We give a necessary and sufficient condition for a homogeneous Markov process taking values in $\R^n$ to enjoy the time-inversion property of degree $α$. The condition sets the shape for the semigroup densities of the process and allows to further extend the class of known processes satisfying the time-inversion property. As an application we recover the result of Watanabe in \cite{Wa1975} for continuous and conservative Markov processes on $\R_+$. As new examples we generalize Dunkl processes and construct a matrix-valued process with jumps related to the Wishart process by a skew-product representation.
We review and study a one-parameter family of functional transformations, denoted by $(S^{(β)})_{β\in \R}$, which, in the case $β<0$, provides a path realization of bridges associated to the family of diffusion processes enjoying the time inversion property. This family includes the Brownian motion, Bessel processes with a positive dimension and their conservative $h$-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary crossing times for these diffusions over a given function $f$ to those over the image of $f$ by the …
T. Szarek [Stud. Math. 189 (2008), § 4] have discussed the relationship between two important notions concerning Markov semigroups: the asymptotic strong Feller property and the e-property, asserting that the former property implies the latter one. In this very short note we rectify this issue exhibiting a simple example of a Markov-Feller semigroup enjoying the asymptotic strong Feller property, for which the e-property is not satisfied. (See also the comment on a connection between the asymptotic strong Feller property and the e-property by T. Szarek, D. Worm [ETDS 32 (2012), § 1]). Addition…
We show that any $\mathbb{R}^d\setminus\{0\}$-valued self-similar Markov process $X$, with index $α>0$ can be represented as a path transformation of some Markov additive process (MAP) $(θ,ξ)$ in $S_{d-1}\times\mathbb{R}$. This result extends the well known Lamperti transformation. Let us denote by $\widehat{X}$ the self-similar Markov process which is obtained from the MAP $(θ,-ξ)$ through this extended Lamperti transformation. Then we prove that $\widehat{X}$ is in weak duality with $X$, with respect to the measure $π(x/\|x\|)\|x\|^{α-d}dx$, if and only if $(θ,ξ)$ is reversible with respect …
We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian …
We introduce quantum Markov categories as a structure that refines and extends a synthetic approach to probability theory and information theory so that it includes quantum probability and quantum information theory. In this broader context, we analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inverses. We prove that each one is a strictly special instance of the latter for certain subcategories, providing a categorical foundation for Bayesian inversion as a generalization of reversing a process. We unif…
Let $X$ be a standard Markov process. We prove that a space inversion property of $X$ implies the existence of a Kelvin transform of $X$-harmonic, excessive and operator-harmonic functions and that the inversion property is inherited by Doob $h$-transforms. We determine new classes of processes having space inversion properties amongst transient processes {satisfying the} time inversion property. {For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly.} We treat in details the examples of free scaled power Bessel proc…
The following electromagnetism (EM) inverse problem is addressed. It consists in estimating local radioelectric properties of materials recovering an object from the global EM scattering measurement, at various incidences and wave frequencies. This large scale ill-posed inverse problem is explored by an intensive exploitation of an efficient 2D Maxwell solver, distributed on High Performance Computing (HPC) machines. Applied to a large training data set, a statistical analysis reduces the problem to a simpler probabilistic metamodel, on which Bayesian inference can be performed. Considering th…
A popular method to compute first-passage probabilities in continuous-time Markov chains is by numerically inverting their Laplace transforms. Past decades, the scientific computing community has developed excellent numerical methods for solving problems governed by partial differential equations (PDEs), making the availability of a Laplace transform not necessary here for computational purposes. In this study we demonstrate that numerical PDE problem solvers are suitable for computing first passage times, and can be very efficient for this purpose. By doing extensive computational experiments…
Let W be either the number of descents or inversions of a permutation. Stein's method is applied to show that W satisfies a central limit theorem with error rate n^(-1/2). The construction of an exchangeable pair (W,W') used in Stein's method is non-trivial and uses a non-reversible Markov chain.
We consider a discrete-time Markov chain, called fragmentation process, that describes a specific way of successively removing objects from a linear arrangement. The process arises in population genetics and describes the ancestry of the genetic material of individuals in a population experiencing recombination. We aim at the law of the process over time. To this end, we investigate sets of realisations of this process that agree with respect to a specific order of events and represent each such set by a rooted (binary) tree. The probability of each tree is, in turn, obtained by Möbius inversi…
This study of occupation time densities for continuous-time Markov processes was inspired by the work of E.Nir et al (2006) in the field of Single Molecule FRET spectroscopy. There, a single molecule fluctuates between two or more states, and the experimental observable depends on the state's occupation time distribution. To mathematically describe the observable there was a need to calculate a single state occupation time distribution. In this paper, we consider a Markov process with countably many states. In order to find a one-stete occupation time density, we use a combination of Fourier a…
Regularization of the superposition principle: Potential theory meets Fokker-Planck equations
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；University POLITEHNICA Bucharest（布加勒斯特理工大学）；Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest（布加勒斯特大学数学与计算机科学系）；Fakultät für Mathematik, Universität Bielefeld（比勒费尔德大学数学系）；Academy for Mathematics and Systems Science, CAS（中国科学院数学与系统科学研究院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)（香港中文大学（深圳）数据科学学院）
