For a>0,let W^a(t) be the a-neighbourhood of standard Brownian motion in R^d starting at 0 and observed until time t.It is well-known that E|W^a(t)|~kappa_a t (t->infty) for d >= 3,with kappa_a the Newtonian capacity of the ball with radius a. We prove that lim_{t->infty} 1/t^{(d-2)/d}log P(|W^a(t)|<=bt) = -I^{kappa_a}(b) in (-infty,0) for all 0<b<kappa_a and derive a variational representation for the rate function I^{kappa_a}.We show that the optimal strategy to realise the above moderate deviation is for W^a(t) to look like a Swiss cheese: W^a(t) has random holes whose sizes are of order 1 and whose density varies on scale t^{1/d}.The optimal strategy is such that t^-1/d W^a(t) is delocalised in the limit as t->infty.This is markedly different from the optimal strategy for large deviations |W^a(t)|<=f(t) with f(t)=o(t),where W^a(t) is known to fill completely a ball of volume f(t) and nothing outside,so that W^a(t) has no holes and f(t)^{-1/d}W^a(t) is localised in the limit as t->infty.We give a detailed analysis of the rate function I^{kappa_a},in particular,its behaviour near the boundary points of (0,kappa_a).It turns out that I^{kappa_a} has an infinite slope at kappa_a and,remarkably,for d>=5 is nonanalytic at some critical point in (0,kappa_a),above which it follows a pure power law.This crossover is associated with a collapse transition in the optimal strategy.We also derive the analogous moderate deviation result for d=2.In this case E|W^a(t)|~2pi t/log t (t->infty),and we prove that lim_{t->infty} 1/log t log P(|W^a(t)|<=bt/log t) =-I^{2pi}(b)in (-infty,0) for all 0<b<2pi.
In this paper we prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall's estimates between the Wiener sausage and the Brownian intersection local times.
We consider the Wiener sausage for a Brownian motion up to time $t$ associated with a closed ball in even dimensional cases. We obtain the asymptotic expansion of the expected volume of the Wiener sausage for large $t$. The result says that the expansion has many $\log$ terms, which do not appear in odd dimensional cases.
We consider the Wiener sausage for a Brownian motion with a constant drift up to time $t$ associated with a closed ball. In the two or more dimensional cases, we obtain the explicit form of the expected volume of the Wiener sausage. The result says that it can be represented by the sum of the mean volumes of the multi-dimensional Wiener sausages without a drift. In addition, we show that the leading term of the expected volume of the Wiener sausage is written as $κt(1+o[1])$ for large $t$ and an explicit form gives the constant $κ$. The expression is of a complicated form, but it converges to …
The expected areas of the Wiener sausages swept by a disc attached to the two-dimensional Brownian Bridge joining the origin to a point x over a time interval [0,t] are computed. It is proved that the leading term of the expectation is given by Ramanujan's function if (x,t) remains in a parabolic region. The corresponding result for unconditioned process is also obtained.
In the present paper, we consider long time behaviors of the volume of the Wiener sausage on Dirichlet spaces. We focus on the volume of the Wiener sausage for diffusion processes on metric measure spaces other than the Euclid space equipped with the Lebesgue measure. We obtain the growth rate of the expectations and almost sure behaviors of the volumes of the Wiener sausages on metric measure Dirichlet spaces satisfying Ahlfors regularity and sub-Gaussian heat kernel estimates. We show that the growth rate of the expectations on a "bounded" modification of the Euclidian space is identical wit…
Let R_n be the radius of the largest disk covered after n steps of a simple random walk. We prove that almost surely limsup_{n \to \infty}(log R_n)^2/(log n log_3 n) = 1/4, where log_3 denotes 3 iterations of the log function. This is motivated by a question of Erdős and Taylor. We also obtain the analogous result for the Wiener sausage, refining a result of Meyre and Werner.
A parallel neighborhood of a path of a Brownian motion is sometimes called the Wiener sausage. We consider almost sure approximations of this random set by a sequence of random polyconvex sets and show that the convergence of the corresponding mean curvature measures holds under certain conditions in two and three dimensions. Based on these convergence results, the mean curvature measures of the Wiener sausage are calculated numerically by Monte Carlo simulations in two dimensions. The corresponding approximation formulae are given.
We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the L{é}vy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.
We consider the statistical mechanics of a random polymer with random walks and disorders in $\mathbb{Z}^d$. The walk collects random disorders along the way and gets nothing if it visits the same site twice. In the continuum and weak disorder regime, the partition function as a random variable converges weakly to a Wiener Chaos expansion when the dimension is lower than the critical dimension, which is four. A finite temperature case in one dimension is also discussed. The last case suggests that the end-point behavior of the polymer is $t^{2/3}$.
