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We will ﬁx such an inner product. Let k be the Lie algebra of K and let p be the orthogonal complement of k in g. It is easy to see that p is Ad(K )-invariant; that is, Ad(k)p ⊂ p for k ∈ K . Let {X 1 , . . , X d } be an orthonormal basis of g such that X 1 , . . , X n form a basis of p and X n+1 , . . , X d form a basis of k. Consider the map y = (y1 , . . , yn ) → π(e φ : Rn n i=1 yi X i ) ∈ M. Restricted to a sufﬁciently small neighborhood V of 0 in Rn , φ is a diffeomorphism and hence y1 , .

3. Riemannian Brownian Motions 47 is a linear endomorphism on g, which is invertible when X is sufﬁciently close to 0. It is easy to show that idg − e−ad(X ) ad(X ) −1 1 = idg + ad(X ) + 2 ∞ cr ad(X )r , r =2 where the last series converges absolutely in the operator norm for X sufd xi (g)X i ] for g contained in a ﬁciently close to 0. Recall that g = exp[ i=1 sufﬁciently small neighborhood U of e. For X = nj=1 x j X j , let idg − e−ad(X ) ad(X ) Z= −1 1 X k = X k + [X, X k ] + O( X 2 ) 2 d = Xk + 1 x j C ijk X i + O(|x|2 ).

Let f be the matrix-valued function on G deﬁned by f (g) = gi j for g ∈ G. 23) leads to the following stochastic integral equation in matrix form: m t gt = g0 + i=1 t + 0 0 gs− Yi ◦ d Wsi + t t gs Z ds + 0 0 gs− (h − Id ) N˜ (ds dh) G gs [h − Id − x(h)] ds (dh). 30) G For any process yt taking values in a Euclidean space, let yt∗ = sup |ys |. 3. 12). Assume E[|g0 |2 ] < ∞ |h − Id |2 (dh) < ∞. 31) G Then, for any t > 0, E (gt∗ )2 < ∞. Moreover, there are Y1 , . . , Ym , Z ∈ g = gl(d, R) and an m-dimensional standard Brownian motion Wt = (Wt1 , .

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