Symmetric matrix notes

# That direction is orthogonal to all the vectors

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Unformatted text preview: induction proceeds. By Blumenthal zero-one law if the determinant is zero for a positive time, then it is so with probability one and there is a deterministic direction in which it is degenerate. That direction is orthogonal to all the vectors generated by all the Lie brackets. Quantitative version. We will estimate E [X −k ] by estimating E [e−λX ] and integrating E [X −k ] = 1 Γ(k ) E [e−λX ]λk−1 dλ We will ﬁx M a bound on σ and b as well as the time interval [0, T ]. C (T, M ) will stand for a constant that may depend on M and T but independent of λ. 2 Lemma 1. Let ξ (t) be a stochastic integral t ξ ( t) = x + 0 t σ (s) · dβ (s) + b(s)ds 0 such that |σ (s)| ≤ M and |b(s)| ≤ M Then for any λ ≥ 0, E exp − λ2 4 t ξ 2 (s)ds + 0 λ 4M t σ 2 (s)ds 0 ≤ C ( t) Proof. Consider the function U (t , x) = exp − where λx2 1 1 tanh λM t + F (λM t) + t 4M 4 2 x F ( x) = 0 Then [1 − tanh x]dx = x − log cosh x ≤ log 2 λx tanh λM t ] 2M λ λ2 x2 tanh2 λM t − tanh λM t ] Uxx = U [ 2 4M 2M λ2 x2 λM 1 Ut = U [− sech2 λM t + (1 − tanh...
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## This note was uploaded on 03/19/2013 for the course MATH GA 2931 taught by Professor Varadhan during the Fall '06 term at NYU.

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