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Quadratic functionals and small ball probabilities for the \(m\)-fold integrated Brownian motion. (English) Zbl 1030.60026

Let \(W(t)\), \(t\geq 0\), be a standard Brownian motion starting at \(0\). Denote by \(X_0(t)= W(t)\), \(X_m(t)= \int^t_0 X_{m-1}(s) ds\), \(t\geq 0\), \(m\geq 1\), the \(m\)-fold integrated Brownian motion for positive integer \(m\). This paper is concerned with the quadratic functionals for the real process \(X_m(t)\) as well as the asymptotic behavior of the small ball probability. A method for estimating the small probability such as \(P(\sup_{0\leq i\leq 1}|X_m(t)|\leq\varepsilon)\), as \(\varepsilon\to 0\), is developed via the appropriate quadratic functionals.

MSC:

60G15 Gaussian processes
60J25 Continuous-time Markov processes on general state spaces
60J60 Diffusion processes
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[40] KNOXVILLE, TENNESSEE 37996 E-MAIL: xchen@math.utk.edu DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF DELAWARE NEWARK, DELAWARE 19716 E-MAIL: wli@math.udel.edu
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