Collision Local Time Research Articles

ON THE COLLISION LOCAL TIME OF BIFRACTIONAL BROWNIAN MOTIONS

Let [Formula: see text], i = 1,2 be two independent bifractional Brownian motions of dimension 1, with indices Hi ∈ (0, 1) and Ki ∈ (0, 1]. We investigate the collision local time of bifractional Brownian motions [Formula: see text] where δ denotes the Dirac delta function at zero. We show that ℓT exists in L2, and it is Hölder continuous of order 1 - min {H1K1,H2K2}, and furthermore, it is also smooth in the sense of Meyer–Watanabe.

Quenched large deviation principle for words in a letter sequence

When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. In a companion paper the annealed and the quenched LDP are applied to the collision local time of transient random walks, and the existence of an intermediate phase for a class of interacting stochastic systems is established.

Phase transitions for the long-time behavior of interacting diffusions

Let ({Xi(t)}i∈ℤd)t≥0 be the system of interacting diffusions on [0, ∞) defined by the following collection of coupled stochastic differential equations: $$dX_{i}(t)=\sum\limits_{j\in \mathbb{Z}^{d}}a(i,j)[X_{j}(t)-X_{i}(t)]\,dt+\sqrt{bX_{i}(t)^{2}} dW_{i}(t),\quad i\in \mathbb{Z}^{d}, t\geq 0.$$ Here, a(⋅, ⋅) is an irreducible random walk transition kernel on ℤd×ℤd, b∈(0, ∞) is a diffusion parameter, and ({Wi(t)}i∈ℤd)t≥0 is a collection of independent standard Brownian motions on ℝ. The initial condition is chosen such that {Xi(0)}i∈ℤd is a shift-invariant and shift-ergodic random field on [0, ∞) with mean Θ∈(0, ∞) (the evolution preserves the mean). We show that the long-time behavior of this system is the result of a delicate interplay between a(⋅, ⋅) and b, in contrast to systems where the diffusion function is subquadratic. In particular, let a(i, j)=½[a(i, j)+a(j, i)], i, j∈ℤd, denote the symmetrized transition kernel. We show that: (A) If a(⋅, ⋅) is recurrent, then for any b>0 the system locally dies out. (B) If a(⋅, ⋅) is transient, then there exist b*≥b2>0 such that: (B1) The system converges to an equilibrium νΘ (with mean Θ) if 0 b*. (B3) νΘ has a finite 2nd moment if and only if 0 b2. The equilibrium νΘ is shown to be associated and mixing for all 0 b2. We further conjecture that the system locally dies out at b=b*. For the case where a(⋅, ⋅) is symmetric and transient we further show that: (C) There exists a sequence b2≥b3≥b4≥⋯>0 such that: (C1) νΘ has a finite mth moment if and only if 0 bm. (C3) b2≤(m−1)bm<2. (C4) lim m→∞(m−1)bm=c=sup m≥2(m−1)bm. The proof of these results is based on self-duality and on a representation formula through which the moments of the components are related to exponential moments of the collision local time of random walks. Via large deviation theory, the latter lead to variational expressions for b* and the bm’s, from which sharp bounds are deduced. The critical value b* arises from a stochastic representation of the Palm distribution of the system. The special case where a(⋅, ⋅) is simple random walk is commonly referred to as the parabolic Anderson model with Brownian noise. This case was studied in the memoir by Carmona and Molchanov [Parabolic Anderson Problem and Intermittency (1994) Amer. Math. Soc., Providence, RI], where part of our results were already established.

On the Collision Local Time of Fractional Brownian Motions*

In this paper, the existence and smoothness of the collision local time are proved for two independent fractional Brownian motions, through L2 convergence and Chaos expansion. Furthermore, the regularity of the collision local time process is studied.

Competing super-Brownian motions as limits of interacting particle systems

We study two-type branching random walks in which the birth or death rate of each type can depend on the number of neighbors of the opposite type. This competing species model contains variants of Durrett's predator-prey model and Durrett and Levin's colicin model as special cases. We verify in some cases convergence of scaling limits of these models to a pair of super-Brownian motions interacting through their collision local times, constructed by Evans and Perkins.

A construction of catalytic super-Brownian motion via collision local time

We give a direct construction of a random measure which is equal in law to the collision local time between a catalytic super-Brownian motion and its catalytic measure. Under a regularity assumption on the catalytic measure, we show that the catalytic super-Brownian motion can be constructed deterministically from this measure.

Competing Species Superprocesses with Infinite Variance

We study pairs of interacting measure-valued branching processes (superprocesses) with alpha-stable migration and $(1+\beta)$-branching mechanism. The interaction is realized via some killing procedure. The collision local time for such processes is constructed as a limit of approximating collision local times. For certain dimensions this convergence holds uniformly over all pairs of such interacting superprocesses. We use this uniformity to prove existence of a solution to a competing species martingale problem under a natural dimension restriction. The competing species model describes the evolution of two populations where individuals of different types may kill each other if they collide. In the case of Brownian migration and finite variance branching, the model was introduced by Evans and Perkins (1994). The fact that now the branching mechanism does not have finite variance requires the development of new methods for handling the collision local time which we believe are of some independent interest.

