We introduce a method of proving maximal inequalities for Hilbert- space-valued differentially subordinate local martingales. As an application, we prove that if \(X=(X_t)_{t\ge 0},\, Y=(Y_t)_{t\ge 0}\) are local martingales such that \(Y\) is differentially subordinate to \(X\), then $$\begin{aligned} ||Y||_1\le \beta ||\sup _{t\ge 0}|X_t|\;||_1, \end{aligned}$$where \(\beta ...

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.

We consider processes of the form [s,T]∋t↦u(t,X t ), where (X,P s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if \(u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})\) with \(\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})\) then there is a quasi-continuous version \(\tilde{u}\) of u such that ...

We relate various concepts of fractal dimension of the limiting set \(\mathcal{C}\) in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in \(\mathcal{C}\) (the “dust”). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of ...

The affine group of a homogeneous tree is the group of all its isometries fixing an end of its boundary. We consider a random walk with law μ on this group and the associated random processes on the tree and its boundary. In the drift-free case there exists on the boundary of the tree a unique μ-invariant Radon measure. In this paper we describe its behaviour at infinity.

Consider a system of particles evolving as independent and identically distributed (i.i.d.) random walks. Initial fluctuations in the particle density get translated over time with velocity \(\vec{v}\), the common mean velocity of the random walks. Consider a box centered around an observer who starts at the origin and moves with constant velocity \(\vec{v}\). To observe ...

We attempt to unify the analysis of several families of naturally occurring multidimensional stochastic processes by studying the underlying combinatorics involved. At equilibrium, the behavior of these processes is determined by the properties of a randomly chosen point of a corresponding polyhedron. How such a randomly chosen point behaves is a difficult question which is ...

We consider the Cauchy problem for systems of viscous conservation laws. We obtain three different but related stochastic representations of weak solutions of the problem: in terms of solutions to systems of usual backward stochastic differential equations, in terms of solutions to some stochastic backward systems, and in terms of solutions to some forward-backward stochastic ...

A theorem of Siebert in its essential part asserts that if μ n (t) are semigroups of probability measures on a Lie group G, and P n are the corresponding generating functionals, then $$\bigl \langle \mu_n(t),f \bigr \rangle \ \xrightarrow[n]{}\ \bigl \langle \mu_0(t),f \bigr \rangle , \quad f\in C_b(G), \ t>0,$$ implies $$\langle \pi_{P_n}u,v\rangle \ \xrightarrow[n]{}\ \langle ...

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ *x ...

Let μ be a Poisson random measure, let \(\mathbb{F}\) be the smallest filtration satisfying the usual conditions and containing the one generated by μ, and let \(\mathbb{G}\) be the initial enlargement of \(\mathbb{F}\) with the σ-field generated by a random variable G. In this paper, we first show that the mutual information between the enlarging random variable G and the ...

We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with ...