One-dimensional Brownian Motion Research Articles

The convergence rate of the equilibrium measure for the hybrid LQG Mean Field Game

In this work, we study the convergence rate of the N-player Linear-Quadratic-Gaussian (LQG) game with a Markov chain common noise towards its asymptotic Mean Field Game. By postulating a Markovian structure via two auxiliary processes for the first and second moments of the Mean Field Game equilibrium and applying the fixed point condition in Mean Field Game, we first provide the characterization of the equilibrium measure in Mean Field Game with a finite-dimensional Riccati system of ODEs. Additionally, with an explicit coupling of the optimal trajectory of the N-player game driven by N dimensional Brownian motion and Mean Field Game counterpart driven by one-dimensional Brownian motion, we obtain the convergence rate O(N−1/2) with respect to 2-Wasserstein distance.

Ergodic properties of Brownian motion under stochastic resetting.

We study the ergodic properties of one-dimensional Brownian motion with resetting. Using generic classes of statistics of times between resets, we find respectively for thin- or fat-tailed distributions the normalized or non-normalized invariant density of this process. The former case corresponds to known results in the resetting literature and the latter to infinite ergodic theory. Two types of ergodic transitions are found in this system. The first is when the mean waiting time between resets diverges, when standard ergodic theory switches to infinite ergodic theory. The second is when the mean of the square root of time between resets diverges and the properties of the invariant density are drastically modified. We then find a fractional integral equationdescribing the density of particles. This finite time tool is particularly useful close to the ergodic transition where convergence to asymptotic limits is logarithmically slow. Our study implies rich ergodic behaviors for this nonequilibrium process which should hold far beyond the case of Brownian motion analyzed here.

Existence and Uniqueness Theorem of Fuzzy Stochastic Ordinary Differential Equations

A fuzzy valued diffusion term, which in a fuzzy stochastic differential equation refers to one-dimensional Brownian motion, is defined by the meaning of the stochastic integral of a fuzzy process. In this paper, the existence and uniqueness theorem of fuzzy stochastic ordinary differential equations, based on the mean square convergence of the mathematical induction approximations to the associated stochastic integral equation, are stated and demonstrated.

Extremal statistics for a one-dimensional Brownian motion with a reflective boundary

In this work, we investigate the extreme value statistics of a one-dimensional Brownian motion (with the diffusion constant D) during a time interval 0,t in the presence of a reflective boundary at the origin, when starting from a positive position x0. We first obtain the distribution P(M|x0,t) of the maximum displacement M and its expectation 〈M〉. In the short-time limit, i.e., t≪td where td=x02/D is the diffusion time from the starting position x0 to the reflective boundary at the origin, the particle behaves like a free Brownian motion without any boundaries. In the long-time limit, t≫td, 〈M〉 grows with t as 〈M〉∼t, which is similar to the free Brownian motion, but the prefactor is π/2 times of the free Brownian motion, embodying the effect of the reflective boundary. By solving the propagator and using a path decomposition technique, we then obtain the joint distribution P(M,tm|x0,t) of M and the time tm at which this maximum is achieved, from which the marginal distribution P(tm|x0,t) of tm is also obtained. For t≪td, P(tm|x0,t) looks like a U-shaped attributed to the arcsine law of free Brownian motion. For t equal to or larger than order of magnitude of tm, P(tm|x0,t) deviates from the U-shaped distribution and becomes asymmetric with respect to t/2. Moreover, we compute the expectation 〈tm〉 of tm, and find that 〈tm〉/t is an increasing function of t. In two limiting cases, 〈tm〉/t→1/2 for t≪td and 〈tm〉/t→(1+2G)/4≈0.708 for t≫td, where G≈0.916 is the Catalan’s constant. Finally, we analytically compute the statistics of the last time tℓ the particle crosses the starting position x0 and the occupation time to spent above x0. We find that 〈tℓ〉/t→1/2 in the short-time and long-time limits, and reaches its maximum at an intermediate value of t. The fraction of the occupation time 〈to〉/t is a monotonic function of t, and tends towards 1 in the long-time limit. All the theoretical results are validated by numerical simulations.

