The theory of infinite dimensional oscillatory integrals and some of its applications are discussed, with special attention to the relations with the original work of K. Itô in this area. A recent general approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented, together with some new developments.

The purpose of this paper is to describe the probabilistic aspects underlying the theory of the hypoelliptic Laplacian, as a deformation of the standard elliptic Laplacian. The corresponding diffusion on the total space of the tangent bundle of a Riemannian manifold is a geometric Langevin process, that interpolates between the geometric Brownian motion and the geodesic flow. Connections with the central limit theorem for the occupation measure by the geometric Brownian motion are emphasized. Spectral aspects of the hypoelliptic deformation are also provided on tori. The relevant hypoelliptic deformation of the Laplacian in the case of Riemann surfaces of constant negative curvature is briefly described, in connection with Selberg's trace formula.

We study the scaling limit and prove the law of large numbers for weakly pinned Gaussian random fields under the critical situation that two possible candidates of the limits exist at the level of large deviation principle. This paper extends the results of [3], [7] for one dimensional fields to higher dimensions: $d\ge3 $, at least if the strength of pinning is sufficiently large.

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near 0. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.

This paper is concerned with the characterization of spaces of square integrable holomorphic functions on a complex manifold, $G$, in terms of the derivatives of the function at a fixed point $o\in G$. The reproducing kernel properties of square integrable holomorphic functions are reviewed and a number of examples are given. These examples include square integrable holomorphic functions relative to Gaussian measures on complex Euclidean spaces along with their generalizations to heat kernel measures on complex Lie groups. These results are intimately related to the Itô's chaos expansion in stochastic analysis and to the Fock space description of free quantum fields in physics.

We give necessary and sufficient conditions for sub-Gaussian estimates of the heat kernel of a strongly local regular Dirichlet form on a metric measure space. The conditions for two-sided estimates are given in terms of the generalized capacity inequality and the Poincaré inequality. The main difficulty lies in obtaining the elliptic Harnack inequality under these assumptions. The conditions for upper bound alone are given in terms of the generalized capacity inequality and the Faber–Krahn inequality.

We prove a version of the Wong-Zakai theorem for one-dimensional parabolic nonlinear stochastic PDEs driven by space-time white noise. As a corollary, we obtain a detailed local description of solutions.

In this paper we consider a general class of second order stochastic partial differential equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and has a homogeneous spatial covariance. Using the techniques of Malliavin calculus we derive the smoothness of the density of the solution at a fixed number of points $(t,x_1), \dots, (t,x_n)$, $t$ > 0, with some suitable regularity and nondegeneracy assumptions. We also prove that the density is strictly positive in the interior of the support of the law.

The Schramm-Loewner evolution (SLE) is a probability measure on random fractal curves that arise as scaling limits of two-dimensional statistical physics systems. In this paper we survey some results about the Hausdorff dimension and Minkowski content of ${\rm SLE}_\kappa$ paths and then extend the recent work on Minkowski content to the intersection of an SLE path with the real line.

We investigate the relations between the Poissonnian loop ensemble arising in the construction of random spanning trees, the free field, and random Eulerian networks.

We give an overview of the recent approach to the integration of rough paths that reduces the problem to an inhomogeneous analogue of the classical Young integration [13]. As an application, we extend an argument of Schwartz [11] to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.

We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by

$S=\sum_{n=1}^{\infty}\frac{I_n}{n^s},$

where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value 1 with probability $1/n^\beta$ and value 0 otherwise. Random series of this type are motivated by the record indicator sequences which have been studied in extreme value theory in statistics. We show that when $s$ > 0 and 0 < $\beta \le 1$ with $s+\beta$ > 1 the distribution of $S$ has a density; otherwise it is purely atomic or not defined because of divergence. In particular, in the case when $s$ > 0 and $\beta=1$, we prove that for every 0 < $s$ < 1 the density is bounded and continuous, whereas for every $s$ > 1 it is unbounded. In the case when $s$ > 0 and 0 < $\beta$ < 1 with $s+\beta$ > 1, the density is smooth. To show the absolute continuity, we obtain estimates of the Fourier transforms, employing van der Corput's method to deal with number-theoretic problems. We also give further regularity results of the densities, and present an example of a non-atomic singular distribution which is induced by the series restricted to the primes.

