We show that for the symmetric simple exclusion process on $/mathbb{Z}^d$ the self-diffusion coefficient of a tagged particle is stable when approximated by simple exclusion processes on large periodic lattices. The proof depends on a similar stability property of the asymptotic variance of additive functionals of mean 0. This requires establishing a property for the Dirichlet space known as the Liouville-D property.

Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the $Z_j:$ an appropriate number of moments and the total variation distance $d_{\mathrm{TV}}(\mathscr{L}(Z_j), \mathscr{L}(Z_j + 1))$. The proofs are based on Stein’s method for signed compound Poisson approximation.

The goal of this paper is to construct canonical Lévy area processes for Banach space valued Brownian motions via dyadic approximations. The significance of the existence of canonical Lévy area processes is that a (stochastic) integration theory can be established for such Brownian motions (in Banach spaces). Existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of Lyons and Lyons and Qian. This investigation involves a careful analysis on the choice of tensor norms, motivated by the applications to infinite dimensional stochastic differential equations.

Let $\tau = (\tau_i : i \in \mathbb{Z})$ denote i.i.d. positive random variables with common distribution $F$ and (conditional on $\tau$) let $X = )X_t : t \geq 0, X_0 = 0)$, be a continuous-time simple symmetric random walk on $\mathbf{Z}$ with inhomogeneous rates $(\tau_i^{-1} : i \in \mathbb{Z})$. When $F$ is in the domain of attraction of a stable law of exponent $\alpha < 1$ [so that $\mathbb{E}(\tau_i) = \infty$ and $X$ is subdiffusive], we prove that $(X, \tau)$, suitably rescaled (in space and time), converges to a natural (singular) diffusion $Z = (Z_t : t \geq 0, Z_0 = 0)$ with a random (discrete) speed measure $\rho$. The convergence is such that the “amount of localization,” $\mathbb{E} \sum_{i \in \mathbb{Z}}[\mathbb{P}(X_t = i|\tau)]^2$ converges as $t \to \infty$ to $\mathbb{E} \sum_{z \in \mathbb{R}}[\mathbb{P}(Z_s = z|\rho)]^2 > 0$, which is independent of $s > 0$ because of scaling/self-similarity properties of $(Z, \rho)$. The scaling properties of $(Z, \rho)$ are also closely related to the “aging” of $(X, \tau)$. Our main technical result is a general convergence criterion for localization and aging functionals of diffusions/walks $Y^{(\epsilon)}$ with (nonrandom) speed measures $\mu^{(\epsilon)} \to \mu$ (in a sufficiently strong sense).

We consider the random fluctuations of the free energy in the $p$-spin version of the Sherrington–Kirkpatrick (SK) model in the high-temperature regime. Using the martingale approach of Comets and Neveu as used in the standard SK model combined with truncation techniques inspired by a recent paper by Talagrand on the $p$-spin version, we prove that the random corrections to the free energy are on a scale $N^{-(p-2)/2}$ only and, after proper rescaling, converge to a standard Gaussian random variable. This is shown to hold for all values of the inverse temperature, $\beta$, smaller than a critical $\beta_p$. We also show that $\beta_p \to \sqrt{2 \ln 2}$ as $p \uparrow + \infty$. Additionally, we study the formal $p \uparrow + \infty$ limit of these models, the random energy model. Here we compute the precise limit theorem for the (properly rescaled) partition function at all temperatures. For $\beta < \sqrt{2 \ln 2}$, fluctuations are found at an exponentially small scale, with two distinct limit laws above and below a second critical value $\sqrt{\ln 2/2:}$ for $\beta$ up to that value the rescaled fluctuations are Gaussian, while below that there are non-Gaussian fluctuations driven by the Poisson process of the extreme values of the random energies. For $\beta$ larger than the critical $\sqrt{2 \ln 2}$, the fluctuations of the logarithm of the partition function are on a scale of 1 and are expressed in terms of the Poisson process of extremes. At the critical temperature, the partition function divided by its expectation converges to 1/2.

We derive a precise Ornstein–Zernike asymptotic formula for the decay of the two-point function $\mathbb{P}_p (0 \leftrightarrow x)$ of the Bernoulli bond percolation on the integer lattice $\mathbb{Z}^d$ in any dimension $d \geq 2$, in any direction $x$ and for any subcritical value of $p < p_c (d)$.

