In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $\mathcal {M}_1(\R^d)$ can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $\R^d$ by adding a point at $\infty$ that results in the compactification of $\mathcal M_1(\R^d)$ by allowing some mass to escape to the point at $\infty$. If one were to use only test functions that are continuous and vanish at $\infty$ then the compactification results in the space of sub-probability distributions $\mathcal {M}_{\le 1}(\R^d)$ by ignoring the mass at $\infty$.
The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $\widetilde{\mathcal M}_1=\widetilde{\mathcal M}_1(\R^d)$ under the action of the translation group $\R^d$ on $\mathcal M_1(\R^d)$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.
We study properties of occupation times by Brownian excursions and Brownian loops in two-dimensional domains. This allows for instance to interpret some Gaussian fields, such as the Gaussian Free Fields as (properly normalized) fluctuations of the total occupation time of a Poisson cloud of Brownian excursions when the intensity of the cloud goes to infinity.
We prove a conditional local limit theorem for discrete-time fractional Brownian motions (dfBm) with Hurst parameter 3/4<H<1. Using results from infinite ergodic theory it is then shown that the properly scaled occupation time of dfBm converges to a Mittag-Leffler distribution.
We derive the asymptotic behavior of the occupation measure of the unit ball, for super-Brownian motion started from the Dirac measure at a distant point x and conditioned to hit the unit ball. In the critical dimension d=4, we obtain a limiting exponential distribution for the ratio of the occupation measure over log(|x|).
We prove second order limit laws for (additive) functionals of the $d$-dimensional fractional Brownian motion with Hurst index $H=\frac{1}{d}$, using the method of moments, extending the Kallianpur-Robbins law.
Sticky Brownian motion on the real line can be obtained as a weak solution of a system of stochastic differential equations. We find the conditional distribution of the process given the driving Brownian motion, both at an independent exponential time and at a fixed time t>0. As a classical problem, we find the distribution of the occupation times of a half-line, and at 0, which is the sticky point for the process.
A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish limit theorems on local and occupation times in n steps.
In this paper we establish the existence of a square integrable occupation density for two classes of stochastic processes. First we consider a Gaussian process with an absolutely continuous random drift, and secondly we handle the case of a (Skorohod) integral with respect to the fractional Brownian motion with Hurst parameter $H>\frac 12$. The proof of these results uses a general criterion for the existence of a square integrable local time, which is based on the techniques of Malliavin calculus.
We derive the probability density function of the positive occupation time of one-dimensional Brownian motion with two-valued drift. Long time asymptotics of the density are also computed. We use the result to describe the transitional probability density function of a general N-dimensional system of stochastic differential equations representing stochastically perturbed sliding motion of a discontinuous, piecewise-smooth vector field on short time frames. A description of the density at larger times is obtained via an asymptotic expansion of the Fokker-Planck equation.
An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$, i.e., the occupation time of the union of the first and third quadr…
We study the transformed path measure arising from the self-interaction of a three-dimensional Brownian motion via an exponential tilt with the Coulomb energy of the occupation measures of the motion by time $t$. The logarithmic asymptotics of the partition function were identified in the 1980s by Donsker and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new technique for studying the path measure itself was introduced, which allows for proving that the normalized occupation measure asymptotically concentrates around the set of all maximizers of the formula. In the pre…
We are making corrections and acknowledging colleagues that pointed out mistakes in our recent paper titled "Occupation and local times for skew Brownian motion with applications to dispersion across an interface" which was published in Annals of Applied Probability (2011) 21(1) 183-214. Specifically the corrections are: 1. The restriction of $γ$ to non negative values in Theorem 1.3 is not needed. But one has probabilistic interpretation only when $γ$ is non negative. 2. State the correct formulas in Corollary 3.3 as their were computational errors in the original formulas. We thank Pierre Et…
Stochastic processes time-changed by an inverse subordinator have been suggested as a way to model the price of assets in illiquid markets, where the jumps of the subordinator correspond to periods of time where one is unable to sell an asset. We develop an excursion theory for time-changed reflected Brownian motion and use this to express the price of certain European options with Parisian barrier condition in terms of solutions of a time-fractional PDE. We provide a general description of the occupation measures of time-changed processes and use this to prove a Ray-Knight theorem for the occ…
We prove a geometric extension of Lévy's occupation time arcsine law near a hypersurface on a Riemannian manifold. The deviation from the classic arcsine law is of the order of the square-root of the time horizon and is expressed explicitly in terms of the mean curvature of the hypersurface and the local time of the underlying standard Brownian motion.
