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 with the one on the Euclidian space equipped with the Lebesgue measure. We give an example of a metric measure Dirichlet space on which a scaled of the means fluctuates.
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.
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.
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 …
We consider the $(1,2)$-Sobolev space $W^{1,2}(U)$ on subsets $U$ in an abstract Wiener space, which is regarded as a canonical Dirichlet space on $U$. We prove that $W^{1,2}(U)$ has smooth cylindrical functions as a dense subset if $U$ is $H$-convex and $H$-open. For the proof, the relations between $H$-notions and quasi-notions are also studied.
In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises in…
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…
Equipping the probability space with a local Dirichlet form with square field operator $Γ$ and generator $A$ allows to improve Monte Carlo computations of expectations, densities, and conditional expectations, as soon as we are able to simulate a random variable $X$ together with $Γ[X]$ and $A[X]$. We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When $X$ is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.
The quasi-invariance is proved for the distributions of Poisson point processes under a random shift map on the path space. This leads to a natural Dirichlet form of jump type on the path space. Differently from the O-U Dirichlet form on the Wiener space satisfying the log-Sobolev inequality, this Dirichlet form merely satisfies the Poincare inequality but not the log-Sobolev one.
Equipping the probability space with a local Dirichlet form with square field operator Γand generator A allows to improve Monte Carlo simulations of expectations and densities as soon as we are able to simulate a random variable X together with Γ[X] and A[X]. We give examples on the Wiener space, on the Poisson space and on the Monte Carlo space. When X is real-valued we give an explicit formula yielding the density at the speed of the law of large numbers.
The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron--Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space ove…
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…
In this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth explicit approximations for a large class of Wiener functionals. We show that any Wiener functional has an underlying robust semimartingale skeleton which under mild conditions converges to it. The discretization is given in terms of discrete-jumping filtrations which allow us to approximate nonsmooth processes by means of a stochastic derivative operator on the Wiener space. As a by-product, we provide a robust semimartingale approximation for weak…
In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $Γ_t(G)$ (resp. $Γ_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $Γ_t(G)$ and $Γ_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Besse…
Consider a set of continuous maps from the interval $[0,1]$ to a domain in ${\mathbb R}^d$. Although the topological boundary of this set in the path space is not smooth in general, by using the theory of functions of bounded variation (BV functions) on the Wiener space and the theory of Dirichlet forms, we can discuss the existence of the surface measure and the Skorokhod representation of the reflecting Ornstein-Uhlenbeck process associated with the canonical Dirichlet form on this set.
We develop an approach for investigating geometric properties of Gaussian multiplicative chaos (GMC) in an infinite dimensional set up. The base space is chosen to be the space of continuous functions endowed with Wiener measure, and the random field is a space-time white noise integrated against Brownian paths. In this set up, we show that in any dimension $d\geq 1$ and for any inverse temperature, the GMC-volume of a ball, uniformly around all paths, decays exponentially with an explicit decay rate. The exponential rate reflects the balance between two competing terms, namely the principal e…
In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bounds of the Ricci curvature are presented r…
We give a extensive account of a recent new way of applying the Dirichlet form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of Lévy processes or solutions of stochastic differential equations with jumps. As in the Wiener case the Dirichlet form approach weakens significantly theregularity assumptions. The main novelty is an explicit formula for the gradient or for the "carré du champ' on the Poisson space called the lent particle formula because based on adding a new particle to the system, computing the derivative …
We study the potential theory of a large class of infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e. excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting. Thus a number of potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as e.g. formulated by R. Carmona in 1980. In particular, we prove the a…
We introduce the setting of extended metric-topological measure spaces as a general "Wiener-like" framework for optimal transport problems and nonsmooth metric analysis in infinite dimension. After a brief review of optimal transport tools for general Radon measures, we discuss the notions of the Cheeger energy, of the Radon measures concentrated on absolutely continuous curves, and of the induced "dynamic transport distances." We study their main properties and their links with the theory of Dirichlet forms and the Bakry-Émery curvature condition, in particular concerning the contractivity pr…
