We study recurrence properties and the validity of the (weak) law of large numbers for (discrete time) processes which, in the simplest case, are obtained from simple symmetric random walk on $\mathbb{Z}$ by modifying the distribution of a step from a fresh point. If the process is denoted as $\{S_n\}_{n \ge0}$, then the conditional distribution of $S_{n+1} - S_n$ given the past through time $n$ is the distribution of a simple random walk step, provided $S_n$ is at a point which has been visited already at least once during $[0,n-1]$. Thus, in this case, $P\{S_{n+1}-S_n = \pm1|S_\ell, \ell\le n\} = 1/2$. We denote this distribution by $P_1$. However, if $S_n$ is at a point which has not been visited before time $n$, then we take for the conditional distribution of $S_{n+1}-S_n$, given the past, some other distribution $P_2$. We want to decide in specific cases whether $S_n$ returns infinitely often to the origin and whether $(1/n)S_n \to0$ in probability. Generalizations or variants of the $P_i$ and the rules for switching between the $P_i$ are also considered.

A model for optimal consumption and investment is posed whose solution is provided by the classical Merton analysis when there is zero transaction cost. A probabilistic argument is developed to identify the loss in value when a proportional transaction cost is introduced. There are two sources of this loss. The first is a loss due to “displacement” that arises because one cannot maintain the optimal portfolio of the zero-transaction-cost problem. The second loss is due to “transaction,” a loss in capital that occurs when one adjusts the portfolio. The first of these increases with increasing tolerance for departure from the optimal portfolio in the zero-transaction-cost problem, while the second decreases with increases in this tolerance. This paper balances the marginal costs of these two effects. The probabilistic analysis provided here complements earlier work on a related model that proceeded from a viscosity solution analysis of the associated Hamilton–Jacobi–Bellman equation.

Basic derivative formulas are presented for hypoelliptic heat semigroups and harmonic functions extending earlier work in the elliptic case. According to our approach, special emphasis is placed on integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with boundary conditions, as well as with difficulties related to finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin calculus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally, some extensions to the nonlinear setting of harmonic mappings are discussed.

We consider large scale behavior of the solution set $\{u(t,x): x\in{\mathbf{Z^d}}\}$ of the parabolic Anderson equation \begin{eqnarray*} u(t,x)&=&1+\kappa\int_0^t\Delta u(s,x)\, ds \\ &&{}+\int_0^tu(s,x)\,\partial W_x(s),\quad x\in{\mathbf{Z^d}},t\ge0, \end{eqnarray*} where $\{W_x : x\in{\mathbf{Z^d}}\}$ is a field of i.i.d. standard, one-dimensional Brownian motions, $\Delta$ is the discrete Laplacian and $\kappa>0.$ We establish that the properly normalized sum, $\sum_{x\in\Lambda _L}u(t,x),$ over spatially growing boxes $\Lambda_L=\{x\in{\mathbf {Z^d}}:\Vert x\Vert<L\}$ has an asymptotically normal distribution if the box $\Lambda_L$ grows sufficiently quickly with $t$ and provided $\kappa$ is sufficiently small depending on dimension. The asymptotic distribution of properly normalized sums over spatially growing disjoint boxes $\Lambda^1_L,\Lambda^2_L$ is asymptotically independent. Thus, on sufficiently large scales the field of solutions averaged over disjoint large boxes looks like an i.i.d. Gaussian field. We identify the variance of this Gaussian distribution in terms of the eigenfunction of the positive eigenvalue of the operator $2\kappa\Delta+\delta_0$.

We study the nonlinear stochastic heat equation driven by space–time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized stochastic convolution via Young-type inequalities.

Given a pair of separable, real Banach spaces $E$ and $F$ and a centered Gaussian measure $\mu $ on $E$, one can ask what sort of Borel measurable maps $\Phi : E\longrightarrow F$ map $\mu $ to a centered Gaussian measure on $F$. Obviously, a sufficient condition is that $\Phi $ be linear. On the other hand, linearity is far more than is really needed. Indeed, it suffices to know that $\Phi $ has the property that \[ \Phi \biggl(\frac{x_1+x_2}{\sqrt2} \biggr)=\frac{\Phi (x_1)+\Phi (x_2)}{\sqrt2} \] for $\mathcal{W}^2$-almost every $(x_1,x_2)\in E^2$. In this article, I will first prove a structure theorem which shows that any map $\Phi $ which satisfies this property arises from a linear map on the Cameron–Martin space associated with $\mu $ on $E$. I will then investigate which linear maps on the Cameron–Martin space determine a $\Phi $, and finally I will discuss some of the properties of $\Phi $ which reflect properties of the linear map from which it is determined.

