The concentration of measure phenomenon is product spaces is a far-reaching abstract generalization of the classical exponential inequalities for sums of independent random variables. We attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon, the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities.

The isoperimetric problem for log-concave product measures in $\mathbb{R}^n$ equipped with the uniform distance is considered. Necessary and sufficient conditions under which standard half-spaces are extremal are presented.

A class of models for riffle shuffles ("$f$-shuffles") related to certain expansive mappings of the unit interval is studied. The main result concerns the cycle structure of the resulting random permutations in $\mathscr{S}_n$ when n is large. It describes the asymptotic distribution of the number of cycles of a given length, relating this distribution to dynamical properties of the associated mapping. This result generalizes a recent result of Diaconis, McGrath and Pitman.

For many random walks on "sufficiently large" finite groups the so-called cut-off phenomenon occurs: roughly stated, there exists a number $k_0$ , depending on the size of the group, such that $k_0$ steps are necessary and sufficient for the random walk to closely approximate uniformity. As a first example on a continuous group, Rosenthal recently proved the occurrence of this cut-off phenomenon for a specific random walk on $SO(N)$. Here we present and [for the case of $O(N)$] prove results for random walks on $O(N), U(N)$ and $Sp(N)$, where the one-step distribution is a suitable probability measure concentrated on reflections. In all three cases the cut-off phenomenon occurs at $k_0 = 1/2 N\log N$.

Rates of convergence of the distribution of the extreme order statistic to its limit distribution are given in the uniform metric and the total variation metric. A second-order regular variation condition is imposed by supposing a von Mises type condition which allows a unified treatment. Rates are constructed from the parameters of the second-order regular variation condition. Some connections with Poisson processes are discussed.

We show that the boundary of a connected component of the complement of a planar Brownian path on a fixed time interval contains almost surely no triple point of this Brownian path.

Existence and uniqueness in law of reflecting Brownian motion in a wedge is proved. The direction of reflection along the sides of the wedge varies in a reasonable fashion, except perhaps at the corner.

For a Brownian sheet on the nonnegative quadrant, we show that any nontrivial curve in the quadrant with the property that the Brownian sheet restricted to the curve gives rise to a differentiable function cannot be differentiable at any point. This result has several implications for level sets of the Brownian sheet. In particular, any Jordan arc contained in a level set must be nowhere differentiable.

Assume that the weak existence and pathwise uniqueness hold for solutions to the equation $dX_t=\sigma(t,X)dB_t + b(t,X)dt$ starting at fixed points. then there exists a Borel measurable function F, such that any solution (X,B) satisfies $X = F(X_0,B)$ a.s. This strengthens a fundamental result of Yamada and Watanbe, where F may depend on the initial distribution $\mu$

Let X be a semimartiangale and let $\Theta$ be the space of all predictable X-integrable process $\vartheta$ such that $\int\vartheta dX$ is in the space $\varsigma^2$ of semimartingales. We consider the problem of approximating a given random variable $H\in L^2(P)$ by the sum of a constant c and a stochastic integral $\int_0^T\vartheta_s dX_s$, with respect to the $L^2(P)$-norm. This problem comes from financial mathematics, where the optimal constant $V_0$ can be interpreted as an approximation price for the contingent clam H. An elementary computation yields $V_0$ as the expectation of H under the variance-optimal signed $\Theta$-martingale measure $\tilde{P}$, and this leads us to study $\tilda{P}$ in more detail. In the case of finite discrete time, we explicitly construct $\tilde{P}$ by backward recursion, and we show that $\tilde{P}$ is typically not a probability, but only a signed measure. In a continuous-time framework, the situation becomes rather different: we prove that $\tilde{P}$ is nonegative is X has continuous paths and satisfies a very mild no-arbitrage condition. As an application, we show how to obtain the optimal integrand $\xi\in\Theta$ in feedback form with the help of a backward stochastic differential equation.

