the particle density, and to identify the total number and the age distribution of particles that are alive. Lawler, Bramson and Griffeath [13] proved that in d ≥ 3 the asymptotic shape of the trap free region at time t approaches a ball with radius R(t) ∼ (λt/ωd) as t → ∞, with ωd the volume of a unit ball in R. Thus, of the λt particles that are born up to time t only o(t) are alive at time t. This comes from the observation that R(t) is much smaller than the diffusive scale √ t. Gravner and Quastel [8] extended this result to d = 2, showing that R(t) ∼ κ∗ √ t as t → ∞, with κ∗ = κ∗(λ) being the unique solution of the equation e−κ2/4 = (π/λ)κ. Thus, a fraction 0 < 1−(π/λ)κ∗ < 1 of the particles born up to time t is alive at time t, and this leads to a hydrodynamic limit behavior of the particle density that is described by a certain Stefan problem. In d = 1 Gravner and Quastel [8] found that R(t) ∼ √2t log t as t → ∞, in which case all except o(t) of the particles born up to time t are alive at time t and the front is pushed outwards by a small group of particles performing a large deviation of order √ t log t √t. Further extensions of these results were obtained for the situation where the injection rate at the origin is time-dependent: λ = λ(t) (Ben Arous and Ramı́rez [4]; Quastel [14]; Ben Arous, Quastel and Ramı́rez [1]). It turns out that there are three regimes – subcritical, critical and supercritical – for which t−d/2 ∫ t 0 λ(s) ds → c as t → ∞ with c = 0, c ∈ (0,∞) and c = ∞, respectively, exhibiting different behavior. In the subcritical regime R(t) ∼ (N(t)/ωd) as t → ∞, with N(t) = ∫ t 0 λ(s) ds, in the critical regime there is diffusive scaling, while in the supercritical regime the growth is driven by large deviations. The results in the latter regime are still incomplete. A further interesting question is to find the tail behavior of the survival probability of particles born at a given time. This question was addressed by Ben Arous and Ramı́rez [2], [3] and again depends on large deviations. An interesting variant of the model is the one where time is discrete, particles and traps annihilate each other upon first contact, and the next particle is released from the origin only when the previous particle annihilates a trap. Diffusion in an annihilating environment 3 This variant, which is called “internal diffusion limited aggregation”, can be viewed as the λ ↓ 0 limit of the original model and was introduced by Diaconis and Fulton [7]. Lawler, Bramson and Griffeath [13] proved that in d ≥ 1 the asymptotic shape of the trap free region is a ball (with volume equal to the number of released particles). Lawler [12] obtained an upper bound on the fluctuations of the trap free region: in d ≥ 2 the difference between the radius of the ‘inner’ and the ‘outer’ ball sandwiching the trap front is at most of order n(log n) for a certain α(d) when their radii are of order n. Blachère [5], [6] has brought this upper bound down to order (logn) for a certain β(d). 1.2. Continuum model Instead of considering the particle vs. trap picture, we will study these problems in terms of a deterministic continuum model consisting of a coupled system of parabolic differential equations for the particle density = (t, x) and the trap density V = V (t, x) in all spatial dimensions d ≥ 1. More precisely, we are interested in the long-time asymptotics of the following Cauchy problem: ∂ ∂t = ∆ − V + λδ0, (0, x) ≡ 0, ∂V ∂t = − V, V (0, x) ≡ 1. (1.1) Here λ > 0 is the intensity of the δ-source at the origin. System (1.1) has a unique weak solution in the class of functions ( , V ) satisfying: (i) is continuous on [0,∞)×(Rd\{0}) and of class C on (0,∞)×(Rd\{0}); (ii) V is continuous on ([0,∞) × R) \ {(0, 0)} and of class C on (0,∞) × (R \ {0}); (iii) 0 ≤ ≤ 0 and 0 ≤ V ≤ 1. Here 0 is the free particle density, i.e. the solution of the majorizing heat equation with δ-source, ∂ 0 ∂t = ∆ 0 + λδ0, 0(0, x) ≡ 0, (1.2) given by 0(t, x) = λ ∫ t 0 (4πs)−d/2 exp {−|x|2/4s} ds. (1.3) Since 0 ≤ V ≤ 1, a comparison of (1.1) with (1.2) yields e−t 0(t, x) ≤ (t, x) ≤ 0(t, x) (1.4) for all (t, x). In particular, (t, x) → 0 as |x| → ∞. Note that, in dimensions d ≥ 2, the functions 0 and have a singularity at x = 0 and V is discontinuous 4 J. Gärtner, F. den Hollander and S.A. Molchanov at (t, x) = (0, 0), while in dimension d = 1 these functions are regular. The total number of particles satisfies ∫ (t, x) dx ≤ ∫ 0(t, x) dx = λt, t ≥ 0. The solution of (1.1) admits the (implicit) Feynman-Kac representation