E. D. Andjel, M. V. Menshikov, V. V. Sisko
We show that certain Markov jump processes associated to crystal growth models are positive recurrent when the parameters satisfy a rather natural condition.
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We discuss complementary recurrence and transience criteria for stochastic processes $(X_n)_{n \ge 0}$ with values in the $d$-dimensional orthant $\mathbb R^d_+$ fulfilling a non-linear stochastic equation of the form $X_{n+1}=MX_n+g(X_n)+ ξ_n$ with a primitive matrix $M$ and random noise $ξ_n$ and obeying a weak Markov property. As examples we discuss bisexual Galton-Watson processes and multivariate Galton-Watson processes, which both may be population size dependent.
Let $\mathcal{K}\subset R^d$, $d\ge2$, be a smooth, bounded domain satisfying $0\in\mathcal{K}$, and let $f(t),\ t\ge0$, be a smooth, continuous, nondecreasing function satisfying $f(0)>1$. Define $D_t=f(t)\mathcal{K}\subset R^d$. Consider a diffusion process corresponding to the generator $\frac12Δ+b(x)\nabla$ in the time-dependent domain $D_t$ with normal reflection at the time-dependent boundary. Consider also the one-dimensional diffusion process corresponding to the generator $\frac12\frac{d^2}{dx^2}+B(x)\frac d{dx}$ on the time-dependent domain $(1,f(t))$ with reflection at the boundary.…
We provide conditions which yield a strong law of large numbers for expressions of the form $1/N\sum_{n=1}^{N}F\big(X(q_1(n)),..., X(q_\ell(n))\big)$ where $X(n),n\geq 0$'s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polinomial growth and certain regularity properties and $q_i,i>m$ are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers wi…
In this note we investigate the behaviour of Brownian motion conditioned on a growth constraint of its local time which has been previously investigated by Berestycki and Benjamini. For a class of non-decreasing positive functions $f(t); t>0$, we consider the Wiener measure under the condition that the Brownian local time is dominated by the function f up to time T. In the case where $f(t)/t^{3/2}$ is integrable we describe the limiting process as T goes to infinity. Moreover, we prove two conjectures in [BB10] in the case for a class of functions f, for which $f(t)/t^{3/2}$ just fails to be i…
Recurrence and capacity of stable branching random walks
本文针对后代分布为临界且吸引α-稳定律的分支随机游走，研究其范围线性增长率，扩展分支容量至α-稳定情形并估计球的对应容量。
Volume Growth and Recurrence of Fractional Powers of the Laplace--Beltrami Operator
该研究针对黎曼流形上由α/2稳定次支配子从属布朗运动得到的过程X^(α)，证明了积分条件∫^∞ dt/V(o,t^(1/α))=∞可推出X^(α)回归，解决了Grigor'yan问题26的充分性部分。
We give a criterion for unlimited growth with positive probability for a large class of multidimensional stochastic models. As a by-product, we recover the necessary and sufficient conditions for recurrence and transience for critical multitype Galton-Watson with immigration processes. We also significantly improve some results on multitype size-dependent Galton-Watson processes.
It has been suggested in 1999 that a certain volume growth condition for geodesically complete Riemannian manifolds might imply that the manifold is stochastically complete. This is motivated by a large class of examples and by a known analogous criterion for recurrence of Brownian motion. We show that the suggested implication is not true in general. We also give counter-examples to a converse implication.
We introduce a three-dimensional, computationally feasible, mesoscopic model for snow crystal growth, based on diffusion of vapor, anisotropic attachment, and a semi-liquid boundary layer. Several case studies are presented that faithfully emulate a wide variety of physical snowflakes.
We study crystal growth inside an infinite octant on a cubic lattice. The growth proceeds through the deposition of elementary cubes into inner corners. After re-scaling by the characteristic size, the interface becomes progressively more deterministic in the long-time limit. Utilizing known results for the crystal growth inside a two-dimensional corner, we propose a hyperbolic partial differential equation for the evolution of the limiting shape. This equation is interpreted as a Hamilton-Jacobi equation which helps in finding an analytical solution. Simulations of the growth process are in e…
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