We consider the supercritical finite measure-valued diffusion, $X(t)$, whose log-Laplace equation is associated with the semilinear equation $u_t =Lu = \beta u - \alpha u^2$, where $\alpha, \beta> 0$ and $L = 1/2 \sum _{i,j=1}^d a_{i,j} (\partial x_i \partial x_j)) = \sum_ {i=1} ^d b_i (\partial / \partial x_i)$. A path $X(\dot)$ is said to survive if $X(t) \not\equiv 0$, for all $t\geq 0$. Since $\beta> 0, P_\mu (X(\dot)$ survives) $>0$, for all $0\not\equiv \mu \in M(R^d)$, where $M(R^d)$ denotes the space of finite measures on $R^d$. We define transience, recurrence and local extinction for the support of the supercritical measure-valued diffusion starting from a finite meausre as follows. The support is recurrent if $P _ \mu (X(t,B)>0$, for some $t \geq 0 | X(\dot)$ survives) =1, for every $0 \not\equiv \mu \in M(R^d)$ and every open set $B \subset R^d$. For $d\geq 2$, the support is transient if $P_\mu(X(t,B)>0$, for some $t \geq 0 |X (\dot)$ survives) $<1$, for every $\mu \in M(R^d)$ and bounded $B\subset R^d$ which satisfy $\supp(\mu)\bigcap \bar{B} = \emptyset$. A similar definition taking into account the topology of $R^1$ is given for $d=1$. The support exhibits local extinction if for each $\mu \in M(R^d)$ and each bounded $B\subset R^d$, there exists a $P_\mu$-almost surely finite random time $\zeta_B$ such that $X(t,B) = 0$, for all $t\geq \zeta_B$. Criteria for transience, recurrence and local extinction are developed in this paper. Also studied is the asymptotic behavior as $t \to \infty$ of $E_\mu \int_0^t \langle \psi, X(s) \rangle ds$, and of $E_\mu \langle g,X(t) \rangle$, for $0\leq g, \psi \in C_c(R^d)$, where $\langle f, X(t) \rangle \not\equiv \int_{R^d} f(x) X(t,dx)$. A number of examples are given to illustrate the general theory.

We consider the exit measures of $(L,\alpha)$-superdiffusions, $1 < \alpha \leq 2$, from a bounded smooth domain D in R ^d. By using analytic results about solutions of the corresponding boundary value problem, we study the states of the exit measures. (Abraham and Le Gall investigated earlier .this problem for a special case $L = \Delta$ and $\alpha = 2$). Also as an application of these analytic results, we give a different proof for the critical Hausdorff. dimension of boundary polarity (established earlier by Le Gall under more restrictive assumptions and by Dynkin and Kuznetsov for general situations).

It is known that any smooth, nondegenerate, second-order elliptic operator on a manifold (dimension $\not= 2$) has the form $\Delta +B$, where B is a vector field and $\Delta$ is the Laplace-Beltrami operator under some Riemannian metric on the manifold. In this paper we give several conditions on the "Ricci curvature" Ric $-\nabla_B^s$ associated with the operator $\Delta + B$ to ensure that the diffusion semigroup generated by $\Delta + B$ conserves probability and possesses the Feller property.

The existence and uniqueness theorem is proved for solutions of the Dirichlet boundary value problems for weakly coupled elliptic systems on bounded domains. The elliptic systems are only assumed to have measurable coefficients and have singular coefficients for the lower-order terms. A probabilistic representation theorem for solutions of the Dirichlet boundary value problems is obtained by using the switched diffusion process associated with the system. A strong positivity result for solutions of the Dirichlet boundary value problems is proved. Formulas expressing resolvents and kernel functions for the system by those of the component elliptic operators are also obtained.

We consider the random fields $X^{\varepsilon}(t, q), \ t\geq 0, \ q\in {\mathcal O},$ goverened by stochastic partial differential equations driven by a Gaussian white noise in space-time, where $\mathcal O$ is a bounded domain in ${\mathbb R}^d$ with regular boundary. To study the continuity of the random fields $X^\varepsilon$ in space and time variables, we prove an analogue of Garsia's theorem. We then derive the large deviation results based on the methods used by the second author in another paper. This article provides an alternative proof of Sower's result for the case of d = 1.

In this paper we prove several theorems concerning the motion of a particle in a random environment. The trajectory of a particle is the solution of the differential equation $dx(t)/dt=V(x(t))$, where $V(x) = v + \varepsilon^{1-\alpha} F(x), \; 0\leq\alpha\leq 1,$ $v$ is a constant vector, $F$ is a mean-zero fluctuation field and $\varepsilon^{1-\alpha}$ is a parameter measuring the size of the fluctuations. We show that both in case of a motion of a single particle and of a particle system considered in the macroscopic coordinate system moving along with velocity $v$ [i.e., $x \sim (x-vt)/\varepsilon^{\alpha}, \; t\sim t/\varepsilon^2$] the diffusion approximation holds provided that $F$ is divergence free. Moreover we show how to renormalize trajectories to obtain a similar result for non-divergence-free fields. These results generalize theorems due to Khasminskii and to Kesten and Papanicolaou.

