We prove the Large Deviation Principle for the empirical process in a system of locally interacting Brownian motions in the nonequilibrium dynamic. Such a phenomenon has been proven only for two lattice systems: the symmetric simple exclusion process and zero-range process. Therefore, we have achieved the third result in this context and moreover the first result for the diffusion-type interacting particle system.
We establish the large deviation principle (LDP) for stochastic flows of interacting Brownian motions. In particular, we consider smoothly correlated flows, coalescing flows and Brownian motion stopped at a hitting moment.
We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.
We prove large deviations principles in large time, for the Brownian occupation time in random scenery. The random scenery is constant on unit cubes, and consist of i.i.d. bounded variables, independent of the Brownian motion. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched'' and ``annealed'' settings.
We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
We find exact small deviation asymptotics with respect to weighted Hilbert norm for some well-known Gaussian processes. Our approach does not require the knowledge of eigenfunctions of the covariance operator of a weighted process. Such a peculiarity of the method makes it possible to generalize many previous results in this area. We also obtain new relations connected to exact small deviation asymptotics for a Brownian excursion, a Brownian meander, and Bessel processes and bridges.
We study small noise large deviation asymptotics for stochastic differential equations with a multiplicative noise given as a fractional Brownian motion $B^H$ with Hurst parameter $H>\frac12$. The solutions of the stochastic differential equations are defined pathwise under appropriate conditions on the coefficients. The ingredients in the proof of the large deviation principle, which include a variational representation for nonnegative functionals of fractional Brownian motions and a general sufficient condition for a LDP for a collection of functionals of a fractional Brownian motions, have …
We establish a large deviation principle for the process of the largest eigenvalue of an Hermitian Brownian motion. By a contraction principle, we recover the LDP for the largest eigenvalue of a rank one deformation of the GUE.
A small ball problem and Chung's law of iterated logarithm for a hypoelliptic Brownian motion in Heisenberg group are proven. In addition, bounds on the limit in Chung's law are established.
A variational representation for functionals of G-Brownian motion is established by a finite-dimensional approximate technique. As an application of the variational representation, we obtain a large deviation principle for stochastic flows driven by G-Brownian motion.
We prove large deviations for $g(t)$-Brownian motion in a complete, evolving Riemannian manifold $M$ with respect to a collection $\{g(t)\}_{t\in [0,1]}$ of Riemannian metrics, smoothly depending on $t$. We show how the large deviations are obtained from the large deviations of the (time-dependent) horizontal lift of $g(t)$-Brownian motion to the frame bundle $FM$ over $M$. The latter is proved by embedding the frame bundle into some Euclidean space and applying Freidlin-Wentzell theory for diffusions with time-dependent coefficients, where the coefficients are jointly Lipschitz in space and t…
Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.
We use Mellin transforms to compute a full asymptotic expansion for the tail of the Laplace transform of the squared $L^2$-norm of any multiply-integrated Brownian sheet. Through reversion we obtain corresponding strong small-deviation estimates.
We prove the the large deviation principle(LDP) for the law of the one-dimensional semilinear stochastic partial differential equations driven by nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of global solution. Then the large deviation principle is obtained via weak convergence method.
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
A quenched large deviation principle for Brownian motion in a non-negative, stationary potential is proved. A sufficient moment condition on the potential is given but unlike the results of Armstrong and Tran (2014) no regularity is assumed. The proof is based on a method developed by Sznitman (1994) for Brownian motion among Poissonian potential. In particular, the LDP holds for potentials with polynomially decaying correlations such as the classical potentials studied by L. Pastur (1977) and R. Fukushima (2008) and the potentials recently introduced by H. Lacoin (2012).
In this paper, we consider forward-backward stochastic differential equation driven by $G$-Brownian motion ($G$-FBSDEs in short) with small parameter $\varepsilon > 0$. We study the asymptotic behavior of the solution of the backward equation and establish a large deviation principle for the corresponding process.
