T. Hida, Brownian Motion (Springer 1980)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Introduction The purpose of this paper is to develop a spectral approach to nonlinear filtering of randomly perturbed dynamical systems. The approach is based on a version of the Wiener Chaos decomposition on the probability space generated by observations (see Ito [1] Zvonkin, Krylov [2] Hida [3], etc. It gives rise to a new numerical scheme for the fundamental equations of nonlinear filtering (Zakai equation, see e.g. 4] 5] The main feature of this scheme is that it allows to separate computations involving observations from those that use parameters of the system. More ....

....2. Set uN (t; x) jffjN ff (t; x) ff (y) Then the error of approximation EjuN Gamma1 (t; x) Gamma u(t; x)j const. r 3. Proofs In this section we prove theorems 1 and 2. Proof of Theorem 1. The proof is based on the celebrated Wiener Chaos Decomposition (see e.g. 1] 2] [3], etc. Below, we reformulate this result in a way convenient for our purpose. Theorem 3 (Wiener Chaos Decomposition, see e.g. 3] Let B s be a Brownian motion ; F ; P ) and j be a measurable functional of the path of fB s ; s Tg such that Ej 1. Let fc i g be an arbitrary orthonormal ....

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T. Hida, Brownian Motion, Springer-Verlag, 1980.

....#[E] 1 #[u m ] q m . Construction of every #[u m ] #[u] x m ) is based on simulation of a Markov chain (j) m , j = 0, 1, Nm with initial point x m = xm , m = 1, M and random length Nm . Here, we shall make use of the strong Markov property of Brownian motion [13]. This makes it possible to avoid the simulation of the whole path, and to jump from the interior point of a domain directly to its boundary. To do this, we have to know the exact distribution of the exit point, or, which is equivalent, the boundary Green s function for this domain. It is clear, ....

T. Hida, Brownian motion, Springer-Verlag, New York, 1979.

....of one parameter families of such transformations are computed, and the Poisson case is also considered. Key words: Wiener measure, Fourier Mehler transform, Brownian motion, time changes. Mathematics Subject Classification (1991) 47B38, 46F25. 1 Introduction The Fourier Mehler transform, cf. [5], 6] 10] is originally a group (F ) 2IR of transformations of random variables on Gaussian space, with the property that F Gamma=2 coincides with the Fourier transform. The adjoint G of the Fourier Mehler transform F also forms a group whose infinitesimal generator is the sum of the ....

T. Hida. Brownian Motion. Springer Verlag, 1981.

.... if N = S C (R) the complexified of the Schwartz test function space S(R) and (x) x 2 , then F (N 0 ) is nothing than the analytic version of the Kubo Takenaka test functions space and the corresponding topological dual is the Hida distribution space, see e.g. KT80a] KT80b] Hid80] The test function space of Kondratiev Streit type (S) fi , fi 2 [0; 1) are obtained choosing (x) x 2 1 fi , see [KS93] Oue94] Oue98] Kuo96] Oba94] More recently, it was introduced a multivariable version of the above spaces, see [Oue00] In fact, we can replace the nuclear space ....

T. Hida. Brownian Motion. Springer-Verlag, 1980. 14

....to Brownian motions fi rK(d; Gamma n ; L 0 c n;d fi with c 2 Gamma n ;d = k 2 n ( 1) 2) d=2 2 n 2 n 2 Gamma2 3 Proofs Proposition 3.1 M k;t are orthogonal Brownian martingales Proof: Orthogonality is obvious. For the martingale property see [13] and [2] it is a consequence of the fact that the kernel functions of M t in (6) do not depend on t (except through the limit of integrations) Xi Their limiting behavior, as 0; is studied in the following lemma (from now on we shall consider only the situations which require ....

T. Hida: "Brownian Motion". Springer, Berlin, 1980.

....) 0 D( d ) where D( d ) 0 is the dual of D( d ) so that in the scalar sense ; D( d ) 0 D( d ) 1 The intention here is to de ne something similar in the Hilbert space sense. It is to introduce a special generalized random process which according to Hida[10], is understood to be a family fX( 2 Eg of random variables on a probability space( B; P ) with parameter set a certain function space E . such that for almost all , X( is a continuous linear functional in . Since the type of measure space that is ....

