In the multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of ℤ is occupied by a particle labeled with some number, and two neighboring particles are interchanged at rate one if their labels are in increasing order. Consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [−1, 1]. We study the joint distribution of these speeds: the TASEP speed process.

We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes.

By combining this result with known stationary measures for TASEPs with finitely many types, we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed.

We also study the partially asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations of ℤ, are symmetric in distribution. This allows us to extend some of our results, including the stationarity and description of all ergodic stationary measures, also to the ASEP.

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on ℤ² is SLE₂. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into ℂ so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is SLE₂. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one.

One example is the infinite component of super-critical percolation on ℤ². Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the loop-erased random walk on the super-critical percolation cluster is SLE₂.

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j-arm exponent is strictly between the polychromatic j-arm and (j + 1)-arm exponents.

Using Huisken’s results about the mean curvature flow on a strictly convex hypersurface and Kendall–Cranston’s coupling, we will build a stochastic process without birth and show that there exists a unique law of such a process. This process has many similarities with the circular Brownian motion studied by Émery and Schachermayer, and Arnaudon. In general this process is not a stationary process; it is linked to some differential equation without initial condition. We will show that this differential equation has a unique solution up to a multiplicative constant.

We study Markov processes where the “time” parameter is replaced by paths in a directed graph from an initial vertex to a terminal one. Along each directed path the process is Markov and has the same distribution as the one along any other directed path. If two directed paths do not interact, in a suitable sense, then the distributions of the processes on the two paths are conditionally independent, given their values at the common endpoint of the two paths. Conditions on graphs that support such processes (e.g., hexagonal lattice) are established. Next we analyze a particularly suitable family of Markov processes, called harnesses, which includes Brownian motion and other Lévy processes, on such time-like graphs. Finally we investigate continuum limits of harnesses on a sequence of time-like graphs that admits a limit in a suitable sense.

The reconstruction problem on the tree has been studied in numerous contexts including statistical physics, information theory and computational biology. However, rigorous reconstruction thresholds have only been established in a small number of models. We prove the first exact reconstruction threshold in a nonbinary model establishing the Kesten–Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten–Stigum bound is not tight for the q-state Potts model when q ≥ 5. Moreover, we determine asymptotics for these reconstruction thresholds.

It is well known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If, in addition, the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition x, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently of x, then the maximal flow is called strongly complete. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a two-dimensional SDE with coefficients which are even bounded (and smooth) and which is not strongly complete thus answering the question in the negative.

We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither Itô’s integrals nor martingale representation formulate are needed. This approach provides new tools for the study of BSDE, and is particularly useful for the study of BSDE with partial information. The approach allows us to study the following type of backward stochastic differential equations: $$dY_t^j = −f_0^j(t, Y_t, L(M)_t) dt − \sum_{i=1}^df_i^j(t, Y_t) dB_t^i + dM_t^j$$ with Y_T = ξ, on a general filtered probability space $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbf{P})$, where B is a d-dimensional Brownian motion, L is a prescribed (nonlinear) mapping which sends a square-integrable M to an adapted process L(M) and M, a correction term, is a square-integrable martingale to be determined. Under certain technical conditions, we prove that the system admits a unique solution (Y, M). In general, the associated partial differential equations are not only nonlinear, but also may be nonlocal and involve integral operators.

By constructing a coupling with unbounded time-dependent drift, dimension-free Harnack inequalities are established for a large class of stochastic differential equations with multiplicative noise. These inequalities are applied to the study of heat kernel upper bound and contractivity properties of the semigroup. The main results are also extended to reflecting diffusion processes on Riemannian manifolds with nonconvex boundary.

We study the dissipation mechanism of a stochastic particle system for the Burgers equation. The velocity field of the viscous Burgers and Navier–Stokes equations can be expressed as an expected value of a stochastic process based on noisy particle trajectories [Constantin and Iyer Comm. Pure Appl. Math. 3 (2008) 330–345]. In this paper we study a particle system for the viscous Burgers equations using a Monte–Carlo version of the above; we consider N copies of the above stochastic flow, each driven by independent Wiener processes, and replace the expected value with 1/N times the sum over these copies. A similar construction for the Navier–Stokes equations was studied by Mattingly and the first author of this paper [Iyer and Mattingly Nonlinearity 21 (2008) 2537–2553].

Surprisingly, for any finite N, the particle system for the Burgers equations shocks almost surely in finite time. In contrast to the full expected value, the empirical mean 1/N ∑₁^N does not regularize the system enough to ensure a time global solution. To avoid these shocks, we consider a resetting procedure, which at first sight should have no regularizing effect at all. However, we prove that this procedure prevents the formation of shocks for any N ≥ 2, and consequently as N → ∞ we get convergence to the solution of the viscous Burgers equation on long time intervals.

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.

A concentration property of the functional −log f(X) is demonstrated, when a random vector X has a log-concave density f on ℝⁿ. This concentration property implies in particular an extension of the Shannon–McMillan–Breiman strong ergodic theorem to the class of discrete-time stochastic processes with log-concave marginals.

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an α-stable law, α ∈ (0, 2). When 1 ≤ α < 2, we show that for a suitable regularly varying sequence κ_n of index 1 − 1/α, the limiting spectral distribution μ_α of κ_nK coincides with the one of the random symmetric matrix of the un-normalized weights (Lévy matrix with i.i.d. entries). In contrast, when 0 < α < 1, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law μ̃_α supported on [−1, 1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of μ_α and μ̃_α. We also study the limiting behavior of the invariant probability measure of K.

For probability distributions on ℝⁿ, we study the optimal sample size N = N(n, p) that suffices to uniformly approximate the pth moments of all one-dimensional marginals. Under the assumption that the marginals have bounded 4p moments, we obtain the optimal bound N = O(n^p/2) for p > 2. This bound goes in the direction of bridging the two recent results: a theorem of Guedon and Rudelson [Adv. Math. 208 (2007) 798–823] which has an extra logarithmic factor in the sample size, and a result of Adamczak et al. [J. Amer. Math. Soc. 23 (2010) 535–561] which requires stronger subexponential moment assumptions.

It is known that the joint limit distribution of independent Wigner matrices satisfies a very special asymptotic independence, called freeness. We study the joint convergence of a few other patterned matrices, providing a framework to accommodate other joint laws. In particular, the matricial limits of symmetric circulants and reverse circulants satisfy, respectively, the classical independence and the half independence. The matricial limits of Toeplitz and Hankel matrices do not seem to submit to any easy or explicit independence/dependence notions. Their limits are not independent, free or half independent.