scholarly journals ON THE UNITARITY OF STOCHASTIC EVOLUTIONS DRIVEN BY THE SQUARE OF WHITE NOISE

2001 ◽  
Vol 04 (04) ◽  
pp. 579-588 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ANDREAS BOUKAS ◽  
HUI-HSUNG KUO
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Using the closed Itô's table for the renormalized square of white noise, recently obtained by Accardi, Hida, and Kuo in Ref. 4, we consider the problem of providing necessary and sufficient conditions for the unitarity of the solutions of a certain type of quantum stochastic differential equations.

scholarly journals STATISTICAL CAUSALITY AND STABLE SUBSPACES OF 

2012 ◽  
Vol 88 (1) ◽  
pp. 17-25 ◽  
Author(s):  
LJILJANA PETROVIĆ ◽  
DRAGANA VALJAREVIĆ
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Weak Solutions ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Statistical Concept ◽  
Extremal Measures ◽  
Probability Spaces ◽  
Necessary And Sufficient

AbstractIn this paper we consider the statistical concept of causality in continuous time between filtered probability spaces, based on Granger’s definitions of causality. Then we consider some stable subspaces of $H^p$ which contain right continuous modifications of martingales $P(A \mid {\mathcal {G}}_t)$. We give necessary and sufficient conditions, in terms of statistical causality, for these spaces to coincide with $H^p$. These results can be applied to extremal measures and regular weak solutions of stochastic differential equations.

scholarly journals Quantum stochastic operator cocycles via associated semigroups

2007 ◽  
Vol 142 (3) ◽  
pp. 535-556 ◽  
Author(s):  
J. MARTIN LINDSAY ◽  
STEPHEN J. WILLS
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
General Principle ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Necessary And Sufficient Conditions ◽  
Contraction Operator ◽  
Stochastic Operator ◽  
Necessary And Sufficient

AbstractA recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.

Oscillation of the Riemann–Weber Version of Euler Differential Equations with Delay

2000 ◽  
Vol 7 (3) ◽  
pp. 577-584
Author(s):  
Jitsuro Sugie ◽  
Mitsuru Iwasaki
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Equations With Delay ◽  
Oscillation Of Solutions

Abstract Our concern is to consider delay differential equations of Euler type. Necessary and sufficient conditions for the oscillation of solutions are given. The results extend some famous facts about Euler differential equations without delay.

Necessary and sufficient conditions for oscillation of nonlinear first-order forced differential equations with several delays of neutral type

Analysis ◽  
2019 ◽  
Vol 39 (3) ◽  
pp. 97-105 ◽  
Author(s):  
Sandra Pinelas ◽  
Shyam S. Santra
Keyword(s):  
Differential Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
First Order ◽  
Several Delays ◽  
Necessary And Sufficient ◽  
T Tau ◽  
First Order Differential Equations

AbstractIn this work, necessary and sufficient conditions are obtained such that every solution of nonlinear neutral first-order differential equations with several delays of the form\bigl{(}x(t)+r(t)x(t-\tau)\bigr{)}^{\prime}+\sum_{i=1}^{m}\phi_{i}(t)H\bigl{(}% x(t-\sigma_{i})\bigr{)}=f(t)is oscillatory or tends to zero as {t\rightarrow\infty.} This problem is considered in various ranges of the neutral coefficient r. Finally, some illustrating examples are presented to show that feasibility and effectiveness of main results.

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

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