Tied-down renewal processes are generalisations of the Brownian bridge, where an event (or a zero crossing) occurs both at the origin of time and at the final observation time $t$. We give an analytical derivation of the two-time correlation function for such processes in the Laplace space of all temporal variables. This yields the exact asymptotic expression of the correlation in the Porod regime of short separations between the two times and in the persistence regime of large separations. We also investigate other quantities, such as the backward and forward recurrence times, as well as the …
Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2 πN 1_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly, as $N$ tends to infinity, towards the occupation measure of the Brownian motion.
Let $B$ denote the range of the Brownian motion in $\mathbb{R}^{d}$ ($d\geq3$). For a deterministic Borel measure $ν$ on $\mathbb{R}^{d}$ we wish to find a random measure $μ$ such that the support of $μ$ is contained in $B$ and it is a solution to the equation $E(μ(A))=ν(A)$ for every Borel set $A$. We discuss when it is possible to find a solution $μ$ and in that case we construct the solution. We study several properties of $μ$ such as the probability of $μ\neq0$ and we establish a formula for the expectation of the double integral with respect to $μ\timesμ$. We calculate $μ$ in terms of the…
In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Leão, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian pa…
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 prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function $g(r)=r^4 (\log \log1/r)^{-3}$ in super-critical dimensions $d\geq 5$. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the $g$-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension $d$.
The strong $L^2$-approximation of occupation time functionals is studied with respect to discrete observations of a $d$-dimensional càdlàg process. Upper bounds on the error are obtained under weak assumptions, generalizing previous results in the literature considerably. The approach relies on regularity for the marginals of the process and applies also to non-Markovian processes, such as fractional Brownian motion. The results are used to approximate occupation times and local times. For Brownian motion, the upper bounds are shown to be sharp up to a log-factor.
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 prove functional limits theorems for the occupation time process of a system of particles moving independently in $R^d$ according to a symmetric $α$-stable Lévy process, and starting off from an inhomogeneous Poisson point measure with intensity measure $μ(dx)=(1+|x|^γ)^{-1}dx,γ>0$, and other related measures. In contrast to the homogeneous case $(γ=0)$, the system is not in equilibrium and ultimately it vanishes, and there are more different types of occupation time limit processes depending on arrangements of the parameters $γ, d$ and $α$. The case $γ<d<α$ leads to an extension of fractio…
Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over the whole real line such that B_i(0)=0. We consider the nth iterated Brownian motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of processes (W_n) do not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W_n converge towards a random probability measure μ_\infty. We then prove that μ_\infty almost surely has a continuous density which must be thought of as the local time process of the infi…
Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In…
Classical arcsine law states that fraction of occupation time on the positive or the negative side in Brownian motion does not converge to a constant but converges in distribution to the arcsine distribution. Here, we consider how a preparation of the system affects the arcsine law, i.e., aging of the arcsine law. We derive aging distributional theorem for occupation time statistics in Brownian motion, where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution. Furthermore, we show that this result can be general…
Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable function. We are interested in finding conditions on $X$ and $f$ which imply that the perpetual integral functional $$ I^X_\infty(f):=\int_0^\infty f(X_t) dt $$ is identical in law with the first hitting time of a point for some other diffusion. This phenomenon may often be explained using random time change. Because of some potential applications in mathematical finance, we are considering mainly the case when $X$ is a Brownian motion with drift $μ>0,$ denoted $\{B^{(μ)}_t: t\geq 0\},$ but it is obvious that the method pre…
We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [6] in terms of the {\it{Pekar variational formula}}, which coincides with the behavior of the partition function of the {\it{polaron problem}} under strong coupling. Based on this, in 1986 Spohn [14] made a heuristic observation that the strong coupling behavior of the polaron…
We derive a Ray-Knight type theorem for the local time process (in the space variable) of a skew Brownian motion up to an independent exponential time. It is known that the local time seen as a density of the occupation measure and taken with respect to the Lebesgue measure has a discontinuity at the skew point (in our case at zero), but the local time taken with respect to the speed measure is continuous. In this paper we discuss this discrepancy by characterizing the dynamics of the local time process in both these cases. The Ray-Knight type theorem is applied to study integral functionals o…
