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On the asymptotic behaviour of the Gaussian functional integral. (Russian) Zbl 0779.60035

For \(n\geq 1\) and \(\beta>0\), let \(y_ 1,\ldots,y_ n\) be continuous functions on \([0,\beta]\) such that \(y(0)=y(\beta)=0\), and let \(x_ 1,\ldots,x_ n\in R\). Let \(\lambda_ 1,\lambda_ 2,\ldots\) be positive numbers such that \(\lambda_ m\leq\lambda_{m+1}\), \(m\geq 1\), and \(\lambda_ k\to\infty\) as \(k\to\infty\), and let \(A_ n\) stand for a positive operator on \(R^ n\) whose eigenvalues are \(\lambda_ 1,\ldots,\lambda_ n\). Further, set \[ F_ n(x,y)=\exp\left[-\int^ \beta_ 0\langle A_ ny(s),x+y(s)\rangle ds\right], \] where \(x=(x_ 1,\ldots,x_ n)\), \(y=(y_ 1,\ldots,y_ n)\). Finally, for a certain set \(C_ \beta\) and a measure \(\mu\) on \(C_ \beta\), the author considers \[ W=\lim_{n\to\infty}(2\pi\beta)^{-n/2}\int_{C_ \beta}F_ n(x,y)d\mu(y). \] On the other hand, for \(\delta>0\) and \(1=\alpha_ 1>\cdots>\alpha_ n\), let \[ C_ \beta(\delta)=\left\{y\in C_ \beta:\sum^ n_{k=1}\alpha_ k\int^ \beta_ 0y^ 2_ k(s)ds>\delta\right\}. \] The author also considers \[ W(\delta)=\lim_{n\to\infty}(2\pi\beta)^{-n/2} \int_{C_ \beta(\delta)}F_ n(x,y)d\mu(y). \] Under several conditions on the sequences \((\lambda_ k)\) and \((\alpha_ k)\), he shows that \(W(\delta)=o(W)\) as \(\delta\to 0\). Unfortunately, the paper is rather poorly written in a frequently encountered disordered style.

MSC:

60G15 Gaussian processes
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