Letters in Mathematical Physics 15 (1988) 345-350. 9 1988 by Kluwer Academic Publishers. 345 A Path Integral Formula for Certain Fourth-Order Elliptic Operators BERNARD GAVEAU and PHILIPPE SAINTY Universit~ Pierre et Marie Curie, Math~matiques, tour 45-46, 5~me ~tage, 4, Place Jussieu 75252 Paris Cedex 05, France (Received: 18 January 1988) Abstract. We define a discretized path integral formula for the operator -A 2 - V. This formula is the generalization of the Feynman-Kac formula for + A - V. I. Introduction In this Letter, we want to construct the analogue of Brownian motion and of the Feynman-Kac formula for a fourth-order elliptic operator like -A 2 - V, where A is the usual Laplace operator in g~d and V is a potential. There have been several attempts at constructing such objects in the past. Let us cite, for example, the general construction of Ehrenpreis [ 1] using a Fourier transform approach, and the construction of Daletskii [2]. More recently, a very detailed study of this kind of path integral was made by Hochberg [3]. The approach of Hochberg is also a Fourier transform approach and he constructs a measure (which is of infinite total variation) on a space of real-valued paths. Last year, the second author constructed a positive probability measure on complex- valued paths valid for the operators dn/dx n (see [4]). Our approach here is different. We start from the Trotter formula for -A z - V, and we want to 'disentangle' the exponential of -A z and the exponential of - Vin the same spirit as in [5] and [6]. This can be done by rewriting the exponential of -A 2 as an expectation value of an exponential of a first-order operator with stochastic coefficients, or what is the same as an expectation value of a translation operator, by a complex random variable. Then, it is easy to disentangle the time-ordered product in Trotter's formula and obtain a Feynman-Kac formula. We are thus 'forced', by our formalism, to construct a complex-valued stochastic process on a Gaussian probability space, so that our construction is different from Hochberg's [3] or from any Fourier transform approach. In one variable, it gives back the construction of the second author [4] but starting from different ideas.