Leonid V. Bogachev

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Random walks provide a simple conventional model to describe various transport processes, for example propagation of heat or diffusion of matter through a medium (for a general reference see, e.g., Hughes (1995)). However, in many practical cases the medium where the system evolves is highly irregular, due to factors such as defects, impurities,(More)
Consider the set Ln of convex polygons Γ with vertices on the integer lattice Z, non-negative inclination of the edges and fixed endpoints 0 = (0, 0) and n = (n1, n2). We study the asymptotic properties of the ensemble Ln, as n1, n2 →∞, with respect to a certain parametric class of probability distributions Pn = P (r) n (0 < r <∞) on the space Ln (in(More)
Received: date / Revised version: date Abstract. We study limiting distributions of exponential sums SN (t) = ∑N i=1 e tXi as t → ∞, N → ∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess supXi = 0 and (B) ess supXi = ∞. We assume that the function h(x) = − log P{Xi > x} (case B) or h(x) = − log P{Xi > −1/x} (case A) is regularly(More)
We consider the occupancy problem where balls are thrown independently at infinitely many boxes with fixed positive frequencies. It is well known that the random number of boxes occupied by the first n balls is asymptotically normal if its variance Vn tends to infinity. In this work, we mainly focus on the opposite case where Vn is bounded, and derive a(More)
We study the limiting distribution of the sum SN (t) = ∑N i=1 e tXi as t→∞, N →∞, where (Xi) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in the theory of random media. Examples include the quenched mean population size of branching random processes with random branching rates and the partition(More)
The distribution μcl of a Poisson cluster process in X = Rd (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X = ⊔ nX n, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure μcl is(More)
Abstract. The distribution μ of a Gibbs cluster point process in X = R (with n-point clusters) is studied via the projection of an auxiliary Gibbs measure defined on the space of configurations in X × X. We show that μ is quasi-invariant with respect to the group Diff0(X ) of compactly supported diffeomorphisms of X and prove an integration-by-parts formula(More)
The distribution μ of a Poisson cluster process in X = R (with n-point clusters) is studied via the projection of an auxiliary Poisson measure in the space of configurations in X, with the intensity measure being the convolution of the background intensity (of cluster centres) with the probability distribution of a generic cluster. We show that μ is(More)
We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form F(z) = ∞l=1F0(zl) (which entails equal weighting among possible parts l ∈ N). Under mild technical assumptions on the function H0(u) = ln(F0(u)), we show that the(More)