Consider the $\sigma$-finite measure-valued diffusion corresponding to the evolution equation $u_t = Lu + \beta (x) u - f(x,u)$, where

$$f(x,u) = \alpha (x)u^2 + \int_0^{\infty} (e^{-ku} - 1 + ku)n(x,dk)$$

and $n$ is a smooth kernel satisfying an integrability condition. We assume that $\beta, \alpha \in C^{\eta}(\mathbb{R}^d)$ with $\eta \in (0,1]$, and $\alpha > 0$. Under appropriate spectral theoretical assumptions we prove the existence of the random measure $$\lim_{t \uparrow \infty} e^{-\lambda_c t} X_t (dx)$$ (with respect to the vague topology), where $\lambda_c$ is the generalized principal eigenvalue of $L + \beta$ on $\mathbb{R}^d$ and it is assumed to be finite and positive, completing a result of Pinsky on the expectation of the rescaled process. Moreover, we prove that this limiting random measure is a nonnegative nondegenerate random multiple of a deterministic measure related to the operator $L + \beta$.

When $\beta$ is bounded from above, $X$ is finite measure-valued. In this case, under an additional assumption on $L + \beta$, we can actually prove the existence of the previous limit with respect to the weak topology.

As a particular case, we show that if $L$ corresponds to a positive recurrent diffusion $Y$ and $\beta$ is a positive constant, then

$$\lim_{t \uparrow \infty} e^{-\beta t} X_t (dx)$$

exists and equals a nonnegative nondegenerate random multiple of the invariant measure for $Y$.

Taking $L = 1/2 \Delta$ on $\mathbb{R}$ and replacing $\beta$ by $\delta_0$ (super-Brownian motion with a single point source), we prove a similar result with $\lambda_c$ replaced by 1/2 and with the deterministic measure $e^{-|x| dx$, giving an answer in theaffirmative to a problem proposed by Engländer and Fleischmann [Stochastic Process. Appl. 88 (2000) 37–58].

The proofs are based upon two new results on invariant curves of strongly continuous nonlinear semigroups.

Let $\mu$ be a probability measure with finite support on a discrete group $\Gamma$ of polynomial volume growth. The main purpose of this paper is to study the asymptotic behavior of the convolution powers $\mu^{*n}$ of $\mu$. If $\mu$ is centered, then we prove upper and lower Gaussian estimates. We prove a central limit theorem and we give a generalization of the Berry–Esseen theorem. These results also extend to noncentered probability measures. We study the associated Riesz transform operators. The main tool is a parabolic Harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of Krylov and Safonov. This inequality implies that the positive $\mu$-harmonic functions are constant. Finally we give a characterization of the $\mu$-harmonic functions which grow polynomially.

We consider the class of one-dimensional stochastic differential equations

$$dX_t = b(X_{t-})dZ_t, \quad t \geq 0,$$

where $b$ is a Borel measurable real function and $Z$ is a strictly $\alpha$-stable Lévy process $(0 < \alpha \leq 2)$. Weak solutions are investigated improving previous results of the author in various ways.

In particular, for the equation driven by a strictly 1-stable Lévy process, a sufficient existence condition is proven.

Also we extend the weak existence and uniqueness exact criteria due to Engelbert and Schmidt for the Brownian case (i.e., $\alpha = 2$) to the class of equations with $\alpha$ such that $1 < \alpha \leq 2$. The results employ some representation properties with respect to strictly stable Lévy processes.

Using the Wiener chaos decomposition, we show that strong solutions of non-Lipschitzian stochastic differential equations are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions.

For $n, k \in \mathbb{N}$ and $r > 0$ let $e_{n,r}(P_k)^r = \inf 1/k \sum_{i=1}^k ||X_i - f(X_i)||^r$, where the infimum is taken over all measurable maps $f : \mathbb{R}^d \to \mathbb{R}^d$ with $|f(\mathbb{R}^d)| \leq n$ and $X_1, \dots, X_k$ are i.i.d. $\mathbb{R}^d$-valued random variables. We analyse the asymptotic a.s. behaviour of the $n$th empirical quantization error $e_{n,r}(P_k)$.

