By Samuel Karlin

This moment direction keeps the advance of the speculation and functions of stochastic methods as promised within the preface of a primary direction. We emphasize a cautious therapy of uncomplicated constructions in stochastic tactics in symbiosis with the research of typical periods of stochastic procedures bobbing up from the organic, actual, and social sciences.

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We can easily rewrite the eigenvalue problem of the Fokker-planck operator, already discussed - -t in Section 3, in the abstract form. ,. of Section 3. In particular, the abstract Fokker-Planck operator is p = ~]ai: + i,K,(q)] . 27) the right-hand sides being the eigenfunctions introduced in Section 3. 28b) hold because of (; = exp[V(q)/(2aJ)] (with af = kaT). Note that I J > is proportional to I Vo > -t and both belong to the zero eigenvalue of F . 26). 1=< < q,(t) qJ(t') qk(t") ... 1=< If we want to use Q,(t) Vo Vo = exp[-iIt] q.

3b) F J(t) ; l' J(t, t') = l' J(t) 1';1(t') , AH where F J(t) = A-1 A A T J (t)F JT J(t). 3c) The kernel function is also defined by TJ(q,t I q',t') =< q I 1'J(t,t') 1 q' >. 4) ° Note that we do not have < J 1 FJ = due to the presence of the source term in F}, and thus the normalization condition, < J 1If> J >= 1 or < fiT J =< J I, does not hold at every time. 5b) 28 apart from constant factors, in which I cI1 > and I cI1' > are arbitrary states, and the superscript I stands for operators in the ''interaction'' picture.

Operator formalism In this section we develop an abstract operator formalism of the theory of stochastic processes, which serves to make theoretical analyses compact and systematic, along the line of thought introduced by some rather old papers (Saito and Namiki 1954; Saito and Namiki 1956; Saito 1957; Namiki 1958; Namiki 1959; Namiki 1961; Namiki 1965, see also, Martin, Siggia and Rose 1973; Kawasaki 1974; Phytian 1976; Garrido and Miguel 1978; Deker 1979; Sagues and Garrido 1984). }, on the vector space.

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