We give a construction of the zero range and bricklayers’ processes in the totally asymmetric, attractive case. The novelty is that we allow jump rates to grow exponentially. Earlier constructions have permitted at most linearly growing rates. We also show the invariance and extremality of a natural family of i.i.d. product measures indexed by particle density. Extremality is proved with an approach that is simpler than existing ergodicity proofs.

Let ({X_i(t)}_i∈ℤ^d)_t≥0 be the system of interacting diffusions on [0, ∞) defined by the following collection of coupled stochastic differential equations: $$dX_{i}(t)=\sum\limits_{j\in \mathbb{Z}^{d}}a(i,j)[X_{j}(t)-X_{i}(t)]\,dt+\sqrt{bX_{i}(t)^{2}}\ dW_{i}(t),\quad i\in \mathbb{Z}^{d},\ t\geq 0.$$ Here, a(⋅, ⋅) is an irreducible random walk transition kernel on ℤ^d×ℤ^d, b∈(0, ∞) is a diffusion parameter, and ({W_i(t)}_i∈ℤ^d)_t≥0 is a collection of independent standard Brownian motions on ℝ. The initial condition is chosen such that {X_i(0)}_i∈ℤ^d is a shift-invariant and shift-ergodic random field on [0, ∞) with mean Θ∈(0, ∞) (the evolution preserves the mean).

We show that the long-time behavior of this system is the result of a delicate interplay between a(⋅, ⋅) and b, in contrast to systems where the diffusion function is subquadratic. In particular, let â(i, j)=½[a(i, j)+a(j, i)], i, j∈ℤ^d, denote the symmetrized transition kernel. We show that:

(A) If â(⋅, ⋅) is recurrent, then for any b>0 the system locally dies out.

(B) If â(⋅, ⋅) is transient, then there exist b_*≥b₂>0 such that:

(B1) The system converges to an equilibrium ν_Θ (with mean Θ) if 0<b<b_*.

(B2) The system locally dies out if b>b_*.

(B3) ν_Θ has a finite 2nd moment if and only if 0<b<b₂.

(B4) The 2nd moment diverges exponentially fast if and only if b>b₂.

The equilibrium ν_Θ is shown to be associated and mixing for all 0<b<b_*. We argue in favor of the conjecture that b_*>b₂. We further conjecture that the system locally dies out at b=b_*.

For the case where a(⋅, ⋅) is symmetric and transient we further show that:

(C) There exists a sequence b₂≥b₃≥b₄≥⋯>0 such that:

(C1) ν_Θ has a finite mth moment if and only if 0<b<b_m.

(C2) The mth moment diverges exponentially fast if and only if b>b_m.

(C3) b₂≤(m−1)b_m<2.

(C4) lim _m→∞(m−1)b_m=c=sup _m≥2(m−1)b_m.

The proof of these results is based on self-duality and on a representation formula through which the moments of the components are related to exponential moments of the collision local time of random walks. Via large deviation theory, the latter lead to variational expressions for b_* and the b_m’s, from which sharp bounds are deduced. The critical value b_* arises from a stochastic representation of the Palm distribution of the system.

The special case where a(⋅, ⋅) is simple random walk is commonly referred to as the parabolic Anderson model with Brownian noise. This case was studied in the memoir by Carmona and Molchanov [Parabolic Anderson Problem and Intermittency (1994) Amer. Math. Soc., Providence, RI], where part of our results were already established.

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus.

We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan’s lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of ℤ^d tending to ℤ^d as time diverges. We finally discuss the relation of our density formula to the Ray–Knight theorem for continuous-time simple random walk on ℤ, which is analogous to the well-known Ray–Knight description of Brownian local times.

By using coupling and Girsanov transformations, the dimension-free Harnack inequality and the strong Feller property are proved for transition semigroups of solutions to a class of stochastic generalized porous media equations. As applications, explicit upper bounds of the L^p-norm of the density as well as hypercontractivity, ultracontractivity and compactness of the corresponding semigroup are derived.