Breakdown of arcsine law for resetting brownian motion

For a one-dimensional Brownian motion starting from the origin, the cumulative distribution of the occupation time V staying above the origin obeys the celebrated arcsine law. In this work, we show how the law is modified for a resetting Brownian motion, where the Brownian is reset to the position x r at random times but with a constant rate r. When x r is exactly equal to zero, we derive the exact expression of the probability distribution P r (V∣0, t) of V during time t, and the moments of V as functions of r and t. P r (V∣0, t) is always symmetric with respect to V = t/2 for arbitrary value of r, but the probability density of V at V = t/2 increases with the increase of r. Interestingly, P r (V∣0, t) at V = t/2 changes from a minimum to a local maximum at a critical value R * ≈ 0.742 338, where R = rt denotes the average number of resetting during time t. Moreover, we consider the case when x r is a random variable and is distributed by a function g(x r ), where g(x r ) is assumed to be symmetric with respect to zero and possesses its maximum at zero. We derive the general expressions of the moments of V when the variance of x r is low. The mean value of V is always equal to t/2, but the fluctuation in x r leads to an increase in the second and third moments of V. Our results provide a quantitative understanding of how stochastic resetting destroys the persistence of Brownian motion.

A singular stochastic control problem with direction switching cost

We introduce a new class of singular stochastic control problems in which the process controller not only chooses the push intensity, at a price proportional to the displacement caused by his action, but he can also change the allowable control direction, paying a fixed cost for each such switching. Singular control of the one-dimensional Brownian motion with quadratic instantaneous cost function and costly direction switching on the infinite time horizon is analyzed in detail, leading to a closed-form solution. This example is used as an illustration of qualitative differences between the class of problems considered here and classic singular stochastic control.

Extreme value statistics and Arcsine laws of Brownian motion in the presence of a permeable barrier

The Arcsine laws of Brownian motion are a collection of results describing three different statistical quantities of one-dimensional Brownian motion: the time at which the process reaches its maximum position, the total time the process spends in the positive half-space and the time at which the process crosses the origin for the last time. Remarkably the cumulative probabilities of these three observables all follow the same distribution, the Arcsine distribution. But in real systems, space is often heterogeneous, and these laws are likely to hold no longer. In this paper we explore such a scenario and study how the presence of a spatial heterogeneity alters these Arcsine laws. Specifically we consider the case of a thin permeable barrier, which is often employed to represent diffusion impeding heterogeneities in physical and biological systems such as multilayer electrodes, electrical gap junctions, cell membranes and fragmentation in the landscape for dispersing animals. Using the Feynman–Kac formalism and path decomposition techniques we are able to find the exact time-dependence of the probability distribution of the three statistical quantities of interest. We show that a permeable barrier has a large impact on these distributions at short times, but this impact is less influential as time becomes long. In particular, the presence of a barrier means that the three distributions are no longer identical with symmetry about their means being broken. We also study a closely related statistical quantity, namely, the distribution of the maximum displacement of a Brownian particle and show that it deviates significantly from the usual half-Gaussian form.

The Bethe ansatz for sticky Brownian motions

We consider a multi-dimensional diffusion whose coordinates behave as one-dimensional Brownian motions, evolving independently when apart, but with a sticky interaction when they coincide. We derive the Kolmogorov backwards equation and show that for a specific choice of interaction it can be solved exactly with the Bethe ansatz. The diffusion in Rn can be viewed as the n-point motions of a stochastic flow of kernels. We use our formulae to study the flow of kernels and show that atoms in the flow are asymptotically exponentially distributed in size at large times.

Relation between the degree and betweenness centrality distribution in complex networks.

