Large Deviations for Brownian Intersection Measures

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 positive measures on some open bounded set $B\subset\R^d$ as $t\to\infty$ before exiting $B$. The rate function is explicit and gives some rigorous meaning, in this asymptotic regime, to the understanding that the intersection measure is the pointwise product of the densities of the normalised occupation times measures of the $p$ motions. Our proof makes the classical Donsker-Varadhan principle for the latter applicable to the intersection measure. A second version of our principle is proved for the motions observed until the individual exit times from $B$, conditional on a large total mass in some compact set $U\subset B$. This extends earlier studies on the intersection measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics

The Dirichlet problem for the Bellman equation at resonance

We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at resonance.Comment: Appendix added. 28 page

Large deviations for many Brownian bridges with symmetrised initial-terminal condition

Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,\beta]$ 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 $\sigma(i)$-th motion, where $\sigma$ 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 $1/\beta$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) and of the mean of the normalised occupation measures of the $N$ motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel-Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers. In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker-Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N}$, where $\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap

Static and dynamical nonequilibrium fluctuations

Various notions of fluctuations exist depending on the way one chooses to measure them. We discuss two extreme cases (continuous measurement versus long inter-measurement times) and we see their relation with entropy production and with escape rates. A simple explanation of why the relative entropy satisfies a Hamilton-Jacobi equation is added.Comment: 10 page

Point-source scalar turbulence

The statistics of a passive scalar randomly emitted from a point source is investigated analytically. Our attention has been focused on the two-point equal-time scalar correlation function. The latter is indeed easily related to the spectrum, a statistical indicator widely used both in experiments and in numerical simulations. The only source of inhomogeneity/anisotropy is in the injection mechanism, the advecting velocity here being statistically homogeneous and isotropic. Our main results can be summarized as follows. 1) For a very large velocity integral scale, a pure scaling behaviour in the distance between the two points emerges only if their separation is much smaller than their distance from the point source. 2) The value we have found for the scaling exponent suggests the existence of a direct cascade, in spite of the fact that here the forcing integral scale is formally set to zero. 3) The combined effect of a finite inertial-range extension and of inhomogeneities causes the emergence of subleading anisotropic corrections to the leading isotropic term, that we have quantified and discussed.Comment: 10 pages, 1 figure, submitted to Journal of Fluid Mechanic

Characterization of the stretched exponential trap-time distributions in one-dimensional coupled map lattices

Stretched exponential distributions and relaxation responses are encountered in a wide range of physical systems such as glasses, polymers and spin glasses. As found recently, this type of behavior occurs also for the distribution function of certain trap time in a number of coupled dynamical systems. We analyze a one-dimensional mathematical model of coupled chaotic oscillators which reproduces an experimental set-up of coupled diode-resonators and identify the necessary ingredients for stretched exponential distributions.Comment: 8 pages, 8 figure

Resolvent Positive Linear Operators Exhibit the Reduction Phenomenon

The spectral bound, s(a A + b V), of a combination of a resolvent positive linear operator A and an operator of multiplication V, was shown by Kato to be convex in b \in R. This is shown here, through an elementary lemma, to imply that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) / \partial a <= s(A) when it exists. Diffusions typically have s(A) <= 0, so that for diffusions with spatially heterogeneous growth or decay rates, greater mixing reduces growth. Models of the evolution of dispersal in particular have found this result when A is a Laplacian or second-order elliptic operator, or a nonlocal diffusion operator, implying selection for reduced dispersal. These cases are shown here to be part of a single, broadly general, `reduction' phenomenon.Comment: 7 pages, 53 citations. v.3: added citations, corrections in introductory definitions. v.2: Revised abstract, more text, and details in new proof of Lindqvist's inequalit

Asymptotics for the Wiener sausage among Poissonian obstacles

We consider the Wiener sausage among Poissonian obstacles. The obstacle is called hard if Brownian motion entering the obstacle is immediately killed, and is called soft if it is killed at certain rate. It is known that Brownian motion conditioned to survive among obstacles is confined in a ball near its starting point. We show the weak law of large numbers, large deviation principle in special cases and the moment asymptotics for the volume of the corresponding Wiener sausage. One of the consequence of our results is that the trajectory of Brownian motion almost fills the confinement ball.Comment: 19 pages, Major revision made for publication in J. Stat. Phy

Nonconventional Large Deviations Theorems

We obtain large deviations theorems for nonconventional sums with underlying process being a Markov process satisfying the Doeblin condition or a dynamical system such as subshift of finite type or hyperbolic or expanding transformation

Dynamics at the angle of repose: jamming, bistability, and collapse

When a sandpile relaxes under vibration, it is known that its measured angle of repose is bistable in a range of values bounded by a material-dependent maximal angle of stability; thus, at the same angle of repose, a sandpile can be stationary or avalanching, depending on its history. In the nearly jammed slow dynamical regime, sandpile collapse to a zero angle of repose can also occur, as a rare event. We claim here that fluctuations of {\it dilatancy} (or local density) are the key ingredient that can explain such varied phenomena. In this work, we model the dynamics of the angle of repose and of the density fluctuations, in the presence of external noise, by means of coupled stochastic equations. Among other things, we are able to describe sandpile collapse in terms of an activated process, where an effective temperature (related to the density as well as to the external vibration intensity) competes against the configurational barriers created by the density fluctuations.Comment: 15 pages, 1 figure. Minor changes and update