Let R_n=max _0≤j≤nS_j−S_n be a random walk S_n reflected in its maximum. Except in the trivial case when P(X≥0)=1, R_n will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of R_n above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of S_n is necessary for passage of R_n above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of R_n above linear and square root boundaries.

Let $\mathscr{A}$ denote the reduced amalgamated free product of a family $\mathsf{A}_{1},\mathsf{A}_{2},\ldots,\mathsf{A}_{n}$ of von Neumann algebras over a von Neumann subalgebra $\mathscr {B}$ with respect to normal faithful conditional expectations $\mathsf {E}_{k}:\mathsf{A}_{k}\to \mathscr {B}$. We investigate the norm in $L_{p}(\mathscr {A})$ of homogeneous polynomials of a given degree d. We first generalize Voiculescu’s inequality to arbitrary degree d≥1 and indices 1≤p≤∞. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from which we generalize recent results of Pisier, Ricard and the authors. All constants in our estimates are independent of n so that we may consider infinitely many free factors. As applications, we study square functions of free martingales. More precisely, we show that, in contrast with the Khintchine and Rosenthal inequalities, the free analogue of the Burkholder–Gundy inequalities does not hold in $L_{\infty}(\mathscr {A})$. At the end of the paper we also consider Khintchine type inequalities for Shlyakhtenko’s generalized circular systems.

In this paper we construct a theory of stochastic integration of processes with values in ℒ(H, E), where H is a separable Hilbert space and E is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an H-cylindrical Brownian motion. Our approach is based on a two-sided L^p-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of ℒ(H, E)-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the Itô isometry, the Burkholder–Davis–Gundy inequalities, and the representation theorem for Brownian martingales.

We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure P^⋆ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to P^⋆ coincides with the variance-optimal martingale measure relative to the original probability measure P.

Let {X_ij}, i, j=…, be a double array of i.i.d. complex random variables with EX₁₁=0,E|X₁₁|²=1 and E|X₁₁|⁴<∞, and let $A_{n}=\frac{1}{N}T_{n}^{{1}/{2}}X_{n}X_{n}^{*}T_{n}^{{1}/{2}}$, where T_n^1/2 is the square root of a nonnegative definite matrix T_n and X_n is the n×N matrix of the upper-left corner of the double array. The matrix A_n can be considered as a sample covariance matrix of an i.i.d. sample from a population with mean zero and covariance matrix T_n, or as a multivariate F matrix if T_n is the inverse of another sample covariance matrix. To investigate the limiting behavior of the eigenvectors of A_n, a new form of empirical spectral distribution is defined with weights defined by eigenvectors and it is then shown that this has the same limiting spectral distribution as the empirical spectral distribution defined by equal weights. Moreover, if {X_ij} and T_n are either real or complex and some additional moment assumptions are made then linear spectral statistics defined by the eigenvectors of A_n are proved to have Gaussian limits, which suggests that the eigenvector matrix of A_n is nearly Haar distributed when T_n is a multiple of the identity matrix, an easy consequence for a Wishart matrix.

We consider a multitype branching process with immigration in a random environment introduced by Key in [Ann. Probab. 15 (1987) 344–353]. It was shown by Key that, under the assumptions made in [Ann. Probab. 15 (1987) 344–353], the branching process is subcritical in the sense that it converges to a proper limit law. We complement this result by a strong law of large numbers and a central limit theorem for the partial sums of the process. In addition, we study the asymptotic behavior of oscillations of the branching process, that is, of the random segments between successive times when the extinction occurs and the process starts again with the next wave of the immigration.

Choose n random, independent points in R^d according to the standard normal distribution. Their convex hull K_n is the Gaussian random polytope. We prove that the volume and the number of faces of K_n satisfy the central limit theorem, settling a well-known conjecture in the field.