Two-dimensional Brownian Motion Research Articles

Rate of escape of conditioned Brownian motion

We study the norm of the two-dimensional Brownian motion conditioned to stay outside the unit disk at all times. By conditioning the process is changed from barely recurrent to slightly transient. We obtain sharp results on the rate of escape to infinity of the process of future minima: we find an integral test on the function g so that the future minima process drops below the barrier exp{lnt×g(lnlnt)} at arbitrary large times; we show that the future minima process exceeds K t×lnlnlntat arbitrary large times with probability 0 [resp., 1] if K is larger [resp., smaller] than some positive constant. For this, we introduce a renewal structure attached to record times and values. Additional results are given for the long time behavior of the norm.

Model of diffuse reflection of optical radiation from the surface of biological tissue, implemented on the basis of two-dimensional fractional Brownian motion process

The numerical model of the diffuse reflection of Gaussian beam from the surface of biological tissue is introduced. The two-dimensional fractional Brownian motion (fBm) with the Hurst index H and the scale parameter σ was used for the simulations of the tissue surface relief. For the surfaces described by fixed σ = 0.1 and H = 0.55, H = 0.803 (corresponds to the surface of a banana fruit), H = 0.9, the angular distributions of the reflected radiation intensity were calculated using a Kirchhoff integral approach. The resulting distributions considerably differ from each other. Therefore, the introduced model can be used for the solution of the inverse problem of finding the fBm parameters of tissue surfaces employing the experimentally measured distribution of the reflected radiation intensity.

Stationary distributions for two-dimensional sticky Brownian motions: Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions, which find applications in many areas including queueing theory and mathematical finance. In this paper, we focus on stationary distributions for sticky Brownian motions. Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions. The kernel method, copula concept and extreme value theory are the main tools used in our analysis.

Pricing Product Options and Using Them to Complete Markets for Functions of Two Underlying Asset Prices

Options paying the product of put and/or call option payouts at different strikes on two underlying assets are observed to synthesize joint densities and replicate differentiable functions of two underlying asset prices. The pricing of such options is undertaken from three perspectives. The first perspective uses a geometric two-dimensional Brownian motion model. The second inverts two-dimensional characteristic functions. The third uses a bootstrapped physical measure to propose a risk charge minimizing hedge using options on the two underlying assets. The options are priced at the cost of the hedge plus the risk charge.

Least-squares estimation for the Vasicek model driven by the complex fractional Brownian motion

In this paper, we study the least-squares estimation problem for the Vasicek model driven by the complex fractional Brownian motion where is a two-dimensional fractional Brownian motion with Hurst parameter We obtain the strong consistency and asymptotic normality of and using the Garsia–Rodemich–Rumsey inequality and complex fourth moment theorems.

Thermally assisted mobility of nanodroplets on surfaces with weak defects

HypothesisThermal activation plays an essential role in contact line dynamics on nanorough surfaces. However, the relation between the aforementioned concept and the sliding motion of nanodroplets remains unclear. As a result, thermally assisted motion of nanodroplets on nanorough surfaces is investigated in this work. ExperimentsSteady slide and random motion of nanodroplets on surfaces with weak defects are investigated by Many-body Dissipative Particle Dynamics. The surface roughness is characterized by the slip length acquired from the velocity profile associated with the flowing film. FindingsThe slip length is found to decline with increasing the defect density. The linear relationship between the sliding velocity and driving force gives the mobility and reveals the absence of contact line pinning. On the basis of the Navier condition, a simple relation is derived and states that the mobility is proportional to the slip length and the reciprocal of the product of viscosity and contact area. Our simulation results agree excellently with the theoretical prediction. In the absence of external forces, a two-dimensional Brownian motion of nanodroplets is observed and its mean square displacement decreases with increasing the defect density. The diffusivity is proportional to the mobility, consistent with the Einstein relation. This consequence suggests that thermal fluctuations are able to overcome contact line pinning caused by weak defects.