Let $\alpha \in (1,2)$ and $X_{\alpha}$ be a symmetric $\alpha$-stable $(S \alpha S)$ process with stationary increments given by the mixed moving average $$X_{\alpha}(t) = \int_X \int_{\mathbb{R}}(G(x, t + u) - G(x, u)) M_{\alpha}(dx, du), \quad t \in \mathbb{R},$$ where $(X, \mathscr{X}, \mu)$ is a standard Lebesgue space, $G : X \times \mathbb{R} \mapsto \mathbb{R}$ is some measurable function and $M_{\alpha}$ is a $(S \alpha S)$ random measure on $X \times \mathbb{R}$ with the control measure $m(dx, du) = \mu (dx) du$. Assume, in addition, that $X_{\alpha}$ is self-similar with exponent $H \in (0,1)$. In this work, we obtain a unique in distribution decomposition of a process $X_{\alpha}$ into three independent processes

$$X_{\alpha} =^d X_{\alpha}^{(1)} + X_{\alpha}^{(2)} + X_{\alpha}^{(3)}.$$

We characterize $X_{\alpha}^{(1)}$ and $X_{\alpha}^{(2)}$ and provide examples of $X_{\alpha}^{(3)}$.

The first process $X_{\alpha}^{(1)}$ can be represented as

$$\int_Y \int_{\mathbb{R}} \int_{\mathbb{R}} e^{\kappa s}(F(y, e^s (t + u)) - F(y, e^s u)) M_{\alpha} (dy, ds, du),$$

where $\kappa = H - \frac{1}{\alpha}, (Y, \mathscr{Y}, \nu)$ is a standard Lebesgue space and $M_{\alpha}$ is a $(S \alpha S)$ random measure on $Y \times \mathbb{R} \times \mathbb{R}$ with the control measure $m(dy, ds, du) = \nu(dy) ds du$. Particular cases include the limit of renewal reward processes, the so-called "random wavelet expansion" and Takenaka process. The second process $X_{\alpha}^{(2)}$ has the representation

$$\int_Z \int_{\mathbb{R}} (G_1(z)((t + u)_+^{\kappa} - u_+^{\kappa}) + G_2 (z)((t + u)_-^{\kappa} - u_-^{\kappa})) M_{\alpha}(dz, du), \quad \text{if $\kappa \not= 0$},$$

$$\int_Z \int_{\mathbb{R}} (G_1(z)(\ln |t + u| - \ln |u|) + G_2 (z)(1_{(0, \infty)}(t + u) - 1_{(0, \infty)}(u))) M_{\alpha}(dz, du), \quad \text{if $\kappa = 0$},$$

where $Z, \mathscr{Z}, v)$ is a standard Lebesgue space and $M_{\alpha}$ is a $S \alpha S$ random measure on $Z \times \mathbb{R}$ with the control measure $m(dz, du) = v(dz)du$. Particular cases include linear fractional stable motions, log-fractional stable motion and $S \alpha S$ Lévy motion. And example of a process $X_{\alpha}^{(3)}$ is $$\int_0^1 \int_{\mathbb{R}} ((t + u)_+^{\kappa} 1_{[0, 1/2)} ({x + \ln |t + u|}) - u_+^{\kappa} 1_{[0, 1/2) ({x + \ln |u|})) M_{\alpha} (dx, du),$$ where $M_{\alpha}$ is a $S \alpha S$ random measure on $[0, 1) \times \mathbb{R}$ with the control measure $m(dx, du) = dxdu$ and ${\cdot}$ is the fractional part function.

We study the Domany–Kinzel model, which is a class of discrete time Markov processes with two parameters $(p_1, p_2) \in [0,1]^2$ and whose states are subsets of $\mathbf{Z}$, the set of integers. When $p_1 = \alpha \beta$ and $p_2 = \alpha (2 \beta - \beta^2)$ with $(\alpha, \beta) \in [0,1]^2$, the process can be identified with the mixed site–bond oriented percolation model on a square lattice with the probabilities of open site a and of open bond $\beta$. For the attractive case, $0 \leq p_1 \leq p_2 \leq 1$, the complete convergence theorem is easily obtained. On the other hand, the case $(p_1, p_2) = (1,0)$ realizes the rule 90 cellular automaton of Wolframin which, starting from the Bernoulli measure with density $\theta$, the distribution converges weakly only if $\theta \in {0, 1/2, 1}$. Using our new construction of processes based on signed measures, we prove limit theorems which are also valid for nonattractive cases with $(p_1, p_2) \not= (1,0)$. In particular, when $p_2 \in [0,1]$ and $p_1$ is close to 1, the complete convergence theorem is obtained as a corollary of the limit theorems.

Wreath products are a type of semidirect product. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group.

Let $(M_t)_{0 \leq t \leq 1} be a continuous martingale with initial law $M_0 \sim \mu_0$, and terminal law $M_1 \sim \mu_1$, and let $S = \sup_{0 \leq t \leq 1} M_t$. In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of $S$. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.