Levy's stochastic area formula in higher dimensions

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Let ( W t ) = ( W 1 t , W 2 t ,…, W d t ), d ⩾ 2, be a d-dimensional standard Brownian motion and let A( t) be a bounded measurable function from R + into the space of d × d skew-symmetric matrices and x( t) such a function into R d . A class of stochastic processes (L t A,x), a particular example of which is Levy's “stochastic area” L t = 1 2 ∝ 0 −t (W 1 s,dW 2 s − W 2 s,dW 1 s) , is dealt with. The joint characteristic function of W t and L 1 A, x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions ( W t , L t A, x ) is given.