Schilder's Theorem
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High Quality Content by WIKIPEDIA articles! Let B be a standard Brownian motion in d-dimensional Euclidean space Rd starting at the origin, 0 Rd; let W denote the law of B, i.e. classical Wiener measure. For 0, let W denote the law of the rescaled process ( )B. Then, on the Banach space C0 = C0([0, T]; Rd) with the supremum norm · , the probability measures W satisfy the large deviations principle with good rate function I : C0 R {+ } given by I(omega) = frac{1}{2} int_{0}^{T} dot{omega}(t) ^{2} , mathrm{d} t if is absolutely continuous, and I( ) = + otherwise. In other words, for every open ...
High Quality Content by WIKIPEDIA articles! Let B be a standard Brownian motion in d-dimensional Euclidean space Rd starting at the origin, 0 Rd; let W denote the law of B, i.e. classical Wiener measure. For 0, let W denote the law of the rescaled process ( )B. Then, on the Banach space C0 = C0([0, T]; Rd) with the supremum norm · , the probability measures W satisfy the large deviations principle with good rate function I : C0 R {+ } given by I(omega) = frac{1}{2} int_{0}^{T} dot{omega}(t) ^{2} , mathrm{d} t if is absolutely continuous, and I( ) = + otherwise. In other words, for every open set G C0 and every closed set F C0, limsup_{varepsilon downarrow 0} varepsilon log mathbf{W}_{varepsilon} (F) leq - inf_{omega in F} I(omega) and liminf_{varepsilon downarrow 0} varepsilon log mathbf{W}_{varepsilon} (G) geq - inf_{omega in G} I(omega).
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