Characteristic operators of Markov processes Differential generators of diffusion processes

1965 ◽  
pp. 131-172
Author(s):  
E. B. Dynkin
Keyword(s):  
Markov Processes ◽  
Diffusion Processes

The age distribution of Markov processes

1977 ◽  
Vol 14 (3) ◽  
pp. 492-506 ◽  
Author(s):  
Benny Levikson
Keyword(s):  
Differential Equations ◽  
Markov Processes ◽  
Difference Equations ◽  
Diffusion Processes ◽  
Genetic Model ◽  
Branching Processes ◽  
Age Distribution ◽  
Renewal Theory ◽  
Gene Frequencies ◽  
Birth And Death Processes

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

scholarly journals A Symmetry-Based Approach for First-Passage-Times of Gauss-Markov Processes through Daniels-Type Boundaries

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 279
Author(s):  
Enrica Pirozzi
Keyword(s):  
Probability Density ◽  
Markov Processes ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Transition Probability ◽  
Passage Time ◽  
First Passage ◽  
Transition Probability Density ◽  
Symmetry Properties ◽  
Time Density

Symmetry properties of the Brownian motion and of some diffusion processes are useful to specify the probability density functions and the first passage time density through specific boundaries. Here, we consider the class of Gauss-Markov processes and their symmetry properties. In particular, we study probability densities of such processes in presence of a couple of Daniels-type boundaries, for which closed form results exit. The main results of this paper are the alternative proofs to characterize the transition probability density between the two boundaries and the first passage time density exploiting exclusively symmetry properties. Explicit expressions are provided for Wiener and Ornstein-Uhlenbeck processes.

The age distribution of Markov processes

1977 ◽  
Vol 14 (03) ◽  
pp. 492-506 ◽  
Author(s):  
Benny Levikson
Keyword(s):  
Differential Equations ◽  
Markov Processes ◽  
Difference Equations ◽  
Diffusion Processes ◽  
Genetic Model ◽  
Branching Processes ◽  
Age Distribution ◽  
Renewal Theory ◽  
Gene Frequencies ◽  
Birth And Death Processes

A limiting distribution for the age of a class of Markov processes is found if the present state of the process is known. We use this distribution to find the age of branching processes. Using the fact that the moments of the age of birth and death processes and of diffusion processes satisfy difference equations and differential equations respectively, we find simple formulas for these moments. For the Wright–Fisher genetic model we find the probability that a given allele is the oldest in the population if all the gene frequencies are known. The proofs of the main results are based on methods from renewal theory.

scholarly journals Uniform convergence of penalized time-inhomogeneous Markov processes

2018 ◽  
Vol 22 ◽  
pp. 129-162 ◽  
Author(s):  
Nicolas Champagnat ◽  
Denis Villemonais
Keyword(s):  
Asymptotic Behavior ◽  
Markov Processes ◽  
Random Environment ◽  
Diffusion Processes ◽  
Exponential Convergence ◽  
General Criterion ◽  
Infinite Time ◽  
Birth And Death Processes ◽  
One Dimensional ◽  
Dimensional Diffusion

We provide a general criterion ensuring the exponential contraction of Feynman–Kac semi-groups of penalized processes. This criterion applies to time-inhomogeneous Markov processes with absorption and killing through penalization. We also give the asymptotic behavior of the expected penalization and provide results of convergence in total variation of the process penalized up to infinite time. For exponential convergence of penalized semi-groups with bounded penalization, a converse result is obtained, showing that our criterion is sharp in this case. Several cases are studied: we first show how our criterion can be simply checked for processes with bounded penalization, and we then study in detail more delicate examples, including one-dimensional diffusion processes conditioned not to hit 0 and penalized birth and death processes evolving in a quenched random environment.