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes. We also show that a fractional Brownian motion and the related Riemann-Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of our large deviation estimates, we derive laws of iterated logarithm for the corresponding local times. The key points of our methods: (1) logarithmic superadditivity of a normalized sequence of moments of…
We study deviation probabilities for the number of high positioned particles in branching Brownian motion, and confirm a conjecture of Derrida and Shi (2016). We also solve the corresponding problem for the two-dimensional discrete Gaussian free field. Our method relies on an elementary inequality for inhomogeneous Galton-Watson processes.
Conditioning a branching Brownian motion to have an atypically low maximum leads to a suppression of the branching mechanism. In this note, we consider a branching Brownian motion conditioned to have a maximum below $\sqrt{2}αt$ ($α<1$). We study the precise effects of an early/late first branching time and a low/high first branching location under this condition. We do so by imposing additional constraints on the first branching time and location. We obtain large deviation estimates, as well as the optimal first branching time and location given the additional constraints.
We compute a closed-form expression for the moment generating function $\hat{f}(x;λ,α)=\frac{1}λ\mathbb{E}_x(e^{αL_τ})$, where $L_t$ is the local time at zero for standard Brownian motion with reflecting barriers at $0$ and $b$, and $τ\sim \mathrm{Exp}(λ)$ is independent of $W$. By analyzing how and where $\hat{f}(x;\cdot,α)$ blows up in $λ$, a large-time large deviation principle (LDP) for $L_t/t$ is established using a Tauberian result and the Gärtner-Ellis Theorem.
Particle approximations for certain nonlinear and nonlocal reaction-diffusion equations are studied using a system of Brownian motions with killing. The system is described by a collection of i.i.d. Brownian particles where each particle is killed independently at a rate determined by the empirical sub-probability measure of the states of the particles alive. A large deviation principle (LDP) for such sub-probability measure-valued processes is established. Along the way a convenient variational representation, which is of independent interest, for expectations of nonnegative functionals of Br…
Consider the intersection measure $\ell^{\mathrm{IS}}_t$ of $p$ independent Brownian motions on $\mathbb{R}^d$. In this article, we prove the large deviation principle for the normalized intersection measure $t^{-p}\ell^{\mathrm{IS}}_t$ as $t\rightarrow \infty$, before exiting a (possibly unbounded) domain $D\subset\mathbb{R}^d$ with smooth boundary. This is an extension of [W. König and C. Mukherjee: Communications on Pure and Applied Mathematics, 66(2):263--306, 2013] which deals with the case $D$ is bounded. Our essential contribution is to prove the so-called super-exponential estimate for…
We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani's lemma and Lamperti's representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609] who studied the corresponding problem for Sinai's random walk in random environment.
In this article we establish a large deviation principle for the family {ν_ε:ε\in (0,1)} of distributions of the scaled stochastic processes {P_{-\log\sqrtε}Z_t}_{t\leq 1}, where (Z_t)_{t\in \lbrack 0,1]} is a square-integrable martingale over Brownian filtration and (P_t)_{t\geq 0} is the Ornstein-Uhlenbeck semigroup. The rate function is identified as well in terms of the Wiener-Itô chaos decomposition of the terminal value Z_{1}. The result is established by developing a continuity theorem for large deviations, together with two essential tools, the hypercontractivity of the Ornstein-Uhlenb…
We investigate the behaviour of a finite chain of Brownian particles, interacting through a pairwise quadratic potential, with one end of the chain fixed and the other end pulled away at slow speed, in the limit of slow speed and small Brownian noise. We study the instant when the chain "breaks", that is, the distance between two neighboring particles becomes larger than a certain limit. There are three different regimes depending on the relation between the speed of pulling and the Brownian noise. We prove weak limit theorems for the break time and the break position for each regime.
