By Daniel W. Stroock

This publication goals to bridge the space among chance and differential geometry. It provides structures of Brownian movement on a Riemannian manifold: an extrinsic one the place the manifold is discovered as an embedded submanifold of Euclidean area and an intrinsic one in response to the "rolling" map. it truly is then proven how geometric amounts (such as curvature) are mirrored via the habit of Brownian paths and the way that habit can be utilized to extract information regarding geometric amounts. Readers must have a robust historical past in research with easy wisdom in stochastic calculus and differential geometry. Professor Stroock is a highly-respected professional in chance and research. The readability and elegance of his exposition extra improve the standard of this quantity. Readers will locate an inviting advent to the learn of paths and Brownian movement on Riemannian manifolds

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**Sample text**

Ii Ii–1 Ii+1 Ii+2 Ii+3 T3,i T2,i T1,i ti–2 ti–1 ti ti+1 ti+2 t Fig. 5: Interval statistics. e. the interspike intervals ({Ii }) are shown with solid lines; the sum of n adjacent ISIs form the nth order intervals Tn,i (dashed lines). Stochastic methods in neuroscience 22 More useful is this relation in the Fourier domain relating the power spectrum to the Fourier transform of the nth-order interval density ρ˜n as follows (see Holden, 1976) ∞ S(f ) = 2Re ∞ ρn (τ ) − r dτ e2πif τ r δ(τ ) + 0 n=1 ∞ ⇒ S(f ) = r 1 − rδ(f ) + ρ˜n (f ) + ρ˜∗n (f ) .

A widely used deﬁnition of the noise intensity of a process with non-negative correlation function is as follows: ∞ dτ C(τ ) = D= 0 S(0) = ∆x2 τcorr . 21). This illustrates that it matters not only how large typical amplitudes of the noise are (as quantiﬁed by the variance) but also for how long the noise acts with roughly the same value (as quantiﬁed by the correlation time). As is clear from the discussion of the correlation time, the deﬁnition of the noise intensity is also meaningful for processes with monotonically decaying correlation but does not apply to processes with a strongly oscillating correlation function.

If we know the probability densities ρn (Tn ) of all nth-order intervals, the statistics of the associated stationary spike train is completely determined. For instance, the conditional probability density can be expressed as follows (see, for instance, Holden, 1976) ∞ r2 (τ |0) = δ(τ ) + ρn (τ ). 62) n=1 The ﬁrst term reﬂects the sure event that we have a spike at τ = 0 (which is our condition); the other terms sum over the probabilities to have the nth spike at ﬁnite τ . e. relates spike train statistics to interval statistics.