We consider the limiting behavior of fluctuations of small noise diffusions with multiple scales around their homogenized deterministic limit. We allow full dependence of the coefficients on the slow and fast motion. These processes arise naturally when one is interested in short time asymptotics of multiple scale diffusions. We do not make periodicity assumptions, but we impose conditions on the fast motion to guarantee ergodicity. Depending on the order of interaction between the fast scale and the size of the noise we get different behavior. In certain cases additional drift terms arise in the limiting process, which are explicitly characterized. These results provide a better approximation to the limiting behavior of such processes when compared to the law of large numbers homogenization limit.
We investigate the asymptotic fluctuation of three interacting particle systems: the geometric q-TASEP, the geometric q-PushTASEP and the q-PushASEP. We prove that the rescaled particle position converges to the GUE Tracy-Widom distribution in the homogeneous case. If the jump rates of the first finitely many particles are perturbed in the first two models, we obtain that the limiting fluctuations are governed by the Baik-Ben Arous-Peche distribution and that of the top eigenvalue of finite GUE matrices.
Consider a supercritical Crump--Mode--Jagers process such that all births are at integer times (the lattice case). Let $\widehatμ(z)$ be the generating function of the intensity of the offspring process, and consider the complex roots of $\widehatμ(z)=1$. The smallest (in absolute value) such root is $e^{-α}$, where $α>0$ is the Malthusian parameter; let $γ_*$ be the second smallest absolute value of a root. We show, assuming some technical conditions, that there are three cases: (i) if $γ_*>e^{-α/2}$, then the second-order fluctuations of the age distribution are asymptotically normal; (ii) i…
A cyclic urn is an urn model for balls of types $0,\ldots,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is $j$, it is then returned to the urn together with a new ball of type $j+1 \mod m$. The case $m=2$ is the well-known Friedman urn. The composition vector, i.e., the vector of the numbers of balls of each type after $n$ steps is, after normalization, known to be asymptotically normal for $2\le m\le 6$. For $m\ge 7$ the normalized composition vector is known n…
Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph $G$, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph the asymptotic shape is known to be a ball in the usual graph metric. In this paper we establish bounds for the fluctuations of the cluster from its asymptotic shape.
We consider the asymptotics of the difference between the empirical measures of the $β$-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik-Kerov-Logan-Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.
A diffusion process associated with the real sub-Laplacian $Δ_b$, the real part of the complex Kohn-Spencer Laplacian $\square_b$, on a strictly pseudoconvex CR manifold has been constructed. In this paper, we investigate diagonal short time asymptotics of the heat kernel corresponding to the diffusion process by using Watanabe's asymptotic expansion and give a representation for the asymptotic expansion of heat kernels which shows a relationship to the geometric structure.
This paper presents estimates for the distribution of the exit time from balls and short time asymptotics for measure metric Dirichlet spaces. The estimates cover the classical Gaussian case, the sub-diffusive case which can be observed on particular fractals and further less regular cases as well. The proof is based on a new chaining argument and it is free of volume growth assumptions.
We impose the uniform probability measure on the set of all discrete Gelfand-Tsetlin patterns of depth $n$ with the particles on row $n$ in deterministic positions. These systems equivalently describe a broad class of random tilings models, and are closely related to the eigenvalue minor processes of a broad class of random Hermitian matrices. They have a determinantal structure, with a known correlation kernel. We rescale the systems by $\frac1n$, and examine the asymptotic behaviour, as $n \to \infty$, under weak asymptotic assumptions for the (rescaled) particles on row $n$: The empirical d…
In this paper we study short time asymptotics of a density function of the solution of a stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H \in (1/2, 1)$ when the coefficient vector fields satisfy an ellipticity condition at the starting point. We prove both on-diagonal and off-diagonal asymptotics under mild additional assumptions. Our main tool is Malliavin calculus, in particular, Watanabe's theory of generalized Wiener functionals.
We write $R_n$ for the minimal position attained after time $n$ by a branching random walk in the boundary case. In this article, we prove that $R_n - \frac{1}{2} \log n$ converges in law toward a shifted Gumbel distribution.
We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite ($ \mathbb{Z}_+ $-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on $ \mathbb{R}_+ $. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-1/4}$, converges as $t \to \infty$ to the Gaussian field corresponding to the solution of the additive stochastic heat equation on $\mathbb{R}_+$ with Neumann boundary condition at zero. This allows us to express the asy…
We investigate the asymptotic behaviour of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctutations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resol…
We study a system of $n$ differential equations, each in dimension $d$. Only the first equation is forced by a Brownian motion and the dependence structure is such that, under a local weak Hörmander condition, the noise propagates to the whole system. We prove upper bounds for the transition density (heat kernel) and its derivatives of any order. Then we give precise short-time asymptotics of the density at a suitable central limit time scale. Both these results account for the different non-diffusive scales of propagation in the various components. Finally, we provide a valuation formula for …
We study the short-time asymptotical behavior of stochastic flows on \mathbb{R} in the \sup-norm. The results are stated in terms of a Gaussian process associated with the covariation of the flow. In case the Gaussian process has a continuous version the two processes can be coupled in such a way that the difference is uniformly $o(\ln\ln t^{-1})$. In case it has no continuous version, an $O(\ln\ln t^{-1})$ estimate is obtained under mild regularity assumptions. The main tools are Gaussian measure concentration and a martingale version of the Slepian comparison principle.