We study the asymptotic behavior of the supremum $M_t$ of the support of a supercritical super-Brownian motion. In our recent paper (Stoch. Proc. Appl. 137 (2021), 1-34), we showed that, under some conditions, $M_t-m(t)$ converges in distribution to a randomly shifted Gumbel random variable, where $m(t)=c_0t-c_1\log t$. In the same paper, we also studied the upper large deviation of $M_t$, i.e., the asymptotic behavior of $\mathbb{P}(M_t>δc_0t) $ for $δ\ge 1$. In this paper, we study the lower large deviation of $M_t$, i.e., the asymptotic behavior of $\mathbb{P}(M_t\le δc_0t|\mathcal{S}) $ fo…
We prove that self-diffusion constants of interacting Brownian particles in $ \mathbb{R}$ always vanish if the particles do not collide with each other. We represent self-diffusion constants by additive functionals of reversible Markov processes as obtained in O. 1998.
In this paper, we investigate two-sided bounds for the small ball probability of a mixed fractional Brownian motion with a general deterministic trend function, in terms of respective small ball probability of a mixed fractional Brownian motion without trend. To maximize the lower bound, we consider various ways to split the trend function between the components of the mixed fractional Brownian motion for the application of Girsanov theorem, and we show that the optimal split is the solution of a Fredholm integral equation. We find that the upper bound for the probability is also a function of…
We study the variational problem that arises from consideration of large deviations for semimartingale reflected Brownian motion (SRBM) in the positive octant. Due to the difficulty of the general problem, we consider the case in which the SRBM has rotationally symmetric parameters. In this case, we are able to obtain conditions under which the optimal solutions to the variational problem are paths that are gradual (moving through faces of strictly increasing dimension) or that spiral around the boundary of the octant. Furthermore, these results allow us to provide an example for which it can …
We consider the last zero crossing time $T_{μ,t}$ of a Brownian motion, with drift $μ\neq 0$ in the time interval $[0, t]$. We prove the large deviation principle of $\{T_{μ\sqrt r t} : r > 0 \}$ as $r$ tends to infinity. Moreover, motivated by the results on moderate deviations in the literature, we also prove a class of large deviation principles for the same random variables with different scalings, which are governed by the same rate function. Finally we compare some aspects of the classical moderate deviation results, and the results in this paper.
Consider a negatively drifted one dimensional Brownian motion starting at positive initial position, its first hitting time to 0 has the inverse Gaussian law. Moreover, conditionally on this hitting time, the Brownian motion up to that time has the law of a 3- dimensional Bessel bridge. In this paper, we give a generalization of this result to a family of Brownian motions with interacting drifts, indexed by the vertices of a conductance network. The hitting times are equal in law to the inverse of a random potential that appears in the analysis of a self-interacting process called the Vertex R…
The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field $ μ$, there exist two natural infinite-volume Dirichlet forms $ (\mathcal{E}^{\mathsf{upr}},\mathcal{D}^{\mathsf{upr}})$ and $(\mathcal{E}^{\mathsf{lwr}},\mathcal{D}^{\mathsf{lwr}})$ on $ L^2(\mathsf{S} ,μ) $ describing interacting Brownian motions each with unlabeled equilibrium state $ μ$. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet for…
We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for $n$-iterated Brownian motions and, more generally, for the iteration of $n$ fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Laws of the iterated logarithm for $α$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18, 434--459.
We study the local mass of a dyadic branching Brownian motion $Z$ evolving in $\mathbb{R}^d$. By 'local mass,' we refer to the number of particles of $Z$ that fall inside a ball with fixed radius and time-dependent center, lying in the 'subcritical' zone. Using the strong law of large numbers for the local mass of branching Brownian motion and elementary geometric arguments, we find large deviation results giving the asymptotic behavior of the probability that the local mass is atypically small on an exponential scale. As corollaries, we obtain an asymptotic result for the probability of absen…
We prove large deviation principles for $\int_0^t γ(X_s)ds$, where $X$ is a $d$-dimensional self-similar Gaussian process and $γ(x)$ takes the form of the Dirac delta function $δ(x)$, $|x|^{-β}$ with $β\in (0,d)$, or $\prod_{i=1}^d |x_i|^{-β_i}$ with $β_i\in(0,1)$. In particular, large deviations are obtained for the functionals of $d$-dimensional fractional Brownian motion, sub-fractional Brownian motion and bi-fractional Brownian motion. As an application, the critical exponential integrability of the functionals is discussed.
