We study families \Phi of coverings which are faithful for the Hausdorff dimension calculation on a given set E (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of E) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones.Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families \scrA of net-coverings which are "extremely non-comparable" to the Hausdorff measure.Ми дослiджуємо сiм'ї \Phi покриттiв, якi є довiрчими для обчислення розмiрностi Хаусдорфа-Безиковича на певнiй множинi E (тобто, спецiальнi вiдносно вузькi сiм'ї покриттiв, яких достатньо для коректного обчислення класичної розмiрностi Хаусдорфа-Безиковича довiльної пiдмножини множини E) i якi є природним узагальненням порiвнянних мережевих покриттiв. В роботi показано, що такi сiм'ї є дуже корисними для обчислення чи оцiнки розмiрностi Хаусдорфа-Безиковича множин та ймовiрнiсних мiр.Нами отримано загальнi необхiднi та достатнi умови довiрчостi для сiмей покриттiв та запропоновано нову технiку доведення довiрчостi/недовiрчостi для сiмей цилiндрiв, породжених рiзними розкладами дiйсних чисел. Маючи додаткову мотивацiю в мультифрактальному аналiзi нескiнченних згорток Бернуллi, ми детально дослiдили розклади Кантора та довели необхiднi та достатнi умови довiрчостi вiдповiдних сiмей покриттiв мережевими цилiндрами. Наскiльки нам вiдомо, цi результати є першими критерiями довiрчостi для класу сiмей покриттiв, що мiстить як довiрчi, так i недовiрчi сiм'ї.Застосовуючи отриманi результати, ми дослiдили тонкi фрактальнi властивостi ймовiрнiсних мiр з незалежними символами розкладiв Кантора i показали, що клас довiрчих мережевих покриттiв суттєво ширше за клас порiвнянних. Ми побудували, зокрема, досить простi приклади довiрчих сiмей \scrA мережевих покриттiв, якi є "екстремально непорiвнянними" вiдносно мiри Хаусдорфа.
We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similarly to the classical lattice growth models the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free branching birth rate.Mathematics subject classification: 60K35, 60J80. IntroductionShape theorems have a long history. Richardson [Ric73] proved the shape theorem for the Eden model. Since then, shape theorems have been proven in various settings, most notably for first passage percolation and permanent and non-permanent growth models. Garet and Marchand [GM12] not only prove a shape theorem for the contact process in random environment, but also have a nice overview of existing results. * Most of literature is devoted to discrete-space models. A continuous-space first passage percolation model was analyzed by Howard and Newman [HN97], see also references therein.A shape theorem for a continuous-space growth model was proven by Deijfen [Dei03], see also Gouéré and Marchand [GM08]. Our model is naturally connected to that model, see the end of Section 2.Questions addressed in this article are motivated not only by probability theory but also by studies in natural sciences. In particular, one can mention a demand to incorporate spatial information in the description and analysis of 1) ecology 2) bacteria populations 3) tumor growth 4) epidemiology 5) phylogenetics among others, see e.g. [WBP + ], [TSH + 13], [VDPP15], and [TM15]. Authors often emphasize that it is preferable to use the continuous-space spaces R 2 and R 3 as the basic, or 'geographic' space, see e.g. [VDPP15]. More on connections between theoretical studies and applications can be found in [MW03].The paper is organized as follows. In Section 2 we describe the model and formulate our results, which are proven in Sections 3 and 4. Section 5 is devoted to computer simulations and conjectures. Technical results, in particular on the construction of the process, are collected in the Section 6. The model, assumptions and resultsWe consider a growth model represented by a continuous-time continuous-space Markov birth process. Let Γ 0 be the collection of finite subsets of R d ,where |η| is the number of elements in η. Γ 0 is also called the configuration space, or the space of finite configurations.The evolution of the spatial birth process on R d admits the following description. Let B(X) be the Borel σ-algebra on the Polish space X. If the system is in state η ∈ Γ 0 at time t, then the probability that a new particle appears (a "birth") in a bounded set B ∈ B(R d ) over timeand with probability 1 no two births happen simultaneously. Here b : R d × Γ 0 → R + is some function which is called the birth rate. Using a slightly different terminology, we can say that the rate at which a birth occurs in B is B b(x, η)dx. We note that it is conventional to call the function b the 'birth rate', even though it is not a rate in the usual se...
We study families Φ of coverings which are faithful for the Hausdorff dimension calculation on a given set E (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of E) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones.Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families A of netcoverings which are "extremely non-comparable" to the Hausdorff measure.
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