We investigate the long-time asymptotics of the fluctuation SPDE in the Kuramoto synchronization model. We establish the linear behavior for large time and weak disorder of the quenched limit fluctuations of the empirical measure of the particles around its McKean-Vlasov limit. This is carried out through a spectral analysis of the underlying unbounded evolution operator, using arguments of perturbation of self-adjoint operators and analytic semigroups. We state in particular a Jordan decomposition of the evolution operator which is the key point in order to show that the fluctuations of the d…
We are concerned with an $M/M$-type join the shortest queue ($M/M$-JSQ for short) with $k$ parallel queues for an arbitrary positive integer $k$, where the servers may be heterogeneous. We are interested in the tail asymptotic of the stationary distribution of this queueing model, provided the system is stable. We prove that this asymptotic for the minimum queue length is exactly geometric, and its decay rate is the $k$-th power of the traffic intensity of the corresponding $k$ server queues with a single waiting line. For this, we use two formulations, a quasi-birth-and-death (QBD for short) …
We consider random walks on quasi one dimensional lattices, as introduced in \cite{FS}. This mathematical setting covers a large class of discrete kinetic models for non-cooperative molecular motors on periodic tracks. We derive general formulas for the asymptotic velocity and diffusion coefficient, and we show how to reduce their computation to suitable linear systems of the same degree of a single fundamental cell, with possible linear chain removals. We apply the above results to special families of kinetic models, also catching some errors in the biophysics literature.
We study a stochastic differential equation in the sense of rough path theory driven by fractional Brownian rough path with Hurst parameter H (1/3 < H <= 1/2) under the ellipticity assumption at the starting point. In such a case, the law of the solution at a fixed time has a kernel, i.e., a density function with respect to Lebesgue measure. In this paper we prove a short time off-diagonal asymptotic expansion of the kernel under mild additional assumptions. Our main tool is Watanabe's distributional Malliavin calculus.
Let $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$, let $\cF$ be a bounded set of real-valued functions on $[0,1]$ with finite $q$-variation. It is proved that $\{n^{-1/2}\sum_{i=1}^nX_if(i/n)\colon\,f\in\cF\}$ converges in outer distribution in the Banach space of bounded functions on $\cF$ as $n\to\infty$. Several applications to a regression model and a multiple change point model are given.
By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant with respect to isometries of the complex plane. Extending the previous results of Sodin and Tsirelson, we compute the variance of linear statistics of random complex zeroes, and find close to optimal conditions on a test-function that yield asymptotic normality of fluctuations of the corresponding linear statistics. We also provide examples of test-functio…
We consider the problem of the error in soliton transmission in long-haul optical fibers caused by the spontaneous emission of noise inherent to amplification. We study two types of noises driving the stochastic focusing cubic one dimensional nonlinear Schrödinger equation which appears in physics in that context. We focus on the fluctuations of the mass and arrival time or timing jitter. We give the small noise asymptotic of the tails of these two quantities for the two types of noises. We are then able to prove several results from physics among which the Gordon--Haus effect which states tha…
We study asymptotics of representations of the unitary groups U(n) in the limit as n tends to infinity and we show that in many aspects they behave like large random matrices. In particular, we prove that the highest weight of a random irreducible component in the Kronecker tensor product of two irreducible representations behaves asymptotically in the same way as the spectrum of the sum of two large random matrices with prescribed eigenvalues. This agreement happens not only on the level of the mean values (and thus can be described within Voiculescu's free probability theory) but also on the…
We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of \textit{out-of-the-money} options is found to be linear in time, the short time asymptotics of \textit{at-the-money} options is shown to depend on the fine structure of the semimartingale.
We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale $N^{2/3}$. Moreover, given several generic positive edges, we establish that the associated extremal ei…
Diffusion processes with small noise conditioned to reach a target set are considered. The AMS algorithm is a Monte Carlo method that is used to sample such rare events by iteratively simulating clones of the process and selecting trajectories that have reached the highest value of a so-called importance function. In this paper, the large sample size relative variance of the AMS small probability estimator is considered. The main result is a large deviations logarithmic equivalent of the latter in the small noise asymptotics, which is rigorously derived. It is given as a maximisation problem e…
We present a model that investigates the spontaneous emergence of randomness in equity market microstructure. The phase space analysis of our model exposes an endogenous source of fluctuation in price and volume. We formulate a control problem for maximizing price regularity and stability while minimizing entanglement with the market, representing the NYSE specialists' affirmative obligation to maintain `fair and orderly markets'.
A description of the short time behavior of solutions of the Allen-Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki in spatial dimension $n=2$ to arbitrary dimensions.
A small-time Edgeworth expansion of the density of an asset price is given under a general stochastic volatility model, from which asymptotic expansions of put option prices and at-the-money implied volatilities follow. A limit theorem for at-the-money implied volatility skew and curvature is also given as a corollary. The rough Bergomi model is treated as an example.
We prove that the fluctuations of mesocopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a Central Limit Theorem for…
We consider a model for interest rates, where the short rate is given by a time-homogenous, one-dimensional affine process in the sense of Duffie, Filipovic and Schachermayer. We show that in such a model yield curves can only be normal, inverse or humped (i.e. endowed with a single local maximum). Each case can be characterized by simple conditions on the present short rate. We give conditions under which the short rate process will converge to a limit distribution and describe the limit distribution in terms of its cumulant generating function. We apply our results to the Vasicek model, the …
We consider a new interacting particle system on the one-dimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any space-like path are described by the Airy_1 and Airy_2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's …
Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This implies universality of mesoscopic fluctuations for a large class of unitary invariant Hermitian ensembles. In particular, this shows that the support of the equilibrium measure need not be connected in order to see Gaussian fluctuations at mesoscopic scales. Our proof is based on the cumulants computations introduced in [48] for the CUE and the sine process and the asymptotics formulae derived by…
A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide variety of laws of fluctuation are possible, beyond the already known cases, including for instance Gumbel and exponential laws at unusual speeds. We study the convergence in law of the spectral radi…
