Spatial branching in random environments and with interaction
The book is concerned with several stochastic models involving spatial motion, branching, random environments and interaction between particles, in domains of \(\mathbb{R}^d\). The following is a description of the book chapters.NEWLINENEWLINEIn Chapter 1, the author calls several notions related to the subject of the book. This greatly facilitates reading of the subsequent chapters. In particular, some attention is paid to stochastic differential equations; martingale change of measure; Brownian motion and more general diffusions; branching processes, branching diffusions and superprocesses. In Chapter 2, the author investigates a dyadic branching diffusion \(Z\) which corresponds to the operator \(Lu+\beta(u^2-u)\) on a non-empty domain \(D\) in \(\mathbb{R}^d\), where \(L\) is a `smooth' second order elliptic differential operator on \(D\) and \(\beta\) is a non-zero Hölder continuous function. Let \(\lambda_c\) denote the generalized principal eigenvalue for the operator \(L+\beta\) on \(D\), i.e., \(\lambda_c\) is the infimum over all real \(\lambda\) for which there exists a positive function \(u\) such that \((L+\beta-\lambda)u=0\) on \(D\). The main result of the chapter, that the author calls the strong law of large numbers for the local mass of branching diffusions, asserts that when \(\lambda_c\in (0,\infty)\) and some additional assumptions hold, the random measures \(e^{-\lambda_c t}Z_t\) converge almost surely in the vague topology as \(t\to\infty\). The principal technical tool here is the `spine change of measure'. This is a generalization of the well-known and commonly used technique in the field of branching processes. Further, the a.s.\ and \(L_p\) convergence of certain martingales play an important role. The martingales can be seen as counterparts of Biggins' martingales which are closely related to branching random walks. In Chapter 3, the author gives examples of branching diffusions which satisfy the strong law of large numbers discussed in the previous chapter. The list of examples includes branching diffusions corresponding to the operators defined on bounded domains \(D\), multidimensional Ornstein-Uhlenbeck processes with quadratic and constant branching rates, and branching Brownian motion. In Chapter 4, the author introduces a branching Brownian motion \((Z_t)_{t\geq 0}\) with `attraction' and `repulsion' between the particles. Considering \(Z_n\) as empirical measure obtained by putting unit point mass at the site of each particle, the main result of the chapter proves that, in the attractive case, \(Z_n\), properly normalized, has an a.s.\ limit as \(n\to\infty\). In the repulsive case, a conjecture is stated concerning the expected asymptotics of \(Z_n\). Some heuristics is also given which supports the conjecture. Another interesting result of the chapter shows that the center of mass of a supercritical super-Brownian motion \(X\) stabilizes on the survival set of \(X\). In Chapter 5, the process in focus is a branching Brownian motion on \(\mathbb{R}^d\) in a Poissonian field of traps. The author finds the asymptotics, as \(t\to\infty\), of the annealed probability of the survival up to time \(t\) of the corresponding particle system. This is closely related to a variational problem which is further investigated in depth. Chapter 6 is devoted to the process similar to that investigated in Chapter 5. However, while in the previous chapter hitting the Poisson trap system led to killing of particles, here the author speaks of mild obstacles, meaning that hitting the Poisson traps only affects the branching rate. The main result of the chapter is a quenched law of large numbers. Further, it is shown that the branching Brownian motion with mild obstacles spreads in space more slowly than the standard branching Brownian motion. Some results of the chapter are partially generalized to the case when the branching Brownian motion is replaced by a branching diffusion or a superprocess. In Chapter 7, the author considers a rather special critical branching random walk in a random environment. Based on computer simulations, a conjecture is set forth concerning the rate of decay for the survival probability of the corresponding particle system.NEWLINENEWLINEAs the author states in the introduction, the book is intended for graduate students in mathematics or statistics, researchers in probability and population biologists with some background in mathematics.NEWLINENEWLINEThe book is well-written. I enjoyed reading it thanks both to the contents and the attractive style of presentation. I got a feeling that the author has invested a lot of efforts to present highly nontrivial results in a clear and understandable way. Many assertions are followed by informal discussions intended to lead the reader into the core of problems. At the end of each chapter, relevant literature is analyzed, and some lines of further research are outlined.
scientific article; zbMATH DE number 6369222