scholarly journals Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment

2022 ◽  
Vol 2022 (1) ◽  
pp. 013206
Author(s):  
Cécile Monthus
Keyword(s):  
Large Deviations ◽  
Markov Processes ◽  
Large Time ◽  
Diffusion Processes ◽  
Large Deviation ◽  
Time Window ◽  
Extinction Rate ◽  
Empirical Density ◽  
Full Optimization ◽  
Absorbing States

Abstract The large deviations at level 2.5 are applied to Markov processes with absorbing states in order to obtain the explicit extinction rate of metastable quasi-stationary states in terms of their empirical time-averaged density and of their time-averaged empirical flows over a large time-window T. The standard spectral problem for the slowest relaxation mode can be recovered from the full optimization of the extinction rate over all these empirical observables and the equivalence can be understood via the Doob generator of the process conditioned to survive up to time T. The large deviation properties of any time-additive observable of the Markov trajectory before extinction can be derived from the level 2.5 via the decomposition of the time-additive observable in terms of the empirical density and the empirical flows. This general formalism is described for continuous-time Markov chains, with applications to population birth–death model in a stable or in a switching environment, and for diffusion processes in dimension d.

Optimal estimation of diffusion processes hidden by general obstacles

2001 ◽  
Vol 38 (4) ◽  
pp. 1067-1073 ◽  
Author(s):  
Hyung Geun Kim ◽  
Dougu Nam
Keyword(s):  
Diffusion Process ◽  
Markov Processes ◽  
Diffusion Processes ◽  
Borel Function ◽  
Optimal Estimation ◽  
Dimensional Diffusion ◽  
Optimal Estimator

Let Xt be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose Xt is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X1) | Xs1Xs∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

scholarly journals Lyapunov-type conditions for non-strong ergodicity of Markov processes

2021 ◽  
Vol 58 (1) ◽  
pp. 238-253
Author(s):  
Yong-Hua Mao ◽  
Tao Wang
Keyword(s):  
Markov Processes ◽  
Riemannian Manifolds ◽  
Diffusion Processes ◽  
Stable Processes ◽  
Strong Ergodicity ◽  
Stable Noise

AbstractWe present Lyapunov-type conditions for non-strong ergodicity of Markov processes. Some concrete models are discussed, including diffusion processes on Riemannian manifolds and Ornstein–Uhlenbeck processes driven by symmetric $\alpha$-stable processes. In particular, we show that any process of d-dimensional Ornstein–Uhlenbeck type driven by $\alpha$-stable noise is not strongly ergodic for every $\alpha\in (0,2]$.

Optimal estimation of diffusion processes hidden by general obstacles

2001 ◽  
Vol 38 (04) ◽  
pp. 1067-1073
Author(s):  
Hyung Geun Kim ◽  
Dougu Nam
Keyword(s):  
Diffusion Process ◽  
Markov Processes ◽  
Diffusion Processes ◽  
Borel Function ◽  
Optimal Estimation ◽  
Dimensional Diffusion ◽  
Optimal Estimator

Let X t be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose X t is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X 1) | X s 1 X s ∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

scholarly journals Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

2011 ◽  
Vol 2011 ◽  
pp. 1-89
Author(s):  
Thibaud Taillefumier ◽  
Jonathan Touboul
Keyword(s):  
Markov Processes ◽  
Diffusion Processes ◽  
Random Coefficients ◽  
Partial Sums ◽  
Stochastic Integration ◽  
Alternative Representation ◽  
Finite Dimensional ◽  
Unit Variance ◽  
Schauder Bases ◽  
Normal Law

The study of the multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Here, inspired by the Lévy-Ciesielski construction of the Wiener process, we propose an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unit variance. We thereby offer a natural multiresolution description of the Gauss-Markov processes as limits of finite-dimensional partial sums of the expansion, that are strongly almost-surely convergent. Moreover, such finite-dimensional random processes constitute an optimal approximation of the process, in the sense of minimizing the associated Dirichlet energy under interpolating constraints. This approach allows for a simpler treatment of problems in many applied and theoretical fields, and we provide a short overview of applications we are currently developing.

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