This paper is a generalization in greater dimensions of a result already due to the authors together with A. N. Shiryayev [Bernoulli 1, No. 1/2, 149-169 (1995; Zbl 0851.60048)]. It deals with an extension of Itô’s formula for Brownian motion: if belongs locally to the Sobolev space and if is a -dimensional Brownian motion, then for any and any starting point except in some polar set, decomposes into
where the denote the (weak) partial derivatives of , and is a quadratic covariation term. Notice that in the above formula, may not be a semimartingale, so that the quadratic covariation term may not have bounded variations as in the classical Itô’s formula. However is a Dirichlet process, and this quadratic covariation term is indeed the process of zero energy appearing in M. Fukushima’s decomposition [“Dirichlet forms and Markov processes” (1980; Zbl 0422.31007)]. It should also be mentioned that the condition on the starting point is not at all a restriction, since it is also needed to define the stochastic integrals along of the . Regarding the proof, the main argument consists in establishing that the quadratic variation term indeed exists, if belongs locally to . This is done by using an approximation in terms of backward and forward stochastic integrals (which leads also, finally, to a change of variable formula of Stratonovich type), and a multidimensional analogue of the 0-1 law of Engelbert and Schmidt [see the paper of R. Höhnle and K.-Th. Sturm, Stochastics Stochastics Rep. 44, No. 1/2, 27-41 (1993; Zbl 0780.60078)].