Natural and Mathematical Sciences

Permanent URI for this collectionhttps://hdl.handle.net/10679/313

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Browsing by Institution Author "ÖZ, Mehmet"

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    Conditional speed of branching Brownian motion, skeleton decomposition and application to random obstacles
    (Institute of Mathematical Statistics, 2017) Öz, Mehmet; Çağlar, M.; Engländer, J.; Natural and Mathematical Sciences; ÖZ, Mehmet
    We study a branching Brownian motion ZZ in RdRd, among obstacles scattered according to a Poisson random measure with a radially decaying intensity. Obstacles are balls with constant radius and each one works as a trap for the whole motion when hit by a particle. Considering a general offspring distribution, we derive the decay rate of the annealed probability that none of the particles of ZZ hits a trap, asymptotically in time tt. This proves to be a rich problem motivating the proof of a more general result about the speed of branching Brownian motion conditioned on non-extinction. We provide an appropriate “skeleton” decomposition for the underlying Galton–Watson process when supercritical and show that the “doomed” particles do not contribute to the asymptotic decay rate.
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    Large deviations for local mass of branching Brownian motion
    (Instituto Nacional de Matematica Pura e Aplicada, 2020) Öz, Mehmet; Natural and Mathematical Sciences; ÖZ, Mehmet
    We study the local mass of a dyadic branching Brownian motion Z evolving in R-d. By 'local mass', we refer to the number of particles of Z that fall inside a ball with fixed radius and time-dependent center, lying in the region where there is typically exponential growth of particles. Using the strong law of large numbers for the local mass of branching Brownian motion and elementary geometric arguments, we find large deviation results giving the asymptotic behavior of the probability that the local mass is atypically small on an exponential scale. As corollaries, we obtain an asymptotic result for the probability of absence of Z in a ball with fixed radius and time-dependent center, and lower tail asymptotics for the local mass in a fixed ball. The proofs are based on a bootstrap argument, which we use to find the lower tail asymptotics for the mass outside a ball with time-dependent radius and fixed center, as well.
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    On the density of branching Brownian motion
    (Hacettepe University, 2023-02-15) Öz, Mehmet; Natural and Mathematical Sciences; ÖZ, Mehmet
    We consider a d-dimensional dyadic branching Brownian motion, and study the density of its support in the region where there is typically exponential growth of particles. Using geometric arguments and an extension of a previous result on the probability of absence of branching Brownian motion in linearly moving balls of fixed size, we obtain sharp asymptotic results on the covering radius of the support of branching Brownian motion, which is a measure of its density. As a corollary, we obtain large deviation estimates on the volume of the r(t)-enlargement of the support of branching Brownian motion when r(t) decays exponentially in time t. As a by-product, we obtain the lower tail asymptotics for the mass of branching Brownian motion falling in linearly moving balls of exponentially shrinking radius, which is of independent interest.
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    On the volume of the shrinking branching Brownian sausage
    (The Institute of Mathematical Statistics and the Bernoulli Society, 2020) Öz, Mehmet; Natural and Mathematical Sciences; ÖZ, Mehmet
    The branching Brownian sausage in R-d was defined in [4] similarly to the classical Wiener sausage, as the random subset of R-d scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a d-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated branching Brownian sausage (BBM-sausage) with radius exponentially shrinking in time. Using a previous result on the density of the support of BBM, and some well-known results on the classical Wiener sausage and Brownian hitting probabilities, we obtain almost sure limit theorems as time tends to infinity on the volume of the shrinking BBM-sausage in all dimensions.
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    Optimal survival strategy for branching Brownian motion in a Poissonian trap field
    (Institut Henri Poincaré, 2019-11) Öz, Mehmet; Engländer, J.; Natural and Mathematical Sciences; ÖZ, Mehmet
    We study a branching Brownian motion Z with a generic branching law, evolving in R-d, where a field of Poissonian traps is present. Each trap is a ball with constant radius. The traps are hard in the sense that the process is killed instantly once it enters the trap field. We focus on two cases of Poissonian fields, a uniform field and a radially decaying field, and consider an annealed environment. Using classical results on the convergence of the speed of branching Brownian motion, we establish precise annealed results on the population size of Z, given that it avoids the trap field, while staying alive up to time t. The results are stated so that each gives an 'optimal survival strategy' for Z. As corollaries of the results concerning the population size, we prove several other optimal survival strategies concerning the range of Z, and the size and position of clearings in R-d. We also prove a result about the hitting time of a single trap by a branching system (Lemma 1), which may be useful in a completely generic setting too.
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    Subdiffusivity of brownian motion among a poissonian field of moving traps
    (Instituto Nacional de Matematica Pura e Aplicada, 2019) Öz, Mehmet; Natural and Mathematical Sciences; ÖZ, Mehmet
    Our model consists of a Brownian particle X moving in N, where a Poissonian field of moving traps is present. Each trap is a ball with constant radius, centered at a trap point, and each trap point moves under a Brownian motion independently of others and of the motion of X. Here, we investigate the 'speed' of X on the time interval [0, t] and on 'microscopic' time scales given that X avoids the trap field up to time t. Firstly, following the earlier work of 'Athreya. et al. (2017), we obtain bounds on the maximal displacement of X from the origin. Our upper bound is an improvement of the corresponding bound therein. Then, we prove a result showing how the speed on microscopic time scales affect the overall macroscopic subdiffusivity on [0, t]. Finally, we show that X moves subdiffusively even on certain microscopic time scales, in the bulk of [0, t]. The results are stated so that each gives an 'optimal survival strategy' for the system. We conclude by giving several related open problems.
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    Survival of branching Brownian motion in a uniform trap field
    (Elsevier, 2016-03) Öz, Mehmet; Natural and Mathematical Sciences; ÖZ, Mehmet
    We study a branching Brownian motion Z evolving in Rd, where a uniform field of Poissonian traps are present. We consider a general offspring distribution for Z and find the asymptotic decay rate of the annealed survival probability, conditioned on non-extinction. The method of proof is to use a skeleton decomposition for the Galton–Watson process underlying Z and to show that the particles of finite line of descent do not contribute to the survival asymptotics. This work is a follow-up to Öz and Çağlar (2013) and solves the problem considered therein completely.