Brownian Limit Research Articles

The Effects of Competition and Monitoring on R&D Investment: A Dynamic Approach

We examine a dynamic model of R&D investment by competing firms with an uncertain payout and uncertain time to development success. The effect of competition on R&D investment depends critically on whether firms are able to monitor, and react to, each other's actions. In the absence of monitoring, competition speeds up investment and erodes option values. When monitoring is allowed, the Pareto-dominant closed-loop equilibrium is identical to the first-best cooperative equilibrium, with investment that is increasingly postponed as more firms enter the market. A novel approach, solving a sequence of pure-jump equilibria, is used to obtain the equilibrium in the Brownian limit.

Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights

Nous nous intéressons à un modèle de permutations aléatoires de $n$ éléments, avec des poids polynomiaux en les longueurs des cycles, qui a été étudié notamment par Ercolani et Ueltschi. En utilisant l’analyse des points selles des transformées de Fourier, nous montrons que la distance en variation totale entre la suite des nombres de cycles de tailles $1,2,\ldots,b$ et une certaine suite $(Z_{1},Z_{2},\ldots,Z_{b})$ de variables de Poisson indépendantes, converge vers $0$ quand $n\to\infty$ si et seulement si $b=o(\ell)$, où $\ell$ désigne la longueur d’un cycle typique de ce modèle. À l’aide de ce résultat, nous établissons ensuite un théorème central limite pour l’ordre de la permutation, étendant ainsi la loi d’Erdős–Turán. Enfin, nous démontrons le caractère brownien pour la limite des petits cycles.

Functional limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is the infinite Karlin “balls-in-boxes” scheme with random probabilities of stick-breaking type. Assuming that the number of placed balls equals n, we prove several functional limit theorems (FLTs) in the Skorohod space D[0,1] endowed with the J1- or M1-topology for the number Kn∗(t) of boxes containing at most [nt] balls, t∈[0,1], and the random distribution function Kn∗(t)/Kn∗(1), as n→∞. The limit processes for Kn∗(t) are of the form (X(1)−X((1−t)−))t∈[0,1], where X is either a Brownian motion, a spectrally negative stable Lévy process, or an inverse stable subordinator. The small value probabilities for the stick-breaking factor determine which of the alternatives occurs. If the logarithm of this factor is integrable, the limit process for Kn∗(t)/Kn∗(1) is a Lévy bridge. Our approach relies upon two novel ingredients and particularly enables us to dispense with a Poissonization-de-Poissonization step which has been an essential component in all the previous studies of Kn∗(1). First, for any Karlin occupancy scheme with deterministic probabilities (pk)k≥1, we obtain an approximation, uniformly in t∈[0,1], of the number of boxes with at most [nt] balls by a counting function defined in terms of (pk)k≥1. Second, we prove several FLTs for the number of visits to the interval [0,nt] by a perturbed random walk, as n→∞. If the stick-breaking factor has a beta distribution with parameters θ>0 and 1, the process (Kn∗(t))t∈[0,1] has the same distribution as a similar process defined by the number of cycles of length at most [nt] in a θ-biased random permutation a.k.a. a Ewens permutation with parameter θ. As a consequence, our FLT with Brownian limit forms a generalization of a FLT obtained earlier in the context of Ewens permutations by DeLaurentis and Pittel (1985), Hansen (1990), Donnelly et al. (1991), and Arratia and Tavaré (1992).

Granular motor in the non-Brownian limit

.In this work we experimentally study a granular rotor which is similar to the famous Smoluchowski–Feynman device and which consists of a rotor with four vanes immersed in a granular gas. Each side of the vanes can be composed of two different materials, creating a rotational asymmetry and turning the rotor into a ratchet. When the granular temperature is high, the rotor is in movement all the time, and its angular velocity distribution is well described by the Brownian limit discussed in previous works. When the granular temperature is lowered considerably we enter the so-called single kick limit, where collisions occur rarely and the unavoidable external friction causes the rotor to be at rest for most of the time. We find that the existing models are not capable of adequately describing the experimentally observed distribution in this limit. We trace back this discrepancy to the non-constancy of the deceleration due to external friction and show that incorporating this effect into the existing models leads to full agreement with our experiments. Subsequently, we extend this model to describe the angular velocity distribution of the rotor for any temperature of the gas, and obtain a very good agreement between the model and experimental data.