We consider the equation

$$\begin{array}{r@{=}l}u_t \, =\, u_{xx}+ u^\gamma\dot{W}, \quad t>0, \; 0\leq x \leq J,\\[1ex]u(0, x) \, =\, u_0 (x),\\[1ex]u(t, 0) \, = \, u(t, J) =0,\end{array}$$

where $\dot{W} = \dot{W}(t,x)$ is two-parameter white noise. We show local existence and uniqueness for unbounded initial conditions satisfying certain conditions. Our results are motivated by earlier work, which showed that, for large $\gamma$, solutions of this equation can blow up. One would wish to show that solutions can be extended beyond blowup, and our results can be viewed as a step in that direction.

We consider standard first-passage percolation on $\mathbb{Z}^2$. Geodesics are nearest-neighbor paths in $\mathbb{Z}^2$, each of whose segments is time-minimizing. We prove part of the conjecture that doubly infinite geodesics do not exist. Our main tool is a result of independent interest about the coalescing of semi-infinite geodesics.

Consider an inhomogeneous contact process on Z ¹ in which the recovery rates $\delta(x)$ at site x are i.i.d. random variables (bounded above) while the infection rate is a constant $\varepsilon$. The condition $u\mathbf{P}(-\log \varepsilon(x) > u) \to = \infty$ as $u \to = \infty$ implies the survival of the process for every $\varepsilon > 0$.

We consider stochastic processes $X=\{X(t),t\in T\}$ represented as a Lévy chaos of finite order, that is, as a finite sum of multiple stochastic integrals with respect to a symmetric infinitely divisible random measure. For a measurable subspace V of $R\sp T$ we prove a very general zero-one law $P(X\in V)=0$ or 1, providing a complete analogue to the corresponding situation in the case of symmetric infinitely divisible processes (single integrals with respect to an infinitely divisible random measure). Our argument requires developing a new symmetrization technique for multi-linear Rademacher forms, as well as generalizing Kanter's concentration inequality to multiple sums.

We examine the structure of the range of a stable process with drift near its initial position and derive an integral test to determine the rate of growth of the size of intervals in the complement of the range. This integral test depends on the random number of excursions that the process makes away from the initial point.

Let $ X_1, X_2,\ldots$ be a sequence of free identically distributed random variables, with common distribution $\mu$. It was shown by Lindsay and Pata, in a more general context, that a sufficient condition for the weak law of large numbers to hold for the sequence $X_1, X_2,\ldots$ is that

$$\lim_{t\to \infty}t\mu(\{x: |x|>t\}) = 0>$$

We show that this condition is necessary as well as sufficient. Even though the condition is identical with the corresponding one for commuting independent variables, the proof of the result uses the analytical techniques of free convolution theory, and it is quite different from the proof of the commutative theorem due to Kolmogorov.

We prove optimal convergence rates in the central limit theorem for sums ${\bf R}^k.$ Assuming a fourth moment, we obtain a Berry-Esseen type bound of $O(N^{-1})$ for the probability of hitting a ball provided that $k\leq 5$. The proof still requires a technical assumption related to the independence of coordinate sums.

We prove the Berry-Esseen bound for the Student t-statistic. Under the assumption of a third moment this bound coincides (up to an absolute constant) with the classical Berry-Esseen bound for the mean. In general the distribution of the Student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean, and sometimes faster. For example, rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law.

We prove functional laws of the iterated logarithm for empirical processes based upon censored data in the neighborhood of a fixed point. We apply these results to obtain strong laws for estimators of local functionals of the lifetime distribution. In particular, we describe the pointwise strong limiting behavior of the kernel density estimator based upon the Kaplan-Meier product-limit estimator.

A short proof of the lower bound in the strong version of the famous Theorem 1A in Kiefer (1970) on the Bahadur-Kiefer process is presented. The proof is elementary and, in particular, does not use strong approximations.

We develop a new technique to prove the conditional CLT for the weighted bootstrap mean. Through 0-1 laws, we show that this conditional CLT can be derived from an unconditional one which easily arises (conditioning with respect to the weights) from the standard Lindeberg CLT.

Jeff Steif has brought to our attention an error on page 975 of our paper. Our argument that inequality (30) implies the immediately following inequality has a gap. A closer look shows an even more serious problem, namely, Lemma 6 as stated is probably not true, since nothing in the weak Bernoulli property precludes the possibility that splitting sets for $x{^n_1}$may depend on past coordinates $\{x_i:i \leq 0\}$. With a modified definition of the splitting concept an alternative version of Lemma 6 is true and this is sufficient to prove our principal theorem, Theorem 4.

The following text replaces the discussion from the paragraph preceding Lemma 5 on page 973 to the end of Section 3 on page 976.