scholarly journals Sequential Generalized Transforms on Function Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jae Gil Choi ◽  
Hyun Soo Chung ◽  
Seung Jun Chang
Keyword(s):  
Brownian Motion ◽  
Banach Algebra ◽  
Function Space ◽  
The Other ◽  
Brownian Motion Process ◽  
Inverse Transform ◽  
Motion Process ◽  
Generalized Brownian Motion Process

We define two sequential transforms on a function spaceCa,b[0,T]induced by generalized Brownian motion process. We then establish the existence of the sequential transforms for functionals in a Banach algebra of functionals onCa,b[0,T]. We also establish that any one of these transforms acts like an inverse transform of the other transform. Finally, we give some remarks about certain relations between our sequential transforms and other well-known transforms onCa,b[0,T].

scholarly journals Relationships of convolution products, generalized transforms, and the first variation on function space

2002 ◽  
Vol 29 (10) ◽  
pp. 591-608 ◽  
Author(s):  
Seung Jun Chang ◽  
Jae Gil Choi
Keyword(s):  
Brownian Motion ◽  
Banach Algebra ◽  
Function Space ◽  
Standard Wiener Process ◽  
Convolution Product ◽  
Brownian Motion Process ◽  
First Variation ◽  
Motion Process ◽  
Generalized Brownian Motion Process ◽  
The First Variation

We use a generalized Brownian motion process to define the generalized Fourier-Feynman transform, the convolution product, and the first variation. We then examine the various relationships that exist among the first variation, the generalized Fourier-Feynman transform, and the convolution product for functionals on function space that belong to a Banach algebraS(Lab[0,T]). These results subsume similar known results obtained by Park, Skoug, and Storvick (1998) for the standard Wiener process.

scholarly journals Generalized analytic Fourier-Feynman transforms with respect to Gaussian processes on function space

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1615-1624
Author(s):  
Seung Chang ◽  
Hyun Chung ◽  
Ae Ko ◽  
Jae Choi
Keyword(s):  
Brownian Motion ◽  
Function Space ◽  
Gaussian Processes ◽  
Brownian Motion Process ◽  
Motion Process ◽  
Generalized Brownian Motion Process

In this article, we introduce a generalized analytic Fourier-Feynman transform and a multiple generalized analytic Fourier-Feynman transform with respect to Gaussian processes on the function space Ca,b[0,T] induced by generalized Brownian motion process. We derive a rotation formula for our multiple generalized analytic Fourier-Feynman transform.

A REPRESENTATION FOR THE INVERSE GENERALISED FOURIER–FEYNMAN TRANSFORM VIA CONVOLUTION PRODUCT ON FUNCTION SPACE

2017 ◽  
Vol 95 (3) ◽  
pp. 424-435
Author(s):  
SEUNG JUN CHANG ◽  
JAE GIL CHOI
Keyword(s):  
Brownian Motion ◽  
Vector Space ◽  
Function Space ◽  
Exponential Type ◽  
Convolution Product ◽  
Brownian Motion Process ◽  
Inverse Transform ◽  
Motion Process

We study a representation for the inverse transform of the generalised Fourier–Feynman transform on the function space $C_{a,b}[0,T]$ which is induced by a generalised Brownian motion process. To do this, we define a transform via the concept of the convolution product of functionals on $C_{a,b}[0,T]$. We establish that the composition of these transforms acts like an inverse generalised Fourier–Feynman transform and that the transforms are vector space automorphisms of a vector space ${\mathcal{E}}(C_{a,b}[0,T])$. The vector space ${\mathcal{E}}(C_{a,b}[0,T])$ consists of exponential-type functionals on $C_{a,b}[0,T]$.

scholarly journals Conditional generalized analytic Feynman integrals and a generalized integral equation

2000 ◽  
Vol 23 (11) ◽  
pp. 759-776 ◽  
Author(s):  
Seung Jun Chang ◽  
Soon Ja Kang ◽  
David Skoug
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Feynman Integral ◽  
Brownian Motion Process ◽  
Feynman Integrals ◽  
Motion Process ◽  
Generalized Brownian Motion Process

We use a generalized Brownian motion process to define a generalized Feynman integral and a conditional generalized Feynman integral. We then establish the existence of these integrals for various functionals. Finally we use the conditional generalized Feynman integral to derive a Schrödinger integral equation.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (01) ◽  
pp. 17-27
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa ), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab ), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process . Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb ).

scholarly journals Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
ZaiTang Huang ◽  
ChunTao Chen
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Stochastic Bifurcation ◽  
Brownian Motion Process ◽  
Van Der Pol ◽  
Limit Circle ◽  
Random Attractors ◽  
Motion Process ◽  
Van Der Pol Equations ◽  
The Stability

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.

L'intégrale du mouvement brownien

1993 ◽  
Vol 30 (1) ◽  
pp. 17-27 ◽  
Author(s):  
Aimé Lachal
Keyword(s):  
Fourier Transform ◽  
Brownian Motion ◽  
Hitting Time ◽  
First Hitting Time ◽  
Brownian Motion Process ◽  
Classical Technique ◽  
Motion Process ◽  
Mouvement Brownien ◽  
The Law ◽  
The Right

Let be the Brownian motion process starting at the origin, its primitive and Ut = (Xt+x + ty, Bt + y), , the associated bidimensional process starting from a point . In this paper we present an elementary procedure for re-deriving the formula of Lefebvre (1989) giving the Laplace–Fourier transform of the distribution of the couple (σ α, Uσa), as well as Lachal's (1991) formulae giving the explicit Laplace–Fourier transform of the law of the couple (σ ab, Uσab), where σ α and σ ab denote respectively the first hitting time of from the right and the first hitting time of the double-sided barrier by the process . This method, which unifies and considerably simplifies the proofs of these results, is in fact a ‘vectorial' extension of the classical technique of Darling and Siegert (1953). It rests on an essential observation (Lachal (1992)) of the Markovian character of the bidimensional process .Using the same procedure, we subsequently determine the Laplace–Fourier transform of the conjoint law of the quadruplet (σ α, Uσa, σb, Uσb).

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