Random walk on random planar maps: Spectral dimension, resistance and displacement

We study simple random walk on the class of random planar maps which can be encoded by a two-dimensional random walk with i.i.d.. increments or a two-dimensional Brownian motion via a “mating-of-trees” type bijection. This class includes the uniform infinite planar triangulation (UIPT), the infinite-volume limits of random planar maps weighted by the number of spanning trees, bipolar orientations, or Schnyder woods they admit, and the γ-mated-CRT map for γ∈(0,2). For each of these maps, we obtain an upper bound for the Green’s function on the diagonal, an upper bound for the effective resistance to the boundary of a metric ball, an upper bound for the return probability of the random walk to its starting point after n steps, and a lower bound for the graph-distance displacement of the random walk, all of which are sharp up to polylogarithmic factors. When combined with work of Lee (2017), our bound for the return probability shows that the spectral dimension of each of these random planar maps is a.s. equal to 2, that is, the (quenched) probability that the simple random walk returns to its starting point after 2n steps is n−1+on(1). Our results also show that the amount of time that it takes a random walk to exit a metric ball is at least its volume (up to a polylogarithmic factor). In the special case of the UIPT, this implies that random walk typically travels at least n1/4−on(1) units of graph distance in n units of time. The matching upper bound for the displacement is proven by Gwynne and Hutchcroft (Probab. Theory Related Fields 178 (2020) 567–611). These two works together resolve a conjecture of Benjamini and Curien (Geom. Funct. Anal. 23 (2013) 501–531) in the UIPT case. Our proofs are based on estimates for the mated-CRT map (which come from its relationship to SLE-decorated Liouville quantum gravity) and a strong coupling of the mated-CRT map with the other random planar map models.

Simultaneous Ruin Probability for Two-Dimensional Fractional Brownian Motion Risk Process over Discrete Grid

We derive the asymptotic behavior of the following ruin probability:P∃t∈Gδ:BHt−c1t>q1uBHt−c2t>q2u,H∈01,u→∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathrm{P}\\left\\{\\exists t\\in G\\left(\\delta \\right):{B}_H(t)-{c}_1t>{q}_1u,{B}_H(t)-{c}_2t>{q}_2u\\right\\},\\kern0.72em H\\in \\left(0,1\\right),u\ o \\infty, $$\\end{document}where BH is a standard fractional Brownian motion, c1, q1, c2, q2> 0, and G(δ) denotes the regular grid {0, δ, 2δ, . } for some δ > 0. The approximation depends on H, δ (only when H ≤ 1/2) and the relations between parameters c1, q1, c2, q2.

Two-dimensional stochastic modeling for predicting bankruptcy for manufacturing companies

Increasing accuracy of the model prediction on business bankruptcy helps reduce substantial losses for owners, creditors, investors and workers, and, further, minimize an economic and social problem frequently. In this study, we propose a stochastic model of financial working capital and cashflow as a two-dimensional Brownian motion X(t) = (X1(t),X2(t)) on the business bankruptcy prediction. The probability of bankruptcy occurring in a time interval [0,T] is defined by the boundary crossing probability of the two-dimensional Brownian motion entering a predetermined threshold domain. Mathematically, we extend the result in Fu and Wu (2016) on the boundary crossing probability of a high dimensional Brownian motion to an unbounded convex hull. The proposed model is applied to a real data set of companies in US and the numerical results show the proposed method performs well.

Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting.

We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic Brownian motion of duration t and diffusion constant D, in the presence of resetting to the origin at a constant rate r. We show that for any t, the mean perimeter is given by 〈L(t)〉=2πsqrt[D/r]f_{1}(rt) and the mean area is given by 〈A(t)〉=2πD/rf_{2}(rt) where the scaling functions f_{1}(z) and f_{2}(z) are computed explicitly. For large t≫1/r, the mean perimeter grows extremely slowly as 〈L(t)〉∝ln(rt) with time. Likewise, the mean area also grows slowly as 〈A(t)〉∝ln^{2}(rt) for t≫1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.

Liouville quantum gravity surfaces with boundary as matings of trees

For γ∈(0,2), the quantum disk and γ-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling SLE16/γ2 curve on a quantum disk or on a γ-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation −cos(πγ2/4). This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case of quantum surfaces with boundary (the disk case for γ∈(2,2) was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.

