%0 Journal Article
%T Probabilistic Representations for the Solution of Higher Order Differential Equations
%A S. Mazzucchi
%J International Journal of Partial Differential Equations
%D 2013
%R 10.1155/2013/297857
%X A probabilistic representation for the solution of the partial differential equation , is constructed in terms of the expectation with respect to the measure associated to a complex-valued stochastic process. 1. Introduction The connection between the solution of parabolic equations associated to second-order elliptic operators and the theory of stochastic processes is a largely studied topic [1]. The main instance is the Feynman-Kac formula, providing a representation of the solution of the heat equation with potential (the continuous functions vanishing at infinity): in terms of an integral with respect to the measure of the Wiener process, the mathematical model of the Brownian motion [2]: If the Laplacian in (1) is replaced with a higher order differential operator, that is, if we consider a Cauchy problem of the form with , , then a formula analogous to (2), giving the solution of (3) in terms of the expectation with respect to the measure associated to a Markov process, is lacking. In fact, such a formula cannot be proved for semigroups whose generator does not satisfy the maximum principle, as in the case of with . In fact in the case where the Cauchy problem (3) is not well posed on the space of continuous bounded functions [3]. In other words it is not possible to find a stochastic process which plays for the parabolic equation (3) the same role that the Wiener process plays for the heat equation. We would like to point out that the problem of the probabilistic representation of the solution of the Cauchy problem (3) presents some similarities with the problem of the mathematical definition of Feynman path integrals (see [4¨C7] for a discussion of this topic). Indeed in both cases it is not possible to implement an integration theory of Lebesgue type in terms of a bounded variation measure on a space of continuous paths [8]. An analogous of the Feynman-Kac formula for the parabolic equation (3), namely, an equation of the form (where should be some ¡°measure¡± on a space of ¡°paths¡± ), can be obtained only under some restrictions on and and by giving up a traditional integration theory in the Lebesgue sense with respect to a bounded variation measure on a space of (real) continuous paths. In the mathematical literature two main approaches have been proposed. The first one [9, 10] realizes formula (4) in terms of the expectation with respect to a signed measure on a space of paths on the interval . It is worthwhile to mention that an analogous of the arc-sine law [10, 11], of the central limit theorem [12], and of Ito formula and Ito stochastic
%U http://www.hindawi.com/journals/ijpde/2013/297857/