We present a general method to construct $m$-symmetric diffusion processes $(X_t, \mathbf{P}_x)$ on any given locally compact metric space $(X, d)$ equipped with a Radon measure $m$. These processes are associated with local regular Dirichlet forms which are obtained as $\Gamma$-limits of approximating nonlocal Dirichlet forms. This general method works without any restrictions on $(X, d, m)$ and yields processes which are well defined for quasi every starting point.

The second main topic of this paper is to formulate and exploit the so-called Measure Contraction Property. This is a condition on the original data $(X, d, m)$ which can be regarded as a generalization of curvature bounds on the metric space $(X, d)$. It is a bound for distortions of the measure $m$ under contractions of the state space $X$ along suitable geodesics or quasi geodesics w.r.t. the metric $d$. In the case of Riemannian manifolds, this condition is always satisfied. Several other examples will be discussed, including uniformly elliptic operators, operators with weights, certain subelliptic operators, manifolds with boundaries or corners and glueing together of manifolds.

The Measure Contraction Property implies upper and lower Gaussian estimates for the heat kernel and a Harnack inequality for the associated harmonic functions. Therefore, the above-mentioned diffusion processes are strong Feller processes and are well defined for every starting point.

We study the behavior for large $|x|$ of Kunita-type stochastic flows $\phi(t, \omega x)$ on $R^d$, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large $|x|$, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.

We investigate the escape rate of the Brownian motion $W_x (t)$ on a complete noncompact Riemannian manifold. Assuming that the manifold has at most polynomial volume growth and that its Ricci curvature is bounded below, we prove that $$\dist (W_x (t), x) \leq \sqrt{Ct \log t}$$ for all large $t$ with probability 1. On the other hand, if the Ricci curvature is nonnegative and the volume growth is at least polynomial of the order $n > 2$ then $$\dist (W_x (t), x) \geq \frac{\sqrt{Ct}}{\log^{1/(n-2)} t \log \log^{(2+\varepsilon)/(n-2)} t} again for all large $t$ with probability 1 (where $\varepsilon > 0$).

We are interested in the almost sure asymptotic behavior of the windings of planar Brownian motion. Both the usual lim sup and Chung’s lim inf versions of the law of the iterated logarithm are presented for the so-called ‘‘big’’ and ‘‘small’’ winding angles. Our method is based on some very accurate estimates of the winding clock. The corresponding problem for a spherically symmetric random walk in $\mathbb{R}^2$ is also studied. A strong approximation using the Brownian big winding process is established. Similar results are obtained.

We construct a Lévy stochastic area for Brownian motion on the Sierpinski gasket. The standard approach via Itô integrals fails because this diffusion has sample paths which are far rougher than those of semimartingales. We thus provide an example demonstrating the restrictions of the semimartingale framework. As a consequence of the existence of the area one has a stochastic calculus and can solve stochastic differential equations driven by Brownian motion on the Sierpinski gasket.

We prove the existence of a unique mild solution for a stochastic evolution equation on a Hilbert space driven by a cylindrical Wiener process. The generator of the corresponding evolution system is supposed to be random and adapted to the filtration generated by the Wiener process. The proof is based on a maximal inequality for the Skorohod integral deduced from the Itô’s formula for this anticipating stochastic integral.

We consider the wave equation in two spatial dimensions driven by space–time Gaussian noise that is white in time but has a nondegenerate spatial covariance. We give a necessary and sufficient integral condition on the covariance function of the noise for the solution to the linear form of the equation to be a real-valued stochastic process, rather than a distribution-valued random variable. When this condition is satisfied, we show that not only the linear form of the equation, but also nonlinear versions, have a real-valued process solution. We give stronger sufficient conditions on the spatial covariance for the solution of the linear equation to be continuous, and we provide an estimate of its modulus of continuity.

The main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure. The exploration process appears as a simple local time functional of a Lévy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Lévy processes and continuous branching processes provides a unified perspective on both theories. In particular, we derive the adequate formulation of the classical Ray–Knight theorem for such Lévy processes. As a consequence of this theorem, we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process.

