Dynamics of a generic Brownian motion: Recursive aspects (original) (raw)

The Descriptive Complexity of Brownian Motion

Advances in Mathematics, 2000

A continuous function x on the unit interval is a generic Brownian motion when every probabilistic event which holds almost surely with respect to the Wiener measure is reflected in x, provided that the event has a suitably effective description. We show that a generic one-dimensional Brownian motion can be computed from an infinite binary string which is complex in the sense of Kolmogorov Chaitin. Conversely, one can construct a Kolmogorov Chaitin random string from the values at the rational numbers of a generic Brownian motion. In this way, we construct a recursive isomorphism between encoded versions of generic Brownian motions and Kolmogorov Chaitin random reals.

Diophantine properties of brownian motion: recursive aspects

2014

We use recent results on the Fourier analysis of the zero sets of Brownian motion to explore the diophantine properties of an algorithmically random Brownian motion ( also known as a complex oscillation). We discuss the construction and definability of perfect sets which are linearly independent over the rationals directly from Martin-Löf random reals. Finally we explore the recent work of Tsirelson on countable dense sets to study the diophantine properties of local minimisers of Brownian motion.

Snakes and spiders: Brownian motion on ℝ-trees

Probability Theory and Related Fields, 2000

We consider diffusion processes on a class of ‫-ޒ‬trees. The processes are defined in a manner similar to that of Le Gall's Brownian snake. Each point in the tree has a real-valued "height" or "generation", and the height of the diffusion process evolves as a Brownian motion. When the height process decreases the diffusion retreats back along a lineage, whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages. The class of ‫-ޒ‬trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation. We give a rigorous construction of the diffusion process, identify its Dirichlet form, and obtain a necessary and sufficient condition for it to be transient. We show that the tail σ-field of the diffusion is always trivial and draw the usual conclusion that bounded space-time harmonic functions are constant. In the transient case, we identify the Martin compactification and obtain the corresponding integral representations of excessive and harmonic functions. Using Ray-Knight methods, we show that the only entrance laws for the diffusion are the trivial ones that arise from starting the process inside the state-space. Finally, we use the Dirichlet form stochastic calculus to obtain a semimartingale description of the diffusion that involves local time additive functionals associated with each branch point of the tree.

Kolmogorov complexity and the geometry of Brownian motion

Mathematical Structures in Computer Science, 2014

In this paper, we continue the study of the geometry of Brownian motions which are encoded by Kolmogorov-Chaitin random reals (complex oscillations). We unfold Kolmogorov-Chaitin complexity in the context of Brownian motion and specifically to phenomena emerging from the random geometric patterns generated by a Brownian motion.

Stochastic dynamics for an infinite system of random closed strings: A Gibbsian point of view

Stochastic Processes and their Applications, 1996

We consider the stochastic dynamics of infinitely many, interacting random closed strings, and show that the law of this process can be characterized as a Gibbs state for some Hamiltonian on the path level, which is represented in terms of the interaction. This is done by means of the stochastic calculus of variations, in particular an integration by parts formula in infinite dimensions. This Gibbsian point of view of the stochastic dynamics allows us to characterize the reversible states as the Gibbs states for the underlying interaction. Under supplementary monotonicity conditions, there is only one stationary distribution, and we prove that there is exactly one Gibbs state.

Brownian motion on the continuum tree

Probability Theory and Related Fields, 1995

We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.

Fourier spectra of measures associated with algorithmically random Brownian motion

Logical Methods in Computer Science, 2014

In this paper we study the behaviour at infinity of the Fourier transform of Radon measures supported by the images of fractal sets under an algorithmically random Brownian motion. We show that, under some computability conditions on these sets, the Fourier transform of the associated measures have, relative to the Hausdorff dimensions of these sets, optimal asymptotic decay at infinity. The argument relies heavily on a direct characterisation, due to Asarin and Pokrovskii, of algorithmically random Brownian motion in terms of the prefix-free Kolmogorov complexity of finite binary sequences. The study also necessitates a closer look at the potential theory over fractals from a computable point of view.

Reflection and coalescence between independent one-dimensional Brownian paths

Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2000

Take two independent one-dimensional processes as follows: (B t , t ∈ [0, 1]) is a Brownian motion with B 0 = 0, and (β t , t ∈ [0, 1]) has the same law as (B 1−t , t ∈ [0, 1]); in other words, β 1 = 0 and β can be seen as Brownian motion running backwards in time. Define (γ t , t ∈ [0, 1]) as being the function that is obtained by reflecting B on β. Then γ is still a Brownian motion. Similar and more general results (with families of coalescing Brownian motions) are also derived. They enable us to give a precise definition (in terms of reflection) of the joint realization of finite families of coalescing/reflecting Brownian motions.

On the computability of a construction of Brownian motion

Mathematical Structures in Computer Science, 2013

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

Excited Brownian Motions

Latin American journal of probability and mathematical statistics

We study a natural continuous time version of excited random walks, introduced by Norris, Rogers and Williams about twenty years ago. We obtain a necessary and sufficient condition for recurrence and for positive speed. This is analogous to results for excited (or cookie) random walks.

A mechanical model of Brownian motion

Communications in Mathematical Physics, 1981

We consider a dynamical system consisting of one large massive particle and an infinite number of light point particles. We prove that the motion of the massive particle is, in a suitable limit, described by the Ornstein-Uhlenbeck process. This extends to three dimensions previous results by Holley in one dimension. 0010-3616/81/0078/0507/S04.80

A Note on Reflecting Brownian Motions

Electronic Communications in Probability, 2002

We give another proof of the following result from a joint paper with Bálint Tóth: "A Brownian motion reflected on an independent time-reversed Brownian motion is a Brownian motion".