AN EFFICIENT MAXIMUM LIKELIHOOD ESTIMATOR FOR TWO-DIMENSIONAL FRACTIONAL BROWNIAN MOTION

For pattern recognition, natural scenes and medical images are often modeled as two-dimensional fractional Brownian motion (2D FBM), which can be easily described by the Hurst exponent ([Formula: see text], a real number between 0 and 1. The Hurst exponent is directly related to the fractal dimension ([Formula: see text]) by [Formula: see text]–[Formula: see text], and hence it very suitably serves as a characteristic index. Therefore, how to estimate the Hurst exponent effectively and efficiently is very important in pattern recognition. In this paper, a partially iterative algorithm, simply called an iterative maximum likelihood estimator (MLE) for 2D DFBM is first proposed and then a more sufficiently iterative algorithm, simply called an efficient MLE for 2D DFBM, is further proposed via a perfect structure of the log-likelihood function related to the Hurst exponent. Except for theoretical knowledge, two practical algorithms are also correspondingly provided for easy applications. Experimental results show that the MLE for 2D DFBM is effective and workable; its accuracy gets higher as the image size increases, and hence its recognition resolution is much finer to identify small difference among patterns. Furthermore, the efficient MLE is much quicker than the iterative MLE, especially at larger image sizes.

Joint scaling limit of site percolation on random triangulations in the metric and peanosphere sense

Recent works have shown that random triangulations decorated by critical (p=1∕2) Bernoulli site percolation converge in the scaling limit to a 8∕3-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by SLE6 in two different ways: The triangulation, viewed as a curve-decorated metric measure space equipped with its graph distance, the counting measure on vertices, and a single percolation interface converges with respect to a version of the Gromov–Hausdorff topology. There is a bijective encoding of the site-percolated triangulation by means of a two-dimensional random walk, and this walk converges to the correlated two-dimensional Brownian motion which encodes SLE6-decorated 8∕3-LQG via the mating-of-trees theorem of Duplantier-Miller-Sheffield (2014); this is sometimes called peanosphere convergence. We prove that one in fact has joint convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to 8∕3-LQG decorated by SLE6 in the metric space sense. This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into C via the so-called Cardy embedding converge to 8∕3-LQG.

Mapping images into ordinal networks.

An increasing abstraction has marked some recent investigations in network science. Examples include the development of algorithms that map time series data into networks whose vertices and edges can have different interpretations, beyond the classical idea of parts and interactions of a complex system. These approaches have proven useful for dealing with the growing complexity and volume of diverse data sets. However, the use of such algorithms is mostly limited to one-dimensional data, and there has been little effort towards extending these methods to higher-dimensional data such as images. Here we propose a generalization for the ordinal network algorithm for mapping images into networks. We investigate the emergence of connectivity constraints inherited from the symbolization process used for defining the network nodes and links, which in turn allows us to derive the exact structure of ordinal networks obtained from random images. We illustrate the use of this new algorithm in a series of applications involving randomization of periodic ornaments, images generated by two-dimensional fractional Brownian motion and the Ising model, and a data set of natural textures. These examples show that measures obtained from ordinal networks (such as average shortest path and global node entropy) extract important image properties related to roughness and symmetry, are robust against noise, and can achieve higher accuracy than traditional texture descriptors extracted from gray-level co-occurrence matrices in simple image classification tasks.

Fractal analysis of recurrence networks constructed from the two-dimensional fractional Brownian motions.

In this study, we focus on the fractal property of recurrence networks constructed from the two-dimensional fractional Brownian motion (2D fBm), i.e., the inter-system recurrence network, the joint recurrence network, the cross-joint recurrence network, and the multidimensional recurrence network, which are the variants of classic recurrence networks extended for multiple time series. Generally, the fractal dimension of these recurrence networks can only be estimated numerically. The numerical analysis identifies the existence of fractality in these constructed recurrence networks. Furthermore, it is found that the numerically estimated fractal dimension of these networks can be connected to the theoretical fractal dimension of the 2D fBm graphs, because both fractal dimensions are piecewisely associated with the Hurst exponent H in a highly similar pattern, i.e., a linear decrease (if H varies from 0 to 0.5) followed by an inversely proportional-like decay (if H changes from 0.5 to 1). Although their fractal dimensions are not exactly identical, their difference can actually be deciphered by one single parameter with the value around 1. Therefore, it can be concluded that these recurrence networks constructed from the 2D fBms must inherit some fractal properties of its associated 2D fBms with respect to the fBm graphs.