Hida, T., Brownian Motion, Springer-Verlag, New York, 1980 16

....ff OE; 2 D 1 ( Proof. First note that Omega OE; G f OE ff 2 Omega OE; G OE ff = 2kOEk 2 2; Gamma1 . Also, in dimensions d = 2; 3, the injection D 1 ( Gamma D Gamma1 ( is of Hilbert Schmidt type. This implies that Q f can be constructed on D 0 1 ( cf. Theorem 3. 1 of [12]. In order to prove the weak convergence, it suffices to show that lim N 1 EP N [exp(iZ N (OE) E Q f [exp(iZ(OE) exp( Gamma 1 2 Omega OE; G f OE ff ) OE 2 D 1 ( where i = p Gamma1. Since P N is gaussian, EP N [exp(iZ N (OE) exp i i 1 p ffl 0 N ....

Hida T., Brownian motion, Springer Verlag, New York, 1980.

....is an infinite dimensional calculus which includes various generalizations of concepts known from finite dimensional analysis. Among them are Fourier transform, differential operators, and generalized functions. For a detailed exposition of white noise analysis we refer to the monographs [BK95] Hid80] HKPS93] HUZ96] Kuo96] and [Oba94] The concepts of regular generalized functions have been introduced in [GKS99] see also [Gro98] Their development was motivated by the results and insights in the theory of stochastic (partial) differential equations of Wick type, see [HUZ96] and the ....

....of a square integrable function F w.r.t. these oe algebras equivalently can be described via the kernels of F . Let P Omega n J denote the projection in L 2 (R n ; C ) given by P Omega n J g = 1 J ng; g 2 L 2 (R n ; C ) The following proposition has been proved in [Hid80] Propositions 4.5 and 4.7: Proposition 2.1 The function F = P 1 n=0 I n (F (n) 2 L 2 ( is measurable w.r.t. oe J if and only if for all n 2 N 0 P Omega n J F (n) F (n) 2 L 2 (R n ; C ) almost everywhere w.r.t. the Lebesgue measure. Furthermore, the conditional ....

T. Hida. Brownian Motion. Springer Verlag, Berlin, Heidelberg, New York, 1980.

....in the configuration space may be defined from potential functions computed in the workspace (see [1] for details) tions. A Brownian motion in R is defined as a Gaussian random process w(t) verifying : w(0) 0; E(w(s) 0 and E(w(s)w(t) min(s; t) for any s and t belonging to R (see [2]) A Brownian motion in R d is a collection of d independent Brownian motions in R. The probability density function due to a Brownian motion in R d at time t is by definition the central Gaussian density of variance t : p(x; t) 2 t) Gammad=2 exp( Gammakxk 2 =2t) It is easy to verify ....

....h; i ) We want to compute the probability distribution of this random motion in the half space y h. For this we express the law of the random time oe( We notice that for every t : foe( tg = fthe rand. mot. is at time t in the rodg We now define P oe (t) P(oe( t) According to [2] P oe (t) Z h 0 k(y; t)dy where k(y; t) 1 p 2 t 1 X n= Gamma1 ( Gamma1) n exp( Gamma (y 2nh) 2 2t ) By definition, the probability density of the random time oe( is : J(t) Gamma dP oe dt (t) As k(y; t) verifies the diffusion equation 1 and k t (0; t) 0, J(t) ....

T. Hida, Brownian Motion, Springer-Verlag, 1980.

....various generalizations of concepts known from finite dimensional analysis, among them are differential operators and Fourier transform. Although we will give a brief introduction to white noise calculus in section 2 the reader unfamiliar with this topic is recommended to the monographs [5] 15] [4] and the introductory articles [12] 17] 19] 21] The idea of realizing Feynman integrals within the White Noise framework goes back to [6] The average over all paths is performed with a Hida distribution as the weight (instead of a measure) The existence of such Hida distributions ....

Hida, T. (1980), Brownian Motion, Applications of Mathematics 11, Springer Verlag, Berlin.

....rather they are distribution valued. A rigorous treatment of generalized random process can be given in the framework of white noise analysis (or more general in Gaussian analysis) For a detailed exposition of Gaussian analysis and for examples of applications we refer to the monographs [BK95] Hid80] HKPS93] H UZ96] Kuo96] and [Oba94] Wick type stochastic differential equations have been introduced in [KP90] L U91] and [L U92] A self contained description of the concepts of Wick type stochastic differential equations, their applications, and more references are presented in ....