We study the lower deviation probability of the position of the rightmost particle in a branching Brownian motion and obtain its large deviation function
Fisher-KPP equation is proved to be the scaling limit of a system of Brownian particles with local interaction. Particles proliferate and die depending on the local concentration of other particles. Opposite to discrete models, controlling concentration of particles is a major difficulty in Brownian particle interaction; local interactions instead of mean field or moderate ones makes it more difficult to implement the law of large numbers properties. The approach taken here to overcome these difficulties is largely inspired by A. Hammond and F. Rezakhanlou [10] implemented there in the mean fr…
Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula which for an infinite sequence of i.i.d real Brownian motions and a Poisson random measure was shown in [5].
Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0\}$ be the local time process at zero of a strictly stable Lévy process $X=\{X_t, t \ge 0\}$ of index $1<α\leq 2$ independent of $W^H$. The $\a$-stable local time fractional Brownian motion $Z^H=\{Z^H(t), t \ge 0\}$ is defined by $Z^H(t) = W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1 - \frac 1 α)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coup…
For a fractional Brownian motion $B^H$ with Hurst parameter $H\in]{1/4},{1/2}[\cup]{1/2},1[$, multiple indefinite integrals on a simplex are constructed and the regularity of their sample paths are studied. Then, it is proved that the family of probability laws of the processes obtained by replacing $B^H$ by $ε^{1/2} B^H$ satisfies a large deviation principle in Hölder norm. The definition of the multiple integrals relies upon a representation of the fractional Brownian motion in terms of a stochastic integral with respect to a standard Brownian motion. For the large deviation principle, the a…
We are interested in the analysis of Gibbs measures defined on two independent Brownian paths in $\mathbb R^d$ interacting through a mutual self-attraction. This is expressed by the Hamiltonian $\int\int_{\mathbb R^{2d}} V(x-y) μ(d x)ν(d y)$ with two probability measures $μ$ and $ν$ representing the occupation measures of two independent Brownian motions. We will be interested in class of potentials $V$ which are {\it{singular}}, e.g., Dirac or Coulomb type interactions in $\mathbb R^3$, or the correlation function of the parabolic Anderson problem with white noise potential. The mutual intera…
We consider a finite or countable collection of one-dimensional Brownian particles whose dynamics at any point in time is determined by their rank in the entire particle system. Using Transportation Cost Inequalities for stochastic processes we provide uniform fluctuation bounds for the ordered particles, their local time of collisions, and various associated statistics over intervals of time. For example, such processes, when exponentiated and rescaled, exhibit power law decay under stationarity; we derive concentration bounds for the empirical estimates of the index of the power law over lar…
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
We consider a system of interacting Brownian particles in R^d with a pairwise potential, which is radially symmetric, of finite range and attains a unique minimum when the distance of two particles becomes a>0. The asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects. If the system is rigidly crystallized, namely if the particles are rigidly arranged in an equal distance a, the crystallization is kept under the evolution in macroscopic time scale. Then, assuming that the crystal has a definite limit shape under a macroscopi…
We study the zero temperature limit for interacting Brownian particles in one dimension with a pairwise potential which is of finite range and attains a unique minimum when the distance of two particles becomes a>0. We say a chain is formed when the particles are arranged in an ``almost equal'' distance a. If a chain is formed at time 0, so is for positive time as the temperature of the system decreases to 0 and, under a suitable macroscopic space-time scaling, the center of mass of the chain performs the Brownian motion with the speed inversely proportional to the total mass. If there are two…