The centrality measures, like betweenness b and degree k in complex networks remain fundamental quantities helping to classify them. It is realized from Barthelemy's paper [Eur. Phys. J. B 38, 163 (2004)10.1140/epjb/e2004-00111-4] that the maximal b-k exponent for the scale-free (SF) networks is η_{max}=2, belonging to SF trees, based on which one concludes δ≥γ+1/2, where γ and δ are the scaling exponents for the distribution functions of the degree and the betweenness centralities, respectively. This conjecture was violated for some special models and systems. Here we present a systematic study on this problem for visibility graphs of correlated time series, and show evidence that this conjecture fails in some correlation strengths. We consider the visibility graph of three models: two-dimensional Bak-Tang-Weisenfeld (BTW) sandpile model, one-dimensional (1D) fractional Brownian motion (FBM), and 1D Levy walks, the two latter cases are controlled by the Hurst exponent H and the step index α, respectively. In particular, for the BTW model and FBM with H≲0.5, η is greater than 2, and also δ<γ+1/2 for the BTW model, while the Barthelemy's conjecture remains valid for the Levy process. We assert that the failure of the Barthelemy's conjecture is due to large fluctuations in the scaling b-k relation resulting in the violation of hyperscaling relation η=γ-1/δ-1 and emergent anomalous behavior for the BTW model and FBM. Universal distribution function of generalized degree is found for these models which have the same scaling behavior as the Barabasi-Albert network.

An optimal sequential procedure for determining the drift of a Brownian motion among three values

We consider a one-dimensional Brownian motion, having a random and unobservable drift which can take one of three known values. Assuming that we monitor the position of the process in real time, the problem is to determine as soon as possible and with minimal probabilities of the wrong terminal decisions, which value the drift has taken. We derive the exact solution to the problem in the Bayesian formulation, under any prior probability distribution on the three values that the drift can assume, when the cost of observation is linear. Remarkably, the optimal stopping boundaries of the present problem are non-monotone.

A stochastic linear-quadratic optimal control problem with jumps in an infinite horizon

&lt;abstract&gt;&lt;p&gt;In this paper, a stochastic linear-quadratic (LQ, for short) optimal control problem with jumps in an infinite horizon is studied, where the state system is a controlled linear stochastic differential equation containing affine term driven by a one-dimensional Brownian motion and a Poisson stochastic martingale measure, and the cost functional with respect to the state process and control process is quadratic and contains cross terms. Firstly, in order to ensure the well-posedness of our stochastic optimal control of infinite horizon with jumps, the $ L^2 $-stabilizability of our control system with jump is introduced. Secondly, it is proved that the $ L^2 $-stabilizability of our control system with jump is equivalent to the non-emptiness of the admissible control set for all initial state and is also equivalent to the existence of a positive solution to some integral algebraic Riccati equation (ARE, for short). Thirdly, the equivalence of the open-loop and closed-loop solvability of our infinite horizon optimal control problem with jumps is systematically studied. The corresponding equivalence is established by the existence of a $ stabilizing\ solution $ of the associated generalized algebraic Riccati equation, which is different from the finite horizon case. Moreover, any open-loop optimal control for the initial state $ x $ admiting a closed-loop representation is obatined.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic behavior for a time-inhomogeneous Kolmogorov type diffusion

We study a kinetic stochastic model with a non-linear time-inhomogeneous friction force and a Brownian-type random force. More precisely, a Kolmogorov type diffusion (V, X) is considered: here, X is the position of the particle, and V is its velocity. The process V is solution to a stochastic differential equation driven by a one-dimensional Brownian motion, with a drift of the form t−βF(v). The function F satisfies some homogeneity condition, and β is a real number. The behavior in large time of the process (V, X) is proved by using stochastic analysis tools.

Branching Brownian motion in a periodic environment and existence of pulsating traveling waves

In this paper, we first study the limits of the additive and derivative martingales of one-dimensional branching Brownian motion in a periodic environment. Then we prove the existence of pulsating traveling wave solutions of the corresponding F-KPP equation in the supercritical and critical cases by representing the solutions probabilistically in terms of the limits of the additive and derivative martingales. We also prove that there is no pulsating traveling wave solution in the subcritical case. Our main tools are the spine decomposition and martingale change of measures.

On two-dimensional extensions of Bougerol’s identity in law

Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of semimartingale eBt,t≥0. The celebrated Bougerol’s identity in law (1983) asserts that, if β={βt}t≥0 is another Brownian motion independent of B, then βAt has the same law as sinhBt for every fixed t>0. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving, as the second coordinates, the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.