Exact asymptotics of component-wise extrema of two-dimensional Brownian motion

We derive the exact asymptotics of $ {\mathbb {P} \left \{ \underset {t\ge 0}{\sup } \left (X_{1}(t) - \mu _{1} t\right )> u, \underset {s\ge 0}{\sup } \left (X_{2}(s) - \mu _{2} s\right )> u \right \} }, u\to \infty , $ where (X1(t), X2(s))t, s≥ 0 is a correlated two-dimensional Brownian motion with correlation ρ ∈ [− 1,1] and μ1, μ2 > 0. It appears that the play between ρ and μ1, μ2 leads to several types of asymptotics. Although the exponent in the asymptotics as a function of ρ is continuous, one can observe different types of prefactor functions depending on the range of ρ, which constitute a phase-type transition phenomena.

On the cumulative Parisian ruin of multi-dimensional Brownian motion risk models

Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalisations of some of the results derived in Dbicki et al. [(2018). Extremal behavior of hitting a cone by correlated Brownian motion with drift. Stochastic Processes and Their Applications 128, 4171–4206]. As a particular interesting case, the two-dimensional Brownian motion risk model is discussed in detail.

A mating-of-trees approach for graph distances in random planar maps

We introduce a general technique for proving estimates for certain random planar maps which belong to the gamma -Liouville quantum gravity (LQG) universality class for gamma in (0,2). The family of random planar maps we consider are those which can be encoded by a two-dimensional random walk with i.i.d. increments via a mating-of-trees bijection, and includes the uniform infinite planar triangulation (UIPT; gamma =sqrt{8/3}); and planar maps weighted by the number of different spanning trees (gamma =sqrt{2}), bipolar orientations (gamma =sqrt{4/3}), or Schnyder woods (gamma =1) that can be put on the map. Using our technique, we prove estimates for graph distances in the above family of random planar maps. In particular, we obtain non-trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the Watabiki (Prog Theor Phys Suppl 114:1–17, 1993) prediction for the Hausdorff dimension of gamma -LQG and we establish the existence of an exponent for certain distances in the map. The basic idea of our approach is to compare a given random planar map M to a mated-CRT map—a random planar map constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev in ESAIM Probab Stat 2:41–108, 1998) of the encoding walk for M and the Brownian motion used to construct the mated-CRT map. This allows us to deduce estimates for graph distances in M from the estimates for graph distances in the mated-CRT map which we proved (using continuum theory) in a previous work. In the special case when gamma =sqrt{8/3}, we instead deduce estimates for the sqrt{8/3}-mated-CRT map from known results for the UIPT. The arguments of this paper do not directly use SLE/LQG, and can be read without any knowledge of these objects.

An almost sure KPZ relation for SLE and Brownian motion

The peanosphere construction of Duplantier, Miller and Sheffield provides a means of representing a $\gamma $-Liouville quantum gravity (LQG) surface, $\gamma \in (0,2)$, decorated with a space-filling form of Schramm’s $\mathrm{SLE}_{\kappa }$, $\kappa =16/\gamma^{2}\in (4,\infty)$, $\eta $ as a gluing of a pair of trees which are encoded by a correlated two-dimensional Brownian motion $Z$. We prove a KPZ-type formula which relates the Hausdorff dimension of any Borel subset $A$ of the range of $\eta $, which can be defined as a function of $\eta $ (modulo time parameterization) to the Hausdorff dimension of the corresponding time set $\eta^{-1}(A)$. This result serves to reduce the problem of computing the Hausdorff dimension of any set associated with an $\mathrm{SLE}$, $\mathrm{CLE}$ or related processes in the interior of a domain to the problem of computing the Hausdorff dimension of a certain set associated with a Brownian motion. For many natural examples, the associated Brownian motion set is well known. As corollaries, we obtain new proofs of the Hausdorff dimensions of the $\mathrm{SLE}_{\kappa}$ curve for $\kappa \neq4$; the double points and cut points of $\mathrm{SLE}_{\kappa }$ for $\kappa >4$; and the intersection of two flow lines of a Gaussian free field. We obtain the Hausdorff dimension of the set of $m$-tuple points of space-filling $\mathrm{SLE}_{\kappa }$ for $\kappa >4$ and $m\geq 3$ by computing the Hausdorff dimension of the so-called $(m-2)$-tuple $\pi /2$-cone times of a correlated planar Brownian motion.

A Ruin Problem for a Two-Dimensional Brownian Motion with Controllable Drift in the Positive Quadrant

In this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. We allow that bot...