This paper presents an overview of some of the applications of the martingale inequalities of D. L. Burkholder to $L^p$-bounds for singular integral operators, concentrating on the Hilbert transform, first and second order Riesz transforms, the Beurling–Ahlfors operator and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. While martingale inequalities can be used to prove the boundedness of a wider class of Calderón–Zygmund singular integrals, the aim of this paper is to show results which give optimal or near optimal bounds in the norms, hence our restriction to the above operators.

Connections of Burkholder’s foundational work on sharp martingale inequalities to other areas of mathematics where either the results themselves or techniques to prove them have become of considerable interest in recent years, are discussed. These include the 1952 conjecture of C. B. Morrey on rank-one convex and quasiconvex functions with connections to problems in the calculus of variations and the 1982 conjecture of T. Iwaniec on the $L^p$-norm of the Beurling–Ahlfors operator with connections to problems in the theory of qasiconformal mappings. Open questions, problems and conjectures are listed throughout the paper and copious references are provided.

We show that the application of linear vertex reinforcement to one dimensional nearest neighbor Markov processes, yielding associated vertex reinforced jump processes, preserves both recurrence and transience. The analog for discrete time linear bond reinforcement is due to Takeshima. This together with another result we prove adds to the numerous known parallels between these two reinforcements. Martingales are the primary tool used to study vertex reinforced jump processes.

We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.

We consider a stochastic perturbation of a moving boundary problem proposed by Ludford and Stewart and studied by Caffarelli and Vazquez. We prove existence and uniqueness.

We show how basic work of Don Burkholder on iterated conditional expectations is intimately connected to a standard tool of scientific computing—Glauber dynamics (also known as the Gibbs sampler). We begin with von Neumann’s alternating projection theorem using an example of Burkholder’s. We then review Burkholder’s theorem. Finally, we introduce Glauber dynamics and show how Burkholder’s theorem can be harnessed to prove convergence. In the other direction, we show how classical convergence rates involving the angle between subspaces can be substantially refined in several cases.

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha\Delta^{\alpha/2}$; $a\in[0, 1]\}$ on ${\mathbb R}^d$ that evolves continuously from $\Delta$ to $\Delta+ \Delta^{\alpha/2}$, where $d\geq1$ and $\alpha\in(0, 2)$. It gives rise to a family of Lévy processes $\{X^a, a\in[0, 1]\}$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. Using a recently obtained uniform boundary Harnack principle with explicit decay rate, we establish sharp bounds for the Green function of the process $X^a$ killed upon exiting a bounded $C^{1,1}$ open set $D\subset{\mathbb R}^d$. Our estimates are uniform in $a\in(0, 1]$ and taking $a\to0$ recovers the Green function estimates for Brownian motion in $D$. As a consequence of the Green function estimates for $X^a$ in $D$, we identify both the Martin boundary and the minimal Martin boundary of $D$ with respect to $X^a$ with its Euclidean boundary. Finally, sharp Green function estimates are derived for certain Lévy processes which can be obtained as perturbations of $X^a$.

We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an equivalent zero-sum game of control and stopping, between an agent (the “stopper”) who chooses the termination time of the game, and an agent (the “controller,” or “nature”) who selects the probability measure.

We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the $p$th power, $p ≥ 2$, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable.

Our methods allow us to include Zakai’s equation for the Kalman–Bucy filter into the general filtering theory.

It is shown that a product preserving bijective (not necessarily real linear or continuous) operator on an appropriate class of complex valued functions must have either the form $[\phi\mapsto \phi\circ u]$ or $[\phi\mapsto\overline{\phi\circ u}]$ where $u$ is a fixed diffeomorphism of the base.

The paper contains the description of the optimal constant $\beta=3.4351\!\ldots$ for which the following inequality holds. Let $X$ be a real-valued martingale, $H$ be a predictable process taking values in $[-1,1]$ and let $Y$ be an Itô integral of $H$ with respect to $X$. Then \[ \Bigl\Vert\sup_{t\geq0}|Y_t|\Bigr\Vert_1\leq\beta\Bigl\Vert\sup_{t\geq0}|X_t|\Bigr\Vert_1. \] A version of this inequality in the discrete-time case is also established. The proof is based on Burkholder's technique, which relates the above estimate to the construction of an upper solution to a corresponding nonlinear three-dimensional problem.

Let $X$ be a separable Banach space, $Y$ a closed, nonreflexive, linear subspace, and $P$ the set of points admitting a nearest approximation in $Y$. Then $P$ is an analytic set, and has three obvious algebraic properties. By adjusting the norm of $X$, any analytic set of this kind can be realized as the set of elements proximal to $Y$.

The uniform infinite planar quadrangulation is an infinite random graph embedded in the plane, which is the local limit of uniformly distributed finite quadrangulations with a fixed number of faces. We study asymptotic properties of this random graph. In particular, we investigate scaling limits of the profile of distances from the distinguished point called the root, and we get asymptotics for the volume of large balls. As a key technical tool, we first describe the scaling limit of the contour functions of the uniform infinite well-labeled tree, in terms of a pair of eternal conditioned Brownian snakes. Scaling limits for the uniform infinite quadrangulation can then be derived thanks to an extended version of Schaeffer’s bijection between well-labeled trees and rooted quadrangulations.

We give a new proof of a result by J. Bourgain which says that if $(\Omega,\mathcal{F},P)$ is a product of probability spaces then $V _d$—the orthonormal in $L_2(\Omega ,\mathcal{F},P)$ projection on the space spanned by those $X \in L_2(\Omega,\mathcal{F},P)$ which depend on most of $d$-variables is a bounded operator in $L_p(\Omega,\mathcal{F},P)$ for $1<p<\infty$. We prove that for $X \in L_p(\Omega,\mathcal{F},P)$ $E|V_d(X)|^p \le C_{p,d} E|X|^p$ with $C_{p,d}= (c\frac{\hat{p}}{\ln{\hat p}})^{dp}$, where ${\hat p}= \max\{p, \frac{p}{p-1}\}$ and $c$ is an universal constant.