The rate of escape of the most visited site of Brownian motion

Let {Ltz} be the jointly continuous local times of a one-dimensional Brownian motion and let Lt∗=supz∈RLtz. Let Vt be any point z such that Ltz=Lt∗, a most visited site of Brownian motion. We prove that if γ>1, then lim inft→∞ |Vt| t∕(logt)γ=∞,a.s., with an analogous result for simple random walk. This proves a conjecture of Lifshits and Shi.

A White Noise Approach to Occupation Times of Brownian Motion

Occupation times of a stochastic process models the amount of time the process spends inside a spatial interval during a certain finite time horizon. It appears in the fiber lay-down process in nonwoven production industry. The occupation time can be interpreted as the mass of fiber material deposited inside some region. From application point of view, it is important to know the average mass per unit area of the final fleece. In this paper we use white noise theory to prove the existence of the occupation times of one-dimensional Brownian motion and provide an expression for the expected value of the occupation times.

Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion

In the framework of a stochastic picture for the one-dimensional branching Brownian motion, we compute the probability density of the number of particles near the rightmost one at a time T, that we take very large, when this extreme particle is conditioned to arrive at a predefined position x T chosen far ahead of its expected position m T . We recover the previously conjectured fact that the typical number density of particles at a distance Δ to the left of the lead particle, when both Δ and are large, is smaller than the mean number density by a factor proportional to , where ζ is a constant that was so far undetermined. Our picture leads to an expression for the probability density of the particle number, from which a value for ζ may be inferred.

Well-posedness and numerical simulations employing Legendre-shifted spectral approach for Caputo–Fabrizio fractional stochastic integrodifferential equations

This paper investigates the well-posedness of a class of FSIDEs utilizing the fractional Caputo–Fabrizio derivative. Herein, the well-posedness proofs are constructed by considering some applicable conditions and combining theories of Banach space, AAT, and FPST. Approximating the solutions of such equations is still challenging for many mathematicians today due to their randomness and the hardness of finding the exact one. For the numerical aim, we introduce some useful properties of the Legendre-shifted polynomials and employ them as a basis of the collocation spectral method. The idea of this scheme is to convert such stochastic equations into algebraic systems subject to [Formula: see text]-measurable independent parameters. The stochastic term is driven by one-dimensional standard Brownian motion which is the most familiar type and for simulating its trajectories we discuss an easy method. We rigorously analyze the convergence of the proposed technique and other error behavior-bound results. Finally, various tangible numerical applications are performed to verify the present scheme’s accuracy and great feasibility and support theoretical results. The acquired results reveal that the methodology used is effective and appropriate to deal with various issues in light of the fractional Caputo–Fabrizio derivative.

Yaglom-type limit theorems for branching Brownian motion with absorption

We consider one-dimensional branching Brownian motion in which particles are absorbed at the origin. We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a critical drift towards the origin so that the process eventually goes extinct with probability one. We establish precise asymptotics for the probability that the process survives for a large time t, building on previous results by Kesten (1978) and Berestycki, Berestycki, and Schweinsberg (2014). We also prove a Yaglom-type limit theorem for the behavior of the process conditioned to survive for an unusually long time, providing an essentially complete answer to a question first addressed by Kesten (1978). An important tool in the proofs of these results is the convergence of a certain observable to a continuous state branching process. Our proofs incorporate new ideas which might be of use in other branching models.

Branching Brownian motion in a periodic environment and uniqueness of pulsating traveling waves

AbstractUsing one-dimensional branching Brownian motion in a periodic environment, we give probabilistic proofs of the asymptotics and uniqueness of pulsating traveling waves of the Fisher–Kolmogorov–Petrovskii–Piskounov (F-KPP) equation in a periodic environment. This paper is a sequel to ‘Branching Brownian motion in a periodic environment and existence of pulsating travelling waves’ (Ren et al., 2022), in which we proved the existence of the pulsating traveling waves in the supercritical and critical cases, using the limits of the additive and derivative martingales of branching Brownian motion in a periodic environment.