A note on a Poissonian functional and a $q$-deformed Dufresne identity

In this note, we compute the Mellin transform of a Poissonian exponential functional, the underlying process being a simple continuous time random walk. It shows that the Poissonian functional can be expressed in term of the inverse of a $q$-gamma random variable. The result interpolates between two known results. When the random walk has only positive increments, we retrieve a theorem due to Bertoin, Biane and Yor. In the Brownian limit ($q \rightarrow 1^-$), one recovers Dufresne's identity involving an inverse gamma random variable. Hence, one can see it as a $q$-deformed Dufresne identity.

The nonequilibrium glassy dynamics of self-propelled particles.

We study the glassy dynamics taking place in dense assemblies of athermal active particles that are driven solely by a nonequilibrium self-propulsion mechanism. Active forces are modeled as an Ornstein-Uhlenbeck stochastic process, characterized by a persistence time and an effective temperature, and particles interact via a Lennard-Jones potential that yields well-studied glassy behavior in the Brownian limit, which is obtained as the persistence time vanishes. By increasing the persistence time, the system departs more strongly from thermal equilibrium and we provide a comprehensive numerical analysis of the structure and dynamics of the resulting active fluid. Finite persistence times profoundly affect the static structure of the fluid and give rise to nonequilibrium velocity correlations that are absent in thermal systems. Despite these nonequilibrium features, for any value of the persistence time we observe a nonequilibrium glass transition as the effective temperature is decreased. Surprisingly, increasing departure from thermal equilibrium is found to promote (rather than suppress) the glassy dynamics. Overall, our results suggest that with increasing persistence time, microscopic properties of the active fluid change quantitatively, but the general features of the nonequilibrium glassy dynamics observed with decreasing the effective temperature remain qualitatively similar to those of thermal glass-formers.

Novel MOEMS Lorentz Force Transducer for Magnetic Fields

This contribution describes a novel magnetic field transducer based on a MOEMS (micro-opto-electo-mechanical-system) readout. The silicon structure is deflected in a static magnetic field due to the Lorentz force This deflection is measured with the mechano-optical transducer [1]. The transduction method is based on the modulation of a perpendicularly introduced light flux. The modulation is achieved by one movable optical grating on a suspended structure and a second grating that is fixed to the foundation of the system. The proof-of-concept device was characterized at ambient pressure exhibiting a sensitivity of 200 mV/T. The noise equivalent magnetic resolution limit is 1.5 μT/√Hz which results mainly from the opto-electrical evaluation circuit. By introducing advanced opto-electronics and using improved MEMS devices it is feasible to reduce this value down to 1 nT/√Hz which is equivalent to the fundamental mechanical Brownian noise limit.

Random walks on weighted, oriented percolation clusters

Abstract. We consider a weighted random walk on the backbone of an oriented percolation cluster. We determine necessary conditions on the weights for Brownian scaling limits under the annealed and the quenched law. This model is a random walk in dynamic random environment (RWDRE), where the environment is mixing, non-Markovian and not elliptic. We provide a generalization of results obtained previously by Birkner et al. (2013).

Generalized fractional Lévy processes with fractional Brownian motion limit

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

Generalized fractional Lévy processes with fractional Brownian motion limit

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

Run-and-tumble particles, telegrapher’s equation and absorption problems with partially reflecting boundaries

Absorption problems of run-and-tumble particles, described by the telegrapher's equation, are analyzed in one space dimension considering partially reflecting boundaries. Exact expressions for the probability distribution function in the Laplace domain and for the mean time to absorption are given, discussing some interesting limits (Brownian and wave limit, large volume limit) and different case studies (semi-infinite segment, equal and symmetric boundaries, totally/partially reflecting boundaries).