Beginning from a space of smooth, cylindrical and non-anticipative processes defined on a Wiener probability space $(\Omega, \mathcal{F}, P)$, we introduce a $P$-weighted Sobolev space, or “$P$-Sobolev space”, of non-anticipative path-dependent processes $u=u(t,\omega)$ such that the corresponding Sobolev derivatives $\mathcal{D}_{t}+(1/2)\Delta_x$ and $\mathcal{D}_{x}u$ of Dupire's type are well defined on this space. We identify each element of this Sobolev space with the one in the space of classical $L_P^p$ integrable Itô's process. Consequently, a new path-dependent Itô's formula is applied to all such Itô processes.

It follows that the path-dependent nonlinear Feynman–Kac formula is satisfied for most $L^p_P$-solutions of backward SDEs: each solution of such BSDE is identified with the solution of the corresponding quasi-linear path-dependent PDE (PPDE). Rich and important results of existence, uniqueness, monotonicity and regularity of BSDEs, obtained in the past decades can be directly applied to obtain their corresponding properties in the new fields of PPDEs.

In the above framework of $P$-Sobolev space based on the Wiener probability measure $P$, only the derivatives $\mathcal{D}_{t}+(1/2)\Delta_x$ and $\mathcal{D}_{x}u$ are well-defined and well-integrated. This prevents us from formulating and solving a fully nonlinear PPDE. We then replace the linear Wiener expectation $E_P$ by a sublinear $G$-expectation $\mathbb{E}^G$ and thus introduce the corresponding $G$-expectation weighted Sobolev space, or “$G$-Sobolev space”, in which the derivatives $\mathcal{D}_{t}u$, $\mathcal{D}_xu$ and $\mathcal{D}^2_{x}u$ are all well defined separately. We then formulate a type of fully nonlinear PPDEs in the $G$-Sobolev space and then identify them to a type of backward SDEs driven by $G$-Brownian motion.

We derive a stochastic representation formula for solutions of heat-type equations on vector bundles with time-dependent Riemannian metric over manifolds whose Riemannian metric is time-dependent as well. As a corollary we obtain a vanishing theorem for bounded ancient solutions under a curvature condition. Our results apply in particular to the case of differential forms.

In this paper, we provide a survey of recent distributional results obtained for Brownian type processes observed up to some random times. We focus on the case of hitting times and inverse local times and consider the situation where the processes are randomly sampled through a uniform random variable. We present various explicit formulas, some of them being quite remarkable.

This note has two goals. First, for those who have heard the term but do not know what it means, it provides a gentle introduction to Malliavin's calculus as it applies to degenerate parabolic partial differential equations. Second, it applies that theory to generalizations of Kolmogorov's example of a highly degenerate operator which is nonetheless hypoelliptic.

We study the level-set percolation of the Gaussian free field on $\mathbb Z^d$, $d \ge 3$. We consider a level $\alpha$ such that the excursion-set of the Gaussian free field above $\alpha$ percolates. We derive large deviation estimates on the probability that the excursion-set of the Gaussian free field below the level $\alpha$ disconnects a box of large side-length from the boundary of a larger homothetic box. It remains an open question whether our asymptotic upper and lower bounds are matching. With the help of a recent work of Lupu [21], we are able to infer some asymptotic upper bounds for similar disconnection problems by random interlacements, or by simple random walk.

Entropy was first introduced in 1865 by Rudolf Clausius in his study of the connection between work and heat. A mathematical definition was given by Boltzmann as the logarithm of the number of micro states that corresponds to a macro state. It plays important roles in statistical mechanics, in the theory of large deviations in probability, as an invariant in ergodic theory and as a useful tool in communication theory. This article explores some of the connections between these different contexts.

A notion of convergence of excursion measures is introduced. It is proved that convergence of excursion measures implies convergence in law of the processes pieced together from excursions. This result is applied to obtain homogenization theorems of jumping-in extensions for positive self-similar Markov processes, for Walsh diffusions and for the Brownian motion on the Sierpiński gasket.