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Generalised Stochastic Processes and Their Distributions
Functionals of Brownian Motion
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a-ﬁeld Borel Brownian motion characteristic functional commutation relations complete orthonormal system complex Gaussian constant continuous convergence Corollary countably dB(u deﬁned denote differential discussion element equation equivalent example exists expression ﬁnd ﬁnite ﬁnite-dimensional ﬁrst ﬁxed ﬂow formula Fourier transform Fourier-Hermite polynomials function f Gaussian distribution Gaussian process Gaussian random variables Gaussian system generalised stochastic process given Hermite polynomials Hilbert space implies independent random variables inequality inﬁnite integral representation interval introduced isomorphic kernel Lemma Lie algebra linear Markov process Markov property martingale measure space norm notation nuclear space obtain one-parameter subgroups orthogonal orthonormal system parameter probability measure probability space PROOF Proposition proved real-valued Remark respectively result sample path satisﬁes sequence space Q stochastic process subset subspace Suppose symmetric system in L2(R Theorem theory tion topology transition probability unitary group unitary operator variance white noise