Fractal properties of the random string processes

2006

Let ut(x),tge0,xinmathbbR\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}ut(x),tge0,xinmathbbR be a random string taking values in mathbbRd{\mathbb{R}}^dmathbbRd, specified by the following stochastic partial differential equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where dotW(x,t)\dot{W}(x,t)dotW(x,t) is an mathbbRd{\mathbb{R}}^dmathbbRd-valued space-time white noise. Mueller and Tribe (2002) have proved necessary and sufficient conditions for the mathbbRd{\mathbb{R}}^dmathbbRd-valued process ut(x):tge0,xinmathbbR\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}ut(x):tge0,xinmathbbR to hit points and to have double points. In this paper, we continue their research by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string process ut(x):tge0,xinmathbbR\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}ut(x):tge0,xinmathbbR. We also consider the Hausdorff and packing dimensions of the range and graph of the string.

On the spread of a branching Brownian motion whose offspring number has infinite variance

We study the impact on shape parameters of an underlying Bienaym\'e-Galton-Watson branching process (height, width and first hitting time), of having a non-spatial branching mechanism with infinite variance. Aiming at providing a comparative study of the spread of an epidemics whose dynamics is given by the modulus of a branching Brownian motion (BBM) we then consider spatial branching processes in dimension d, not necessarily integer. The underlying branching mechanism is then either a binary branching model or one presenting infinite variance. In particular we evaluate the chance p(x) of being hit if the epidemics started away at distance x. We compute the large x tail probabilities of this event, both when the branching mechanism is regular and when it exhibits very large fluctuations. Online first, 9 Feb 2015, Physica D Nonlinear Phenomena, under the new title: On extreme events for non-spatial and spatial branching Brownian motions.

Random perturbations of stochastic processes with unbounded variable length memory

Electronic Journal of Probability, 2008

We consider binary infinite order stochastic chains perturbed by a random noise. This means that at each time step, the value assumed by the chain can be randomly and independently flipped with a small fixed probability. We show that the transition probabilities of the perturbed chain are uniformly close to the corresponding transition probabilities of the original chain. As a consequence, in the case of stochastic chains with unbounded but otherwise finite variable length memory, we show that it is possible to recover the context tree of the original chain, using a suitable version of the algorithm Context, provided that the noise is small enough. * This work is part of PRONEX/FAPESP's project Stochastic behavior, critical phenomena and rhythmic pattern identification in natural languages (grant number 03/09930-9), CNRS-FAPESP project Probabilistic phonology of rhythm and CNPq's projects Stochastic modeling of speech (grant number 475177/2004-5) and Rhythmic patterns, prosodic domains and probabilistic modeling in Portuguese Corpora (grant number 485999/2007-2). AG is partially supported by a CNPq fellowship (grant 308656/2005-9) and FL is supported by a FAPESP fellowship (grant 06/56980-0).

The Brownian Web: Characterization and Convergence

2003

The Brownian Web (BW) is the random network formally consisting of the paths of coalescing one-dimensional Brownian motions starting from every space-time point in mathbbRtimesmathbbR{\mathbb R}\times{\mathbb R}mathbbRtimesmathbbR. We extend the earlier work of Arratia and of T\'oth and Werner by providing characterization and convergence results for the BW distribution, including convergence of the system of all coalescing random walkssktop/brownian web/finale/arXiv submits/bweb.tex to the BW under diffusive space-time scaling. We also provide characterization and convergence results for the Double Brownian Web, which combines the BW with its dual process of coalescing Brownian motions moving backwards in time, with forward and backward paths ``reflecting'' off each other. For the BW, deterministic space-time points are almost surely of ``type'' (0,1)(0,1)(0,1) -- {\em zero} paths into the point from the past and exactly {\em one} path out of the point to the future; we determine the Hausdorff dimension for all types that actually occur: dimension 2 for type (0,1)(0,1)(0,1), 3/2 for (1,1)(1,1)(1,1) and (0,2)(0,2)(0,2), 1 for (1,2)(1,2)(1,2), and 0 for (2,1)(2,1)(2,1) and (0,3)(0,3)(0,3).

Brownian motion on nested fractals

1990

The purpose of this paper is to construct Brownian motion on a reasonably general class of self-similar fractals. To this end, I introduce an axiomatically defined class of ,.nested fractals .. , which satisfy certain symmetry and connectivity conditions, and which also are (in the physicists' terminology) finitely ramified. On each one of these nested fractals, a Brownian motion is constructed and shown to be a strong Markov prOcess with continuous paths. If the Laplacian 6 on the fractal is defined as the infinitesimal generator of the Brownian motion, and n(a) denotes the number of eigenvalues of-6 less than a, I prove that () d•logv/log~ n a ~ a as a~~. where d is the Hausdorff dimension of the fractal, and v and ~ are two parameters describing its self-similarity structure. In general, logv/log~*l/2 and hence Weyl's conjecture can only hold for fractals in a modified form.

Two recursive decompositions of Brownian bridge

2004

studied asymptotic distributions as n → ∞, of various functionals of a uniform random mapping of the set {1, . . . , n}, by constructing a mapping-walk and showing these random walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the Lévy-Itô theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index α ∈ (0, 1).

Random Walk, Brownian Motion, and Martingales

Graduate Texts in Mathematics, 2021

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.