Let $\mathscr{T}(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for Brownian motion in $\mathbb{R}^3$. We prove that $\sup_{|x|\leq1}\mathscr{T}(x,r)/(r^{2}|\log r|)\rightarrow16/\pi^2$ a.s. as $r\rightarrow0$, thus solving a problem posed by Taylor in 1974. Furthermore, for any $a \in(0,16/\pi^2)$, the Hausdorff dimension of the set of “thick points” $x$ for which $\lim\sup_{r \to 0}\mathscr{T}(x,r)/(r^2|\log r|) = a$ is almost surely $2-a\pi^2 /8$; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for Brownian occupation measure. Analogous results hold for Brownian motion in any dimension $d \ge 3$. These results are related to the LIL of Ciesielski and Taylor for the Brownian occupation measure of small balls in the same way that Lévy’s uniform modulus of continuity, and the formula of Orey and Taylor for the dimension of “fast points ”are related to the usual LIL. We also show that the lim inf scaling of $\mathscr{T}(x,r)$ is quite different: we exhibit nonrandom $c_1,c_2 \ge 0$, such that $c_1 < \sup_x\lim \inf _{r \to 0}\mathscr{T}(x,y)/r^2 < c_2$ a.s. In the course of our work we provide a general framework for obtaining lower bounds on the Hausdorff dimension of random fractals of “limsup type.”

We study the existence and uniqueness of the solution of a function-valued stochastic evolution equation based on a stochastic semigroup whose kernel $p(s,t,y,x)$ is Brownian in $s$ and $t$.The kernel $p$ is supposed to be measurable with respect to the increments of an underlying Wiener process in the interval $[s, t]$. The evolution equation is then anticipative and, choosing the Skorohod formulation,we establish existence and uniqueness of a continuous solution with values in $L^2(\mathbb{R}^d)$.

As an application we prove the existence of a mild solution of the stochastic parabolic equation

du_t = \Delta_x u dt + v(dt, x) \cdot \nabla u + F(t, x, u) W(dt, x),

where $v$ and $W$ are Brownian in time with respect to a common filtration. In this case, p is the formal backward heat kernel of $\Delta_x + v(dt, x) \cdot \nabla_x$ .

We study Cauchy’s problem for certain second-order linear parabolic stochastic differential equation (SPDE)driven by a cylindrical Brownian motion.Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes.

The superprocess and its occupation time process are represented as Hilbert space valued solutions of stochastic evolution equations by using the Fourier transform of the process. For appropriate parameter values, the existence of density valued solutions follows. Pathwise regularity of the processes is obtained.As a new tool we develop a maximal inequality. We also extend the Tanaka-like evolution equations developed for local time processes and provide an Ito formula for certain functionals of the superprocess.

The distribution of the integral of the absolute value of the Brownian bridge was expressed by Cifarelli and independently by Johnson and Killeen in the form of a series. Rice obtained the corresponding probability density by numerical integration. Here we determine the exact tail asymptotics of this distribution, as well as the exact asymptotics of its density function for the large values of the argument.

For the system of $d$-dim stochastic differential equations,

dX^{\varepsilon} (t) = b(X^{\varepsilon}(t)) dt + \varepsilon dW(t), \quad t \in [0, 1]

X^{\varepsilon} (0) = x^0 \in R^d

where $b$ is smooth except possibly along the hyperplane $x_1 = 0$, we shall consider the large deviation principle for the lawof the solution diffusion process and its occupation time as $\varepsilon\rightarrow0$. In other words, we consider $P(\|X^\varepsilon-\varphi\|<\delta,\|u^{\varepsilon}-\psi\|\<\delta)$ where $u^\varepsilon(t)$ and $\psi(t)$ are the occupation times of $X^\varepsilon$ and $\varphi$ in the positive half space $\{x\in R^d: x_1>0\}$, respectively. As a consequence, an unified approach of the lower level large deviation principle for the law of $X^\varepsilon(\cdot)$ can be obtained.

Under a natural invariance assumption on the Lévy measure we construct compound Poisson processes and more general isotropic Lévy processes on Riemannian manifolds by projection of a suitable horizontal process in the bundle of orthonormal frames.We characterize such Lévy processes through their infinitesimal generators and we show that they can be realized as the limit of a sequence of Brownian motions which are interlaced with jumps along geodesic segments.