Convex hulls of random walks: Large-deviation properties.

We study the convex hull of the set of points visited by a two-dimensional random walker of T discrete time steps. Two natural observables that characterize the convex hull in two dimensions are its perimeter L and area A. While the mean perimeter 〈L〉 and the mean area 〈A〉 have been studied before, analytically and numerically, and exact results are known for large T (Brownian motion limit), little is known about the full distributions P(A) and P(L). In this paper we provide numerical results for these distributions. We use a sophisticated large-deviation approach that allows us to study the distributions over a larger range of the support, where the probabilities P(A) and P(L) are as small as 10(-300). We analyze (open) random walks as well as (closed) Brownian bridges on the two-dimensional discrete grid as well as in the two-dimensional plane. The resulting distributions exhibit, for large T, a universal scaling behavior (independent of the details of the jump distributions) as a function of A/T and L/√[T], respectively. We are also able to obtain the rate function, describing rare events at the tails of these distributions, via a numerical extrapolation scheme and find a linear and square dependence as a function of the rescaled perimeter and the rescaled area, respectively.

Time-driven quantum master equations and their compatibility with the fluctuation dissipation theorem

We contribute to a long-standing debate on the supposed failure of the fluctuation dissipation theorem (FDT) for the Davies master equation (DME), an important class of Lindblad quantum master equations, describing time-driven quantum systems weakly coupled to a heat bath. First we propose two simple and natural criteria on the driving which guarantee compatibility with the FDT. We show through our setting that, contrary to what is often stated in the literature, the DME is fully compatible with the FDT. We thus argue that the cause of the dispute lies in the adopted perturbation scheme, rather than in the Lindblad character of the master equation itself. We confirm our statement by proving that the Grabert master equation, first proposed by Grabert [Projection Operator Techniques in Nonequilibrium Statistical Mechanics (Springer, Berlin, 1982)] as an alternative linear dynamics fulfilling the FDT, is nothing else than the incriminated DME. Our criteria for the FDT can also be used in the analysis of the nonlinear thermodynamical master equation, first obtained in the Brownian motion limit [H. Grabert, Z. Phys. B 49, 161 (1982)] and later independently rediscovered and generalized on purely thermodynamic grounds [H. C. \"Ottinger, Europhys. Lett. 94, 10006 (2011)].

Reinforced Brownian Motion: A Prototype

We analyze the Brownian Motion limit of a prototypical unit step reinforced random-walk on the half-line. A reinfoced random walk is one which changes the weight of any edge (or vertex) visited to increase the frequency of return visits. The generating function for the discrete case is first derived for the joint probability distribution of \(S_N\) (the location of the walker at the \(N^{th}\) step) and \(A_N\), the maximum location the walker achieved in \(N\) steps. Then the bulk of the analysis concerns the statistics of the limiting Brownian walker, and of its “environment”, both parametrized by the amplitude \(\delta \) of the reinforcement. The walker marginal distribution can be interpreted as that of free diffusion with a source serving as a diffusing soft confinement, details depending very much on the value of \(-1< \delta < \infty \).

Sub-hertz-linewidth diode laser stabilized to an ultralow-drift high-finesse optical cavity

An external cavity diode laser stabilized to a high-finesse rigid cavity made of ultralow-expansion (ULE) glass maintained at a zero-crossing temperature of −3.3 °C showed a linear frequency drift of 25 mHz/s, which was the lowest ever reported with ULE glass. The linewidth of the beat note between two independent laser systems stabilized to independent ULE glass cavities was narrower than 1 Hz, and the Allan deviation of the beat note around 1 s of averaging time was close to the Brownian thermomechanical noise limit.

