Poisson kernels of half-spaces in real hyperbolic spaces.
Poisson point process limits in size-biased Galton-Watson trees.
Poisson point processes attached to symmetric diffusions
Poisson snake and fragmentation.
Poisson stochastic integration in Hilbert spaces
Polarization, conformal invariants and Brownian motion.
Policy iteration for continuous-time average reward Markov decision processes in Polish spaces.
Polynomial asymptotic stability of damped stochastic differential equations.
Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times
Let X be a one-dimensional positive recurrent diffusion with initial distribution ν and invariant probability μ. Suppose that for some p>1, ∃a∈ℝ such that ∀x∈ℝ, and , where Ta is the hitting time of a. For such a diffusion, we derive non-asymptotic deviation bounds of the form ℙν(|(1/t)∫0tf(Xs) ds−μ(f)|≥ε)≤K(p)(1/tp/2)(1/εp)A(f)p. Here f bounded or bounded and compactly supported and A(f)=‖f‖∞ when f is bounded and A(f)=μ(|f|) when f is bounded and compactly supported. We also give, under...
Polynomial deviation bounds for recurrent Harris processes having general state space
Consider a strong Markov process in continuous time, taking values in some Polish state space. Recently, Douc et al. [Stoc. Proc. Appl. 119, (2009) 897–923] introduced verifiable conditions in terms of a supermartingale property implying an explicit control of modulated moments of hitting times. We show how this control can be translated into a control of polynomial moments of abstract regeneration times which are obtained by using the regeneration method of Nummelin, extended to the time-continuous...
Population-size-dependent branching processes.
Portfolio selection with jumps under regime switching.
Positively and negatively excited random walks on integers, with branching processes.
Positivity of Brownian transition densities.
Potential theory for a family of several Markov processes
Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains
The purpose of the paper is to extend results of the potential theory of the classical Schrödinger operator to the α-stable case. To obtain this we analyze a weak version of the Schrödinger operator based on the fractional Laplacian and we prove the Conditional Gauge Theorem.
Potential theory of hyperbolic Brownian motion in tube domains
Let X = X(t); t ≥ 0 be the hyperbolic Brownian motion on the real hyperbolic space ℍⁿ = x ∈ ℝⁿ:xₙ > 0. We study the Green function and the Poisson kernel of tube domains of the form D × (0,∞)⊂ ℍⁿ, where D is any Lipschitz domain in . We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in . We give formulas and uniform estimates for the set . The constants in the estimates depend only on the dimension of the space.
Potential theory of one-dimensional geometric stable processes
The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter α ∈ (0,2]. This process has an infinitesimal generator of the form . As an application we prove the global scale invariant Harnack inequality as well as the boundary Harnack principle.
Potential Theory on the Infinite Dimensional Torus.
Potentials of a Markov process are expected suprema
Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and Meziou (2006) on the max-plus decomposition...