We show that a sequence of voter models, suitably rescaled in space and time,converges weakly to super-Brownian motion. The result includes both nearest neighbor and longer range voter models and complements a limit theorem of Mueller and Tribe in one dimension.

In this paper, some new forms of Cheeger’s inequalities are established for general (maybe unbounded) symmetric forms (Theorems 1.1 and 1.2): the resulting estimates improve and extend the ones obtained by Lawler and Sokal for bounded jump processes. Furthermore, some existence criteria for spectral gap of general symmetric forms or general reversible Markov processes are presented (Theorems 1.4 and 3.1), based on Cheeger’s inequalities and a relationship between the spectral gap and the first Dirichlet and Neumann eigenvalues on local region.

A functional law of the iterated logarithm is obtained for symmetric stable processes with stationaryindependent increments.This extends the classical liminf results of Chung for Brownian motion, and of Taylor for such remaining processes. It also extends an earlier result of Wichura on Brownian motion.Proofs depend on small ball probability estimates and yield the small ball probabilities of the weighted sup-norm for these processes.

Some invariance principles for additive functionals of simple exclusion with finite-range translation-invariant jump rates $p(i, j) = p(j - i)$ in dimensions $d \geq1$ are established. A previous investigation concentrated on the case of $p$ symmetric. The principal tools to take care of nonreversibility, when $p$ is asymmetric, are invariance principles for associated random variables and a “local balance”estimate on the asymmetric generator of the process.

As a by-product,we provide upper and lower bounds on some transition probabilities for mean-zero asymmetric second-class particles,which are not Markovian, that show they behave like their symmetric Markovian counterparts.Also some estimates with respect to second-class particles with drift are discussed.

In addition,a dichotomy between the occupation time process limits in $d =1$ and $d \geq 2$ for symmetric exclusion is shown. In the former, the limit is fractional Brownian motion with parameter 3/4, and in the latter, the usual Brownian motion.

We consider a system of particles,each of which performs a continuous time random walk on $\mathbb{Z}^d$ . The particles interact only at times when a particle jumps to a site at which there are a number of other particles present. If there are $j$ particles present, then the particle which just jumped is removed from the system with probability $p_j$. We show that if $p_j$ is increasing in $j$ and if the dimension $d$ is at least 6 and if we start with one particle at each site of $\mathbb{Z}^d$, then $p(t):= P\{there is at least one particle at the origin at time t\}\sim C(d)/t$. The constant $C(d)$ is explicitly identified. We think the result holds for every dimension $d \geq 3$ and we briefly discuss which steps in our proof need to be sharpened to weaken our assumption $d \geq 6$.

The proof is based on a justification of a certain mean field approximation for $dp(t)/dt$.The method seems applicable to many more models of coalescing and annihilating particles.

Consider first passage Ising percolation on $Z^2$. Let $\beta$ denote the reciprocal temperature and let $h$ denote an external magnetic field. Denote by $\beta_c$ the critical temperature and, for $\beta\<\beta_c$, let

h_c(\beta) = h_c = \sup\{h:\theta(\beta, h) = 0\},

where $\theta(\beta,h)$ is the probability that the origin is connected by an infinite (+)-cluster. With these definitions let us consider first passage Ising percolation on $Z^2$. Let $a_{0,n}$ denote the first passage time from $(0,0)$ to $(n,0)$. It follows from a subadditive argument that

\lim_{n \to \infty} \frac{a_{0, n}}{n} = \nu \text{ a.s. and in } L_1.

It is known that $\nu > 0$ if $\beta < \beta_c$ and $|h| < h_c(\beta)$. Here we will estimate the speed of the convergence,

\nu n \leq Ea_{0,n} \leq \nu n + C(n \log^5 n)^{1/2}

for some constant $C$. Define $\mu_{\beta,h}$ to be the unique Gibbs measure for $\beta<\beta_c$. We also prove that there exist $\tilde{C},\tilde{\alpha}>0$ such that

\mu_{\beta, h}(|a_{0,n} - Ea_{0,n} \geq x) \leq \tilde{C} \exp(\\tilde{a}\frac{x^2}{n \log^4 n})

In addition to $a_{0,n)$, we shall also discuss other passage times.

