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Register a new userLocalization for Linear Stochastic Evolutions
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Authors
Nobuo Yoshida
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We consider a discrete-time stochastic growth model on the $d$-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of the binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of replica overlap. The main novelty of this paper is that we obtain this equivalence even for models with positive probability of extinction at finite time. In the course of the proof, we characterize, in a general setting, the event on which an exponential martingale vanishes in the limit.
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We consider a simple discrete-time Markov chain with values in $[0,infty)^{Z^d}$. The Markov chain describes various interesting examples such as oriented percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the total number of particles in this framework. The main results are roughly as follows: If $d ge 3$ and the Markov chain is not too random, then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, $d=1,2$, or the Markov chain is random enough, then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the Markov chain with proper normalization.
We consider a class of continuous-time stochastic growth models on $d$-dimensional lattice with non-negative real numbers as possible values per site. The class contains examples such as binary contact path process and potlatch process. We show the equivalence between the slow population growth and localization property that the time integral of the replica overlap diverges. We also prove, under reasonable assumptions, a localization property in a stronger form that the spatial distribution of the population does not decay uniformly in space.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d ge 3$ and the environment is not too random, then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d le 2$, or the environment is ``random enough, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of replica overlap. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.
Let $D={mathbb H}setminus bigcup_{j=1}^N C_j$ be a standard slit domain, where ${mathbb H}$ is the upper half plane and $C_j,1le jle N,$ are mutually disjoint horizontal line segments in ${mathbb H}$. A stochastic Komatu-Loewner evolution denoted by ${rm SKLE}_{alpha,b}$ has been introduced in cite{CF} as a family ${F_t}$ of random growing hulls with $F_tsubset D$ driven by a diffusion process $xi(t)$ on $partial {mathbb H}$ that is determined by certain continuous homogeneous functions $alpha$ and $b$ defined on the space ${cal S}$ of all labelled standard slit domains. We aim at identifying the distribution of a suitably reparametrized ${rm SKLE}_{alpha,b}$ with that of the Loewner evolution on ${mathbb H}$ driven by the path of a certain continuous semimartingale and thereby relating the former to the distribution of ${rm SLE}_{alpha^2}$ when $alpha$ is a constant. We then prove that, when $alpha$ is a constant, ${rm SKLE}_{alpha,b}$ up to some random hitting time and modulo a time change has the same distribution as ${rm SLE}_{alpha^2}$ under a suitable Girsanov transformation. We further show that a reparametrized ${rm SKLE}_{sqrt{6},-b_{rm BMD}}$ has the same distribution as ${rm SLE}_6$, where $b_{rm BMD}$ is the BMD-domain constant indicating the discrepancy of $D$ from ${mathbb H}$ relative to Brownian motion with darning (BMD in abbreviation). A key ingredient of the proof is a hitting time analysis for the absorbing Brownian motion on ${mathbb H}.$ We also revisit and examine the locality property of ${rm SLE}_6$ in several canonical domains. Finally K-L equations and SKLEs for other canonical multiply connected planar domains than the standard slit one are recalled and examined.
We consider stochastic resonance for a diffusion with drift given by a potential, which has two metastable states and two pathways between them. Depending on the direction of the forcing, the height of the two barriers, one for each path, will either oscillate alternating or in synchronisation. We consider a simplified model given by a continuous time Markov Chains with two states. This was done for alternating and synchronised wells. The invariant measures are derived for both cases and shown to be constant for the synchronised case. A PDF for the escape time from an oscillatory potential is studied. Methods of detecting stochastic resonance are presented, which are linear response, signal-noise ratio, energy, out-of-phase measures, relative entropy and entropy. A new statistical test called the conditional Kolmogorov-Smirnov test is developed, which can be used to analyse stochastic resonance. An explicit two dimensional potential is introduced, the critical point structure derived and the dynamics, the invariant state and escape time studied numerically. The six measures are unable to detect the stochastic resonance in the case of synchronised saddles. The distribution of escape times however not only shows a clear sign of stochastic resonance, but changing the direction of the forcing from alternating to synchronised saddles an additional resonance at double the forcing frequency starts to appear. The conditional KS test reliably detects the stochastic resonance. This paper is mainly based on the thesis Stochastic Resonance for a Model with Two Pathways
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