On the Absolute Continuity of Transition Probabilities of a One-Dimensional Diffusion Process

1961 ◽  
Vol 6 (4) ◽  
pp. 404-410 ◽  
Author(s):  
A. D. Venttzel’
Keyword(s):  
Diffusion Process ◽  
Absolute Continuity ◽  
Transition Probabilities ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
The Absolute

scholarly journals LQG Homing in a Finite Time Interval

2011 ◽  
Vol 2011 ◽  
pp. 1-3 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Optimal Control ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
Time Interval ◽  
Finite Time Interval ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Fixed Constant

LetX(t)be a controlled one-dimensional diffusion process having constant infinitesimal variance. We consider the problem of optimally controllingX(t)until timeT(x)=min{T1(x),t1}, whereT1(x)is the first-passage time of the process to a given boundary andt1is a fixed constant. The optimal control is obtained explicitly in the particular case whenX(t)is a controlled Wiener process.

Co-dimension Two Bifurcations in the Presence of Noise

1991 ◽  
Vol 58 (1) ◽  
pp. 259-265 ◽  
Author(s):  
N. Sri Namachchivaya
Keyword(s):  
Normal Form ◽  
Diffusion Process ◽  
Stochastic Averaging ◽  
Singularly Perturbed ◽  
Stability Conditions ◽  
Exit Times ◽  
Double Zero ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stochastic Bifurcations

Some results pertaining to co-dimension two stochastic bifurcations are presented. The normal form associated with non-semi-simple double-zero eigenvalues is considered. The method of stochastic averaging applicable for singularly perturbed stochastic differential equations is used to further reduce the problem to a one dimensional diffusion process. Probability density, most probable values, stability conditions in probability, and mean exit times are obtained for the reduced system.

scholarly journals Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

2014 ◽  
Vol 46 (01) ◽  
pp. 186-202 ◽  
Author(s):  
Laura Sacerdote ◽  
Ottavia Telve ◽  
Cristina Zucca
Keyword(s):  
Diffusion Process ◽  
Joint Distribution ◽  
Continuous Functions ◽  
Time Dependent ◽  
Hitting Times ◽  
Convergence Properties ◽  
System Of Integral Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
First Hitting Times

Consider a one-dimensional diffusion process on the diffusion interval I originated in x 0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t 0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t 0. We study the joint distribution of the two random variables T a and T b , the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of T a and T b in terms of ℙ(T a < t, T a < T b ) and ℙ(T b < t, T a > T b ), and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

Analysis of Chloride Penetration into the Corner Zone of Concrete Structure Member

2007 ◽  
Vol 348-349 ◽  
pp. 397-400
Author(s):  
Xiao Yong Wang ◽  
Han Seung Lee ◽  
Hai Moon Jung
Keyword(s):  
Service Life ◽  
Diffusion Process ◽  
Concrete Structure ◽  
Chloride Concentration ◽  
Chloride Penetration ◽  
Chloride Diffusion ◽  
Recent Analysis ◽  
Two Dimensional ◽  
One Dimensional ◽  
Dimensional Diffusion

Chloride penetration into concrete is the main cause of steel corrosion in concrete structures exposed to chloride-rich environments. In general, conditions on the diffusion process are dominant among various penetration mechanisms, such as ionic diffusion, capillary sorption, and so on. In recent analysis of current literature, chloride diffusion is as a simplified one-dimensional diffusion process. However, for the rebar in the corner zone of concrete beam, the diffusion belongs to a two-dimensional diffusion. Based on a galerkin finite element method, a two-dimensional diffusion differential equation is built and solved numerically and the different chloride concentration is compared to one dimensional diffusion and two-dimensional diffusion process. The service life of concrete structure members under two-dimensional chloride penetration is predicted by compared with a critical threshold chloride concentration. Compared with general one-dimensional chloride attack, the service life is considerably reduced in a corner zone due to two-dimension penetration.

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