Mutually catalytic branching in the plane: Finite measure states

We study a pair of populations in $\mathbb{R}^{2}$ which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a diffusion rate sufficiently large compared with the branching rate, the model is constructed as the unique pair of finite measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit, global extinction of one type is shown. The process constructed is a rescaled limit of the corresponding $\mathbb{Z}^{2}$-lattice model studied by D. A. Dawson and E. A. Perkins and resolves the large scale mass-time-space behavior of that model.

Mutually Catalytic Branching in The Plane: Infinite Measure States

A two-type infinite-measure-valued population in $R^2$ is constructed which undergoes diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. For a collision rate sufficiently small compared with the diffusion rate, the model is constructed as a pair of infinite-measure-valued processes which satisfy a martingale problem involving the collision local time of the solutions. The processes are shown to have densities at fixed times which live on disjoint sets and explode as they approach the interface of the two populations. In the long-term limit (in law), local extinction of one type is shown. Moreover the surviving population is uniform with random intensity. The process constructed is a rescaled limit of the corresponding $Z^2$-lattice model studied by Dawson and Perkins (1998) and resolves the large scale mass-time-space behavior of that model under critical scaling. This part of a trilogy extends results from the finite-measure-valued case, whereas uniqueness questions are again deferred to the third part.

On the hot spots of a catalytic super-Brownian motion

Consider the catalytic super-Brownian motion Xϱ (reactant) in ℝd, d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L[ϱ,Xϱ] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for Xϱ and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of Xϱ. In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds Ks(dx). At fixed time s, the collision measures Ks(dx) of ϱs and Xsϱ have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts.

Finite time extinction of superprocesses with catalysts

Consider a catalytic super-Brownian motion $X =X^\Gamma$ with finite variance branching. Here “catalytic ” means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $\Gamma$ on $\mathsf{R}$ of index $0 <\gamma<1$. Consequently, here the catalyst is located in a countable dense subset of $\mathsf{R}$. Starting with a finite reactant mass $X_0$ supported by a compact set, $X$ is shown to die in finite time.We also deal with two other cases, with a power low catalyst and with a super-random walk on $\mathsf{Z^d}$ withan i.i.d.catalyst. Our probabilistic argument uses the idea of good and bad historical paths of reactant “particles ”during time periods $[T_n, T_{n +1}$. Good paths have a signi .cant collision local time with the catalyst, and extinction can be shown by individual time change according to the collision local time and a comparison with Feller’s branching diffusion. On the other hand, the remaining bad paths are shown to have a small expected mass at time $T_{n +1}$ which can be controlled by the hitting probability of point catalysts and the collision local time spent on them.

Collision Measure and Collision Local Time for (α, d , β) Superprocesses

Existence and uniqueness of the solutions for some nonlinear evolution equations with measure-valued boundary conditions is established. This gives the existence of the collision local time and the collision measure for two independent (α1, d, β1) and (α2, d, β2) superprocesses without using any moment conditions on the mass processes. We obtain expressions for the Laplace transforms of the collision local time and the collision measure.

Collision Local Times, Historical Stochastic Calculus, and Competing Species

Branching measure-valued diffusion models are investigated that can be regarded as pairs of historical Brownian motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed-point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure-valued diffusions. This existence and uniqueness is used to establish well-posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.

Longtime behavior of a branching process controlled by branching catalysts

The model under consideration is a catalytic branching model constructed in Dawson and Fleischmann (1997), where the catalysts themselves undergo a spatial branching mechanism. The key result is a convergence theorem in dimension d = 3 towards a limit with full intensity (persistence), which, in a sense, is comparable with the situation for the “classical” continuous super-Brownian motion. As by-products, strong laws of large numbers are derived for the Brownian collision local time controlling the branching of reactants, and for the catalytic occupation time process. Also, the catalytic occupation measures are shown to be absolutely continuous with respect to Lebesgue measure. © 1997 Elsevier Science B.V.

A Continuous Super-Brownian Motion in a Super-Brownian Medium

A continuous super-Brownian motion $$X^Q $$ is constructed in which branching occurs only in the presence of catalysts which evolve themselves as a continuous super-Brownian motion $$Q$$ . More precisely, the collision local time $$L_{[W,Q]}$$ (in the sense of Barlow et al. (1)) of an underlying Brownian motion path W with the catalytic mass process $$Q$$ goerns the branching (in the sense of Dynkin's additive functional approach). In the one-dimensional case, a new type of limit behavior is encountered: The total mass process converges to a limit without loss of expectation mass (persistence) and with a nonzero limiting variance, whereas starting with a Lebesgue measure $$\ell$$ , stochastic convergence to $$\ell$$ occurs.

Collision Local Times and Measure-Valued Processes

AbstractWe consider two independent Dawson-Watanabe super-Brownian motions,Y1andY2. These processes are diffusions taking values in the space of finite measures on ℝd. We show that ifd ≤ 5then with positive probability there exist timestsuch that the closed supports ofintersect; whereas ifd&gt; 5 then no such intersections occur. For the cased≤ 5, we construct a continuous, non-decreasing measure–valued processL(Y1,Y2), thecollision local time,such that the measure defined by, is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition ofL(Y1,Y2). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions withpoint interactions.In the course of our proofs we obtain smoothness results for the random measuresthat are uniform int.These theorems use a nonstandard description ofYiand are of independent interest.