scholarly journals About Lyapunov functionals construction for difference equations with continuous time

2004 ◽  
Vol 17 (8) ◽  
pp. 985-991 ◽  
Author(s):  
L. Shaikhet
Keyword(s):  
Difference Equations ◽  
Continuous Time ◽  
Lyapunov Functionals

GENERAL METHOD OF LYAPUNOV FUNCTIONALS CONSTRUCTION IN STABILITY INVESTIGATIONS OF NONLINEAR STOCHASTIC DIFFERENCE EQUATIONS WITH CONTINUOUS TIME

2005 ◽  
Vol 05 (02) ◽  
pp. 175-188 ◽  
Author(s):  
LEONID SHAIKHET
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Continuous Time ◽  
Type Theorem ◽  
Volterra Equations ◽  
Lyapunov Functionals ◽  
Stochastic Difference Equation ◽  
Stochastic Difference Equations ◽  
After Effect ◽  
General Method

The general method of Lyapunov functionals construction has been developed during the last decade for stability investigations of stochastic differential equations with after-effect and stochastic difference equations. After some modification of the basic Lyapunov type theorem this method was successfully used also for difference Volterra equations with continuous time. The latter often appear as useful mathematical models. Here this method is used for a stability investigation of some nonlinear stochastic difference equation with continuous time.

scholarly journals Hyperspace fermions, Möbius transformations, Krein space, fermion doubling, dark matter

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200383
Author(s):  
George Jaroszkiewicz
Keyword(s):  
Dark Matter ◽  
Difference Equations ◽  
Continuous Time ◽  
Krein Space ◽  
Equations Of Motion ◽  
Möbius Transformations ◽  
Physical Limit ◽  
Kreĭn Space ◽  
The Difference ◽  
Mobius Transformations

We develop an approach to classical and quantum mechanics where continuous time is extended by an infinitesimal parameter T and equations of motion converted into difference equations. These equations are solved and the physical limit T  → 0 then taken. In principle, this strategy should recover all standard solutions to the original continuous time differential equations. We find this is valid for bosonic variables whereas with fermions, additional solutions occur. For both bosons and fermions, the difference equations of motion can be related to Möbius transformations in projective geometry. Quantization via Schwinger’s action principle recovers standard particle-antiparticle modes for bosons but in the case of fermions, Hilbert space has to be replaced by Krein space. We discuss possible links with the fermion doubling problem and with dark matter.

An invariance principle for solutions to stochastic difference equations

1981 ◽  
Vol 18 (2) ◽  
pp. 548-553
Author(s):  
Harry A. Guess
Keyword(s):  
Differential Equation ◽  
Population Genetics ◽  
Brownian Motion ◽  
Weak Convergence ◽  
Difference Equations ◽  
Continuous Time ◽  
Martingale Problem ◽  
Martingale Differences ◽  
Invariance Principles ◽  
Stochastic Difference Equations

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

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