By Elton P. Hsu

Likelihood concept has develop into a handy language and a useful gizmo in lots of components of contemporary research. the most goal of this booklet is to discover a part of this connection about the family among Brownian movement on a manifold and analytical features of differential geometry. A dominant subject matter of the booklet is the probabilistic interpretation of the curvature of a manifold. The booklet starts off with a quick evaluation of stochastic differential equations on Euclidean house. After providing the fundamentals of stochastic research on manifolds, the writer introduces Brownian movement on a Riemannian manifold and reports the impact of curvature on its habit. He then applies Brownian movement to geometric difficulties and vice versa, utilizing many famous examples, e.g., short-time habit of the warmth kernel on a manifold and probabilistic proofs of the Gauss-Bonnet-Chern theorem and the Atiyah-Singer index theorem for Dirac operators. The e-book concludes with an creation to stochastic research at the course house over a Riemannian manifold.

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VN∗ ; Z, U0 ) and Yt = f (Ut ) = πUt . Differentiating, we have dYt = Vα∗ f (Ut ) ◦ d Ztα = Vα (Yt ) ◦ d Ztα . This is the equation for X. , U is the horizontal lift of X. 4 dWt = Ut−1 Pβ (Xt ) ◦ dXtβ . In our case dXt = Vα (Xt ) ◦ d Ztα , or in components dXtβ = Vαβ (Xt ) ◦ d Ztα , where Vαβ are the component of the vector Vα in RN . Since Vα is tangent to M , it is easy to verify that N Vαβ Pβ . Vα = β=1 It follows that   N dWt = Ut−1  Vαβ (Xt )Pβ (Xt ) ◦ d Ztα = Ut−1 Vα (Xt ) ◦ d Ztα , β=1 as desired.

Since x˙ t ∈ Txt M , we have u−1 ˙ t ∈ Rd . 7) wt = u−1 ˙ s ds. s x 0 Note that w depends on the choice of the initial frame u0 at x0 but in a simple way: if {vt } is another horizontal lift of {xt } and u0 = v0 g for a g ∈ GL(d, R), then the anti-development of {vt } is {gwt }. From ut w˙ t = x˙ t we have Hw˙ t (ut ) = ut w˙ t = x˙ t = u˙ t . 8) u˙ t = Hi (ut ) w˙ ti . We can also start from a curve {wt } in Rd and a frame u0 at a point x0 . The unique solution of the above equation is a horizontal curve {ut } in F(M ) and is called the development of {wt } in F(M ), and its projection t → xt = πut is called the development of {wt } in M .

8) interpreted properly—this usually means replacing the usual integral by the corresponding Stratonovich stochastic integral—we can develop a semimartingale W on Rd into a horizontal semimartingale U on F(F ), and then project it down to a semimartingale on M (“rolling without slipping”). Conversely, we can lift a semimartingale X on M to a horizontal semimartingale U on F(M ) and then to a semimartingale W on Rd . , πU0 = X0 ), the correspondence W ←→ X is one-to-one. Because euclidean semimartingales are easier to handle than manifold-valued semimartingales, we can use this geometrically defined correspondence to our advantage.

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