For $d\geq1$ and $0<\beta<\alpha<2$, consider a family of pseudo differential operators $\{\Delta^{\alpha} + a^\beta \Delta^{\beta/2}$; $a \in[0, 1]\}$ on $\mathbb{R}^d$ that evolves continuously from $\Delta^{\alpha/2}$ to $ \Delta^{\alpha/2}+ \Delta^{\beta/2}$. It gives arise to a family of Lévy processes $\{X^a, a\in[0, 1]\}$ on $\mathbb{R}^d$, where each $X^a$ is the independent sum of a symmetric $\alpha$-stable process and a symmetric $\beta$-stable process with weight $a$. For any $C^{1,1}$ open set $D\subset\mathbb{R}^d$, we establish explicit sharp two-sided estimates, which are uniform in $a\in(0, 1]$, for the transition density function of the subprocess $X^{a, D}$ of $X^a$ killed upon leaving the open set~$D$. The infinitesimal generator of $X^{a, D}$ is the nonlocal operator $\Delta^{\alpha} + a^\beta\Delta^{\beta/2}$ with zero exterior condition on $D^c$. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for $X^{a, D}$ and uniform boundary Harnack principle for $X^a$ in $D$ with explicit decay rate.

We will show how to get Burkholder’s function from (Ann. Probab. 12 (1984) 647–702) by using Monge–Ampère equation. This method is quite different from those in the series of Burkholder’s papers (Ann. Probab. 12 (1984) 647–702, An extension of classical martingale inequality (1986) Marcel Dekker, Astérisque 157–158 (1988) 75–94, In Harmonic analysis and partial differential equations (1989) 1–23 Springer, In École d’Ete de Probabilités de Saint-Flour XIX (1991) 1–66 Springer, Ann. Probab. 22 (1994) 995–1025, Studia Math. 91 (1988) 79–83).

We give a direct construction of the conformally invariant measure on self-avoiding loops in Riemann surfaces (Werner measure) from chordal $\mathrm{SLE}_{8/3}$. We give a new proof of uniqueness of the measure and use Schramm’s formula to construct a measure on boundary bubbles encircling an interior point. After establishing covariance properties for this bubble measure, we apply these properties to obtain a measure on loops by integrating measures on boundary bubbles. We calculate the distribution of the conformal radius of boundary bubbles encircling an interior point and deduce from it explicit upper and lower bounds for the loop measure.

A primary motivating application of the author’s work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions. That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings. The search for a Martin-type ideal boundary for the placement of those measures resulted in a new almost everywhere regular boundary that supported the representing measures for a large proper subclass of all nonnegative harmonic functions. In this note, we outline the construction of the rich measure spaces that are now called Loeb measure spaces in the literature. We then review the application of these measure spaces to the construction of representing measures. We finish with the problem of constructing an appropriate boundary associated with the nonstandard construction of general representing measures that supports all of those measures.

Let $B$ be a Brownian motion. We show that there is a process $H$ predictable in the natural filtration of $B$, such that $H⋅S$ is a Brownian motion in its own filtration, where $S_t = B_t + t$. In other words, $H$ hides the constant drift. This gives a positive answer to a question posed by Marc Yor.

Pathwise nonuniqueness is established for nonnegative solutions of the parabolic stochastic pde \[ \frac{\partial X}{\partial t}=\frac{\Delta}{2}X+X^p\dot W+\psi,\quad X_0 \equiv0 \] where $\dot W$ is a white noise, $\psi\ge0$ is smooth, compactly supported and nontrivial, and $0<p<1/2$. We further show that any solution spends positive time at the $0$ function.

We introduce and analyze a notion of orthogonality and dimension for spaces of $\mathbb{C}^n$-martingales. In particular, the space of martingale transforms of heat-extensions of $L^2(\mathbb{R}^{2m})$ functions is shown to be the orthogonal sum of $2$ conformal subspaces. We show that a theorem and proof of D. L. Burkholder for the computation of $L^p$-norm of martingale transforms applies specially for $n$-conformal and for pairwise conformal $n$-martingales. This leads to estimates of the $L^p$-norms of singular integral operators associated with the second-order Riesz transforms, in particular of the Beurling–Ahlfors operator.