We consider super processes whose spatial motion is the $d$-dimensional Brownian motion and whose branching mechanism $ψ$ is critical or subcritical; such processes are called $ψ$-super Brownian motions. If $d\!>\!2\bgamma/(\bgamma\!-\!1)$, where $\bgamma\!\in\!(1,2]$ is the lower index of $ψ$ at $\infty$, then the total range of the $ψ$-super Brownian motion has an exact packing measure whose gauge function is $g(r)\! =\! (\log\log1/r) / φ^{-1} ( (1/r\log\log 1/r)^{2})$, where $φ\! =\! ψ^\prime\! \circ \! ψ^{\!-1}$. More precisely, we show that the occupation measure of the $ψ$-super Brownian…
We study the ergodicity of stochastic reaction-diffusion equation driven by subordinate Brownian motions. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution's law. These properties imply that this stochastic system admits a unique invariant measure according to Doob's and Krylov-Bogolyubov's theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $α$-stable type noises do not satisfy Freidlin-W…
Let $S_n$ be a lattice random walk with mean zero and finite variance, and let $Λ^a_n$ be its occupation measure at level $a$. In this note, we prove local limit theorems for $\Pr[S_n=x,Λ^a_n=\ell]$ and $\Pr[S_n=x|Λ^a_n=\ell]$ in the cases where $a$, $|x-a|$ and $\ell$ are either zero or at least of order $\sqrt n$. The asymptotic description of these quantities matches the corresponding probabilities for Brownian motion and its local time process. This note can be seen as a generalization of previous results by Kaigh (1975) and Uchiyama (2011). In similar fashion to these results, our method …
Let $B^{H}$ be a fractional Brownian motion in $\mathbb{R}^{d}$ of Hurst index $H\in\left(0,1\right)$, $f:\left[0,1\right]\longrightarrow\mathbb{R}^{d}$ a Borel function and $A\subset\left[0,1\right]$ a Borel set. We provide sufficient conditions for the image $(B^{H}+f)(A)$ to have a positive Lebesgue measure or to have a non-empty interior. This is done through the study of the properties of the density of the occupation measure of $(B^{H}+f)$. Precisely, we prove that if the parabolic Hausdorff dimension of the graph of $f$ is greater than $Hd$, then the density is a square integrable funct…
In this paper, we study one dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator's domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the…
Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/β$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Di…
We study the occupation measure of various sets for a symmetric transient random walk in $Z^d$ with finite variances. Let $μ^X_n(A)$ denote the occupation time of the set $A$ up to time $n$. It is shown that $\sup_{x\in Z^d}μ_n^X(x+A)/\log n$ tends to a finite limit as $n\to\infty$. The limit is expressed in terms of the largest eigenvalue of a matrix involving the Green's function of $X$ restricted to the set $A$. Some examples are discussed and the connection to similar results for Brownian motion is given.
We consider two models of random diffusion in random environment in two dimensions. The first example is the self-repelling Brownian polymer, this describes a diffusion pushed by the negative gradient of its own occupation time measure (local time). The second example is a diffusion in a fixed random environment given by the curl of massless Gaussian free field. In both cases we show that the process is superdiffusive: the variance grows faster than linearly with time. We give lower and upper bounds of the order of t log log t, respectively, t log t. We also present computations for an anisotr…
Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$ and the weighted local time ${\mathscr L}^H(\cdot,t)$. In this paper, we consider the integral functional $$ {\mathcal C}^H_t(a):=\lim_{\varepsilon\downarrow 0}\int_0^t1_{\{|B^H_s-a|>\varepsilon\}}\frac1{B^H_s-a}ds^{2H}\equiv \frac1π{\mathscr H}{\mathscr L}^H(\cdot,t)(a) $$ in $L^2(Ω)$ with $ a\in {\mathbb R}, t\geq 0$ and ${\mathscr H}$ denoting the Hilbert transform. We show that $$ {\mathcal C}^H_t(a)=2\left((B^H_t-a)\log|B^H_t-a|-B^H_t+a\log|a| -\int_0^t\log|B^H_s-a|δB^H_s\right) $$ for all $a\in {\mathbb R}, t\geq 0$ which…
We derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of Z^d, with d bigger or equal to 3. As an application, we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box. As a step in this program, we obtain a similar large deviation principle for the occupation-time measure of Brownian interlacements at a fixed level in a large box of R^d, and we derive a new identity for the Laplace transform of the occupation-time measur…
We study the distribution of additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate and placed back to a given reset position. Our goal is two-fold: (1) For general functionals, we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting. (2) For three examples of functionals (positive occupation time, area and absolute area), we investigate the effect of resetting by com…
If $L^x$ is the total occupation local time of $d$-dimensional super-Brownian motion, $X$, for $d=2$ and $d=3$, we construct a random measure $\mathcal{L}$, called the boundary local time measure, as a rescaling of $L^x e^{-λL^x} dx$ as $λ\to \infty$, thus confirming a conjecture of \cite{MP17} and further show that the support of $\mathcal{L}$ equals the topological boundary of the range of $X$, $\partial\mathcal{R}$. This latter result uses a second construction of a boundary local time $\widetilde{\mathcal{L}}$ given in terms of exit measures and we prove that $\widetilde{\mathcal{L}}=c\mat…
We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of …
We extend results on time-rescaled occupation time fluctuation limits of the $(d,α, β)$-branching particle system $(0<α\leq 2, 0<β\leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>α/ β$ only, since the particle system becomes locally extinct if $d\le α/ β$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>α/β$. We also give high-density limits for the …
We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $\mathbb {R}^d$ under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number $N$ of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measu…
