Journal of Theoretical Probability

http://link.springer.com/journal/10959

List of Papers (Total 46)

Maximal Inequalities for Martingales and Their Differential Subordinates

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 ...

A Note on Wiener–Hopf Factorization for Markov Additive Processes

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.

On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators

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 ...

Dimension (In)equalities and Hölder Continuous Curves in Fractal Percolation

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 ...

Random Walks on the Affine Group of a Homogeneous Tree in the Drift-Free Case

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.

Current Fluctuations for Independent Random Walks in Multiple Dimensions

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 ...

A Combinatorial Analysis of Interacting Diffusions

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 ...

Stochastic Representation of Weak Solutions of Viscous Conservation Laws: A BSDE Approach

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 ...

Convergence of Semigroups of Complex Measures on a Lie Group

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 ...

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

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 ...

Initial Enlargement of Filtrations and Entropy of Poisson Compensators

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 ...

Distribution of Global Measures of Deviation Between the Empirical Distribution Function and Its Concave Majorant

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 ...