Lyons, Pemantle and Peres asked whether the asymptoticlower speed in an infinite tree is bounded by the asymptoticspeed in the regular tree with the same average number of branches. In the more general setting of random walks on graphs, we establish a bound on the expected value of the exit time from a vertex set in terms of the size and distance from the origin of its boundary, and prove this conjecture. We give sharp bounds for limiting speed (or, when applicable,sublinear rate of escape) in terms of growth properties of the graph. For trees, we get a bound for the speed in terms of the Hausdorff dimension of the harmonicmeasure on the boundary. As a consequence, two conjectures of Lyons, Pemantle and Peres are resolved, and a new bound is given for the dimension of the harmonicmeasure defined by the biased random walk on a Galton–Watson tree.

The uniform model is a reversible interactingparticle system that evolves on the homogeneous tree. Occupied sites become vacant at rate one provided the number of occupied neighbors does not exceed one.Vacant sites become occupied at rate $\beta$ times the number of occupied neighbors. On the binary tree, it has been shown that the survival threshold $\beta_c$ is 1/4. In particular, for $\beta \leq 1/4$, the expected extinction time is finite.Otherwise, the uniform model survives locally. We show that the survival probability decays faster than a quadratic near $\beta_c$. This contrasts with the behavior of the survival probability for the contact process on homogeneous trees, which decays linearly.We also provide a lower bound that implies that the rate of decay is slower than a cubic. Tools associated with reversibility, for example, the Dirichlet principle and Thompson's principle, are used to prove this result.

We prove concentration inequalities for some classes of Markov chains and $\Phi$-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures of Talagrand. The method is based on information inequalities put forwardby Marton in case of contracting Markov chains. Using a simple duality argument on entropy, our results also include the family of logarithmic Sobolev inequalities for convex functions. Applications to bounds on supremum of dependent empirical processes complete this work.

We construct a super iterated Brownian motion (SIBM) from a historical version of iterated Brownian motion (IBM) using an iterated Brownian snake (IBS). It is shown that the range of super iterated Brownian motion is qualitatively quite different from that of super Brownian motion in that there are points with explosions in the branching. However, at a fixed time the support of SIBM has an exact Hausdorff measure function that is the same (up to a constant) as that of super Brownian motion at a fixed time.

We study limit distributions of sums $S_N^{(G)} = \sum_{t=1}^N G(X_t)$ of nonlinear functions $G(x)$ in stationary variables of the form $X_t = Y_t + Z_t$, where ${Y_t}$ is a linear (moving average) sequence with long-range dependence, and ${Z_ t}$ is a (nonlinear) weakly dependent sequence. In particular, we consider the case when ${Y_ t}$ is Gaussian and either (1)${Z_t}$ is a weakly dependent multilinear form in Gaussian innovations, or (2) ${Z_t}$ is a finitely dependent functional in Gaussian innovations or (3)${Z_t}$ is weakly dependent and independent of $Y_t$ . We show in all three cases that the limit distribution of $S^(G)_N$ is determined by the Appell rank of $G( x)$, or the lowest $k\geq 0$ such that $a_k = \partial^k E\{G(X_0+c)\}/\partial c^k|_{c=0 \not= 0$.

Given any closed subset $C$ of $\bar{R^d}$, containing a pair of antipodal points at $\infty$, there is a sequence of independent and identically distributed random variables $\mathbf{X}_i}$ such that the set of limit points (in the topology of $\bar{R^d}$ of $\{(\mathbf{X}_1 + \cdots + \mathbf{X}_t)/t\}_{t \geq 1}$ equals $C$. Here $\bar{R^d}$ is the compact space gotten by “adjoining the sphere, $S^{d -1}\infty$ at infinity.”

We derive Berry–Esséen bounds for a class of Studentized statistics. The results are applied to Studentized $U$-statistics, Studentized $L$-statistics and Studentized functions of the sample mean to give the Berry–Esséen bounds under conditions weaker than those obtained by alternative methods.