Absolute continuity of symmetric Markov processes (original) (raw)
Stochastic calculus for symmetric Markov processes
The Annals of Probability, 2008
Using time-reversal, we introduce a stochastic integral for zeroenergy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an Itô formula for Dirichlet processes is obtained.
Perturbation of symmetric Markov processes
Probability Theory and Related Fields, 2007
We present a path-space integral representation of the semigroup associated with the quadratic form obtained by a lower order perturbation of the L 2-infinitesimal generator L of a general symmetric Markov process. An illuminating concrete example for L is ∆ D − (−∆) s D , where D is a bounded Euclidean domain in R d , s ∈]0, 1[, ∆ D is the Laplacian operator in D with zero Dirichlet boundary condition and −(−∆) s D is the fractional Laplacian operator in D with zero exterior condition. The strong Markov process corresponding to L is a Lévy process that is the sum of Brownian motion in R d and an independent symmetric (2s)-stable process in R d killed upon exiting domain D. This probabilistic representation is a combination of Feynman-Kac and Girsanov formulas. Crucial to the development is to use the extension of Nakao's stochastic integral for zero-energy additive functionals and the associated Itô formula, both of which were recently developed in [3].
Fluctuation theory and exit systems for positive self-similar Markov processes
Annals of Probability, 2012
For a positive self-similar Markov process, X, we construct a local time for the random set, , of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R, H ) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set and the process X sampled on the local time scale. The process (R, H ) is described in terms of a ladder process linked to the Lévy process associated to X via Lamperti's transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R, H ) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying Lévy process oscillates.
Continuity of symmetric stable processes
Journal of Multivariate Analysis, 1989
The path continuity of a symmetric p-stable process is examined in terms of any stochastic integral representation for the process. When 0 < p < 1, we give necessary and suflicient conditions for path continuity in terms of any (every) representation. When 1 &p<2, we extend the known sutliciency condition in terms of metric entropy and offer a conjecture for the stable version of the Dudley-Fernique theorem. Finally, necessary and sufficient conditions for path continuity are given in terms of continuity at a point for 0 < p < 2.
Stochastic Processes and their Applications, 2014
In this paper, we study sharp Dirichlet heat kernel estimates for a large class of symmetric Markov processes in C 1,η open sets. The processes are symmetric pure jump Markov processes with jumping intensity κ(x, y)ψ 1 (|x−y|) −1 |x−y| −d−α , where α ∈ (0, 2). Here, ψ 1 is an increasing function on [0, ∞), with ψ 1 (r) = 1 on 0 < r ≤ 1 and c 1 e c2r β ≤ ψ 1 (r) ≤ c 3 e c4r β on r > 1 for β ∈ [0, ∞], and κ(x, y) is a symmetric function confined between two positive constants, with |κ(x, y) − κ(x, x)| ≤ c 5 |x − y| ρ for |x − y| < 1 and ρ > α/2. We establish two-sided estimates for the transition densities of such processes in C 1,η open sets when η ∈ (α/2, 1]. In particular, our result includes (relativistic) symmetric stable processes and finite-range stable processes in C 1,η open sets when η ∈ (α/2, 1].
On the convergence of sequences of stationary jump Markov processes
Statistics & Probability Letters, 1983
This paper presents two main results: first, a Liapunov type criterion for the existence of a stationary probability distribution for a jump Markov process; second, a Liapunov type criterion for existence and tightness of stationary probability distributions for a sequence of jump Markov processes. If the corresponding semigroups TN(t ) converge, under suitable hypotheses on the limit semigroup, this last result yields the weak convergence of the sequence of stationary processes (Tjv (t), *r N) to the stationary limit one.
A note on discontinuous time changes of Markov processes
Stochastic Processes and their Applications, 1986
The "time change" of a Markov process via the inverse of a discontinuous additive functional A, can be accomplished in two steps. First, perform a time change via the inverse of the strictly increasing discontinuous additive functional obtained by replacing the continuous part of A, by t. The second step is an ordinary time change via the inverse of a continuous additive functional. Decomposing the time change in this way is useful in studying the time changed process.
A conditional approach to the anticipating Girsanov transformation
Probability Theory and Related Fields, 1993
We study the law of a stochastic differential equation where the drift anticipates the future behavior of the Brownian path co, for example the endpoint. We first investigate anticipation of the endpoint, using a conditional Girsanov transformation and methods of Malliavin calculus. A combination with results of Buckdahn [2] leads to new versions of the anticipating Girsanov transformation of Ramer and Kusuoka, and in particular to explicit formulas for the Carleman-Fredholm determinant.
On solutions of Kolmogorov's equations for jump Markov processes
arXiv (Cornell University), 2013
This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller's seminal paper. In particular, this paper extends Feller's results for continuous Q-functions to measurable Q-functions and provides additional results.
On jump-diffusion processes with regime switching: martingale approach
2015
We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy approach is also developed. It is shown that in contrast to the case of Lévy processes, for this model an Esscher transformation does not produce the minimal relative entropy.