We show that $$\frac{1}{\sqrt{\varepsilon}}{\int_{-\infty}^{\infty} f(u)k_{\varepsilon}N_{\tau}^{X_{\varepsilon}}(u)du - \int_0^{\tau} f(X_t)a_t dt}$$ converges in law (as a continuous process) to $c_{\psi} \int_0^{\tau}f(X_t)a_t dB_t$ where $X_t = \int_0^t a_s dW_s + \int_0^t b_x ds$, with $W$ a standard Brownian motion, $a$ and $b$ regular and adapted processes, $X_{\varepsilon}(t) = \int_{-\infty}^{\infty}(1/ \varepsilon) \psi ((t - u)/ \varepsilon)X_u du, \psi$ a smooth kernel, $N_t^g (u)$ the number of roots of the equation $g(s) = u, s \epsilon (o, t], k_{\varepsilon} = \sqrt{\pi \varepsilon /2/ \parallel \psi \parallel_2$, $f$ a smooth function, a standard Brownian motion independent of $W$ and $c_{\psi}$ constant depending only on $\psi$. .

We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Itô’s equations the rate is $1/ \sqrt{n}$; we provide a necessary and sufficient condition for this rate to be $1/ \sqrt{n}$ when the driving semimartingale is a continuous martingale, or a continuous semimartingale under a mild additional assumption; we also prove that in these cases the normalized error processes converge in law.

The rate can also differ from $1/ \sqrt{n}$: this is the case for instance if the driving process is deterministic, or if it is a Lévy process without a Brownian component. It is again $1/ \sqrt{n}$ when the driving process is Lévy with a nonvanishing Brownian component, but then the normalized error processes converge in law in the finite-dimensional sense only, while the discretized normalized error processes converge in law in the Skorohod sense, and the limit is given an explicit form.

We prove that, under some general assumptions, the range of any nonconstant harmonic morphism from a simply connected open set $U$ in $\mathbf{R}^n$ to $\mathbf{R}^3$, $n > 3$, cannot avoid three concurrent half-lines, which is an extension to Picard’s little theorem. To this end, we will prove two results concerning the windings of Brownian motion around three concurrent half-lines in $\mathbf{R}^3$ and the recurrence of some domains linked with the harmonic morphism.

Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n \times N$ with i.i.d. complex standardized entries having finite fourth moment and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. It is known that, as $n \to \infty$, if $n/N$ converges to a positive number and the empirical distribution of the eigenvalues of $T_n$ converges to a proper probability distribution, then the empirical distribution of the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper we prove that, under certain conditions on the eigenvalues of $T_n$, for any closed interval outside the support of the limit, with probability 1 there will be no eigenvalues in this interval for all $n$ sufficiently large.

The Gaussian correlation conjecture states that for any two symmetric, convex sets in $n$-dimensional space and for any centered, Gaussian measure on that space, the measure of the intersection is greater than or equal to the product of the measures. In this paper we obtain several results which substantiate this conjecture. For example, in the standard Gaussian case, we show there is a positive constant, $c$ , such that the conjecture is true if the two sets are in the Euclidean ball of radius $c \sqrt{n}$. Further we show that if for every $n$ the conjecture is true when the sets are in the Euclidean ball of radius $\sqrt{n}$, then it is true in general. Our most concrete result is that the conjecture is true if the two sets are (arbitrary) centered ellipsoids.

An H-geodesic is a doubly infinite path which locally minimizes the passage time in the i.i.d. first passage percolation model on a half-plane H. Under the assumption that the bond passage times are continuously distributed with a finite mean, we prove that, with probability 1, H-geodesics do not exist. As a corollary we show that, with probability 1, any geodesic in the analogous model on the whole plane $\mathbf{Z}^2$ has to intersect all straight lines with rational slopes.

It is shown that an infinite subset of $Z^N$ is either recurrent for each aperiodic $N$-dimensional random walk with mean zero and finite variance, or transient for each of such random walks. This is an exact extension of the result by Spitzer in three dimensions to that in the dimensions $N \geq 4$.

We establish the central limit theorem for quadratic forms $\Sigma_{t, s=1}^N b(t - s)P_{m, n} (X_t, X_s)$ of the bivariate Appell polynomials $P_{m, n} (X_t, X_x))$ under time-domain conditions. These conditions relate the weights $b(t)$ and the covariances of the sequences $(P_{m, n} (X_t, X_s))$ and $(X_t)$. The time-domain approach, together with the spectral domain approach developed earlier, yields a general set of conditions for central limit theorems.

The domain of attraction of a 1-stable law on $\mathbf{R}^d$ is characterized by the expansions of the characteristic functions of its elements.

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