....hx; n i 2 F n o ; i 2 N ; F i 2 B(R) where B(R) denotes the Borel oe algebra on R. The canonical Gaussian measure on (N 0 ; C oe (N 0 ) is given by its characteristic function Z N 0 exp(ihx; i) d (x) exp( Gamma 1 2 j j 2 ) 2 N ; via Minlos theorem, see e.g. BK95] Hid80] and [HKPS93] N 0 ; C oe (N 0 ) is the basic probability space throughout this work. The integral R N 0 f(x) d (x) of a measurable function f defined on N 0 is called the expectation of f if f is integrable, i.e. the integral R N 0 jf(x)j d (x) is finite. The space of the ....

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T. Hida. Brownian Motion. Springer Verlag, 1980.

....of convoluted generalized functions Phi G H also fulfill the modified Wightman axioms. The paper is organized as follows. In Section 2 we introduce the concepts of Gaussian and white noise analysis as far as necessary for our considerations. For a detailed exposition we refer to the monographs [Hid80] BK95] HKPS93] Oba94] H UZ96] and [Kuo96] In the framework of white noise analysis various aspects of QFT have been discussed, see [AHP 90a] AHP 90b] AHPS89] PS90] and [HKPS93] Section 3 of this note is attended to represent Euclidean QFT in the framework of white noise ....

....the vector space S 0 we consider the oe algebra C oe (S 0 ) generated by cylinder sets. The canonical Gaussian measure on (S 0 ; C oe (S 0 ) is given by its characteristic function Z S 0 exp(ih ; fi) d ( exp( Gamma 1 2 jf j 2 ) f 2 S; via Minlos theorem, see e.g. BK95] Hid80] and [HKPS93] If we chose H = L 2 (R d ) the space of real valued square integrable functions w.r.t. the Lebesgue measure on R d , this is the Gaussian white noise measure. For H = H Gamma1;2 (R d ) the Sobolev space of order ( Gamma1; 2) this is the measure corresponding to the ....

T. Hida. Brownian Motion. Springer Verlag, 1980.

....motions fi rK(d; Gamma n ; L 0 c Gamma n ;d fi with c 2 Gamma n ;d = k 2 n ( 1) 2 ) d=2 2 n 2 n 2 Gamma2 3 Proofs Proposition 3.1 M k;t are orthogonal Brownian martingales Proof: Orthogonality is obvious. For the martingale property see [13] and [2] it is a consequence of the fact that the kernel functions of M t in (2.6) do not depend on t (except through the limit of integrations) Xi Their limiting behavior, as 0; is studied in the following lemma (from now on we shall consider only the situations which require ....

Hida T.: "Brownian Motion". Springer, Berlin, 1980.

....the stochastic Neumann boundary value problem LU(x) Gamma c(x)U(x) 0; x 2 D ae R ; fl(x) Delta rU(x) GammaW (x) x 2 D where L is a uniformly elliptic linear partial differential operator and W (x) x 2 R , is d parameter white noise. 1. Introduction Since the seminal book by Hida [4] appeared in 1980, there has been a rapid development of white noise theory and its applications. In particular, white noise theory has found many spectacular applications in mathematical physics. See, e.g. 5, 7] and references therein. In addition, white noise calculus turns out to be useful in ....

T. Hida, Brownian Motion, Springer, Berlin 1980.

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T. Hida, Brownian Motion (Springer 1980)

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Hida, T. (1980) Brownian Motion. Springer, New York.

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T. Hida, Brownian Motion, Springer-Verlag, New York, Berlin, 1980.

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Hida T., Brownian Motion, Springer-Verlag, Berlin (1980).

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Hida, T.: Brownian Motion. Springer-Verlag, 1980

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Hida, T. (1980). Brownian Motion. Springer Verlag, New York.

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Hida, T. (1980). Brownian motion. Springer-Verlag.

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T. Hida, Brownian Motion, Springer-Verlag, New York, 1980.

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T. Hida, Brownian Motion (Springer 1980)

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Hida, T. (1980), Brownian Motion. Springer, New York.

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Hida, T. (1980). Brownian Motion. Springer Verlag, New York.

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