The space-time distribution, $Q_A(x,dt dξ)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtdξ)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt dξ)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joini…
In the context of a heat kernel diffusion which admits a Gaussian type estimate with parameter beta on a local Dirichlet space, we consider the log asymptotic behavior of the negative exponential moments of the Wiener sausage. We show that the log asymptotic behavior up to time t^{beta}V(x,t) is V(x,t), which is analogous to the Euclidean result. Here V(x,t) represents the mass of the ball of radius t about a point x of the local Dirichlet space. The proof uses a known coarse graining technique to obtain the upper asymptotic, but must be adapted to for use without translation invariance in thi…
We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Martínez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the…
The branching Brownian sausage in $\mathbb{R}^d$ was defined by Engländer in [Stoch. Proc. Appl. 88 (2000)] similarly to the classical Wiener sausage, as the random subset of $\mathbb{R}^d$ scooped out by moving balls of fixed radius with centers following the trajectories of the particles of a branching Brownian motion (BBM). We consider a $d$-dimensional dyadic BBM, and study the large-time asymptotic behavior of the volume of the associated exponentially shrinking branching Brownian sausage (BBM-sausage). Using a previous result on the density of the support of BBM, and some well-known resu…
Let $(ξ(s))_{s\geq 0}$ be a standard Brownian motion in $d\geq 1$ dimensions and let $(D_s)_{s \geq 0}$ be a collection of open sets in $\R^d$. For each $s$, let $B_s$ be a ball centered at 0 with $\vol(B_s) = \vol(D_s)$. We show that $\E[\vol(\cup_{s \leq t}(ξ(s) + D_s))] \geq \E[\vol(\cup_{s \leq t}(ξ(s) + B_s))]$, for all $t$. In particular, this implies that the expected volume of the Wiener sausage increases when a drift is added to the Brownian motion.
We study two objects concerning the Wiener sausage among Poissonian obstacles. The first is the asymptotics for the \textit{replica overlap}, which is the intersection of two independent Wiener sausages. We show that it is asymptotically equal to their union. This result confirms that the localizing effect of the media is so strong as to completely determine the motional range of particles. The second is an estimate on the \textit{covering time}. It is known that the Wiener sausage avoiding Poissonian obstacles up to time $t$ is confined in some `clearing' ball near the origin and almost fills…
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.
We provide asymptotic bounds on the survival probability of a moving polymer in an environment of Poisson traps. Our model for the polymer is the vector-valued solution of a stochastic heat equation driven by additive spacetime white noise; solutions take values in ${\mathbb R}^d, d \geq 1$. We give upper and lower bounds for the survival probability in the cases of hard and soft obstacles. Our bounds decay exponentially with rate proportional to $T^{d/(d+2)}$, the same exponent that occurs in the case of Brownian motion. The exponents also depend on the length $J$ of the polymer, but here our…
We consider a Hamiltonian involving the range of the simple random walk and the Wiener sausage so that the walk tends to stretch itself. This Hamiltonian can be easily extended to the multidimensional cases, since the Wiener sausage is well-defined in any dimension. In dimension one, we give a formula for the speed and the spread of the endpoint of the polymer path. It can be easily showed that if the self-repelling strength is stronger, the end point is going away faster. This strict monotonicity of speed has not been proven in the literature for the one-dimensional case.
A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are consid…
The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to establish the existence of a homogenized limit for such a diffusion when subject to Dirichlet conditions on the boundaries of a sparse and independent array of obstacles. There is a constant effective long-time loss rate at the obstacles. The dependence of this rate on the form and intensity of the obstacles and on the velocity field is investigated. A Monte Car…
We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thäle (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.
We use a new method via $p$-Wasserstein bounds to prove Cramér-type moderate deviations in (multivariate) normal approximations. In the classical setting that $W$ is a standardized sum of $n$ independent and identically distributed (i.i.d.) random variables with sub-exponential tails, our method recovers the optimal range of $0\leq x=o(n^{1/6})$ and the near optimal error rate $O(1)(1+x)(\log n+x^2)/\sqrt{n}$ for $P(W>x)/(1-Φ(x))\to 1$, where $Φ$ is the standard normal distribution function. Our method also works for dependent random variables (vectors) and we give applications to the combinat…
In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$α$ Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by $\sqrtα$. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as $α$ goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term o…