Time Correlation Functions of Brownian Motion and Evaluation of Friction Coefficient in the Near-Brownian-Limit Regime

The exponentially decaying behavior of the momentum-momentum and momentum-force time correlation functions of Brownian motion at large times has been extensively used for the numerical evaluation of the friction coefficient $\gamma$ from molecular dynamics simulations. We perform numerical analysis on these methods and address issues related to the appropriate choice of large time and the rate of convergence of these methods. To this end, we obtain asymptotic expansions of the time correlation functions with respect to the reduced mass $\mu$ of the Brownian particle. For two important limit procedures of achieving the Brownian limit, certain forms of the asymptotic expansion of the Mori memory function $K(t)$ are introduced by physical arguments, and then the asymptotic expansions of the time correlation functions are expressed in terms of the limit of $K(t)$, i.e., $K_0(t)=\lim_{\mu\rightarrow\infty}K(t)$, and the next-order correction term $K_1(t)$. For the infinite mass limit case, we show that the num...

What Exactly does a Random Walk Simulation Simulate?

One of the most common statements about random walks or Brownian motion is that they are statistically self-similar. Self-similarity is commonly depicted as an individual trajectory at different magnifications or plots of square displacement versus time at different magnifications. The first question is, what exactly does self-similarity mean here? That the trajectory itself and certain functionals of it are self-similar? That all functionals of it are self-similar? That all functionals of a random walk are self-similar in the limit as the random walk approaches Brownian motion? This problem is a form of the standard question of how a random walk (finite step length) approaches the limit of Brownian motion (infinitesimal step length). This problem is well understood mathematically but for applications it is useful to understand how the approach to the Brownian limit affects simulations. When does the histogram of a functional of a random walk approximate the corresponding distribution for Brownian motion? Second, what exactly does a random walk simulation yield? In the simulation the step size is chosen from a Rayleigh distribution with mean-square radius = 2d D₀ Δt, and the angle is random. This is exactly what the propagator for Brownian motion yields for a time interval Δt. Thus the usual random walk simulation is periodically sampled Brownian motion. Periodic sampling — in engineering terminology, downsampling — is a form of low-pass filtering. Likewise, an experimental single-particle tracking trajectory is to a first approximation periodically sampled Brownian motion, but there are deviations. At short distances, the dynamics is Newtonian, not Brownian, and at longer distances, complex behavior may occur, such as the Alder-Wainwright long-time tails from hydrodynamics, and the collective motions found by Vattulainen and collaborators. Supported in part by NIH grant GM038133.

Joint temporal and contemporaneous aggregation of random-coefficient AR(1) processes

We discuss joint temporal and contemporaneous aggregation of N independent copies of AR(1) process with random-coefficient a∈[0,1) when N and time scale n increase at different rate. Assuming that a has a density, regularly varying at a=1 with exponent −1<β<1, different joint limits of normalized aggregated partial sums are shown to exist when N1/(1+β)/n tends to (i) ∞, (ii) 0, (iii) 0<μ<∞. The limit process arising under (iii) admits a Poisson integral representation on (0,∞)×C(R) and enjoys ‘intermediate’ properties between fractional Brownian motion limit in (i) and sub-Gaussian limit in (ii).

Suppressed Dispersion for a Randomly Kicked Quantum Particle in a Dirac Comb

I study a model for a massive one-dimensional particle in a singular periodic potential that is receiving kicks from a gas. The model is described by a Lindblad equation in which the Hamiltonian is a Schrödinger operator with a periodic δ-potential and the noise has a frictionless form arising in a Brownian limit. I prove that an emergent Markov process in an adiabatic limit governs the momentum distribution in the extended-zone scheme. The main result is a central limit theorem for a time integral of the momentum process, which is closely related to the particle’s position. When normalized by \(t^{\frac{5}{4}}\) the integral process converges to a time-changed Brownian motion whose diffusion rate depends on the momentum process. The scaling \(t^{\frac{5}{4}}\) contrasts with \(t^{\frac{3}{2}}\), which would be expected for the case of a smooth periodic potential or for a comparable classical process. The difference is a wave effect driven by momentum reflections that occur when the particle’s momentum is kicked near the half-spaced reciprocal lattice of the potential.

Brownian limits, local limits and variance asymptotics for convex hulls in the ball

Schreiber and Yukich [Ann. Probab. 36 (2008) 363–396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^{d}$, $d\geq 2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the $k$-face and intrinsic volume functionals.