scholarly journals Exponential ergodicity for regime-switching diffusion processes in total variation norm

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jun Li ◽  
Fubao Xi
Keyword(s):  
Probability Measure ◽  
Total Variation ◽  
Regime Switching ◽  
Diffusion Processes ◽  
Invariant Probability Measure ◽  
Stability And Convergence ◽  
Time Behavior ◽  
Total Variation Norm ◽  
Switching Diffusion ◽  
Variation Norm

<p style='text-indent:20px;'>We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.</p>

scholarly journals Existence and Uniqueness of a Quasistationary Distribution for Markov Processes with Fast Return from Infinity

2014 ◽  
Vol 51 (3) ◽  
pp. 756-768 ◽  
Author(s):  
Servet Martínez ◽  
Jaime San Martín ◽  
Denis Villemonais
Keyword(s):  
Total Variation ◽  
Existence And Uniqueness ◽  
Initial Distribution ◽  
Total Variation Norm ◽  
Birth And Death Processes ◽  
Long Time Behaviour ◽  
Long Time ◽  
Unique Distribution ◽  
Variation Norm ◽  
Quasistationary Distributions

We study the long-time behaviour of a Markov process evolving in N and conditioned not to hit 0. Assuming that the process comes back quickly from ∞, we prove that the process admits a unique quasistationary distribution (in particular, the distribution of the conditioned process admits a limit when time goes to ∞). Moreover, we prove that the distribution of the process converges exponentially fast in the total variation norm to its quasistationary distribution and we provide a bound for the rate of convergence. As a first application of our result, we bring a new insight on the speed of convergence to the quasistationary distribution for birth-and-death processes: we prove that starting from any initial distribution the conditional probability converges in law to a unique distribution ρ supported in N* if and only if the process has a unique quasistationary distribution. Moreover, ρ is this unique quasistationary distribution and the convergence is shown to be exponentially fast in the total variation norm. Also, considering the lack of results on quasistationary distributions for nonirreducible processes on countable spaces, we show, as a second application of our result, the existence and uniqueness of a quasistationary distribution for a class of possibly nonirreducible processes.

scholarly journals Total Variation Applications in Computer Vision

2018 ◽  
pp. 543-570
Author(s):  
Vania Vieira Estrela ◽  
Hermes Aguiar Magalhães ◽  
Osamu Saotome
Keyword(s):  
Computer Vision ◽  
Total Variation ◽  
Total Variation Norm ◽  
Mathematical Background ◽  
Tv Regularization ◽  
Variation Norm ◽  
Image Processing Algorithms ◽  
Processing Algorithms ◽  
Set Up

The objectives of this chapter are: (i) to introduce a concise overview of regularization; (ii) to define and to explain the role of a particular type of regularization called total variation norm (TV-norm) in computer vision tasks; (iii) to set up a brief discussion on the mathematical background of TV methods; and (iv) to establish a relationship between models and a few existing methods to solve problems cast as TV-norm. For the most part, image-processing algorithms blur the edges of the estimated images, however TV regularization preserves the edges with no prior information on the observed and the original images. The regularization scalar parameter λ controls the amount of regularization allowed and it is essential to obtain a high-quality regularized output. A wide-ranging review of several ways to put into practice TV regularization as well as its advantages and limitations are discussed.

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