The transport equation and zero quadratic variation processes

Abstract

We analyze the transport equation driven by a zero quadratic variation process. Using the stochastic calculus via regularization and the Malliavin calculus techniques, we prove the existence, uniqueness and absolute continuity of the law of the solution. As an example, we discuss the case when the noise is a Hermite process.2010 Mathematics Subject Classification. Primary 60H15: Secondary 60H05, 60H07.In this paper we analyze a stochastic transport equation. The following onedimensional Cauchy problem is considered: given an initial-data u 0 , find u(t, x; ω) ∈ R, satisfyinggiven vector field, and the noise (Z t ) t≥0 is a stochastic process with zero quadratic variation. Problem (2) may be understood as a model for the concentration (density) of a pollutant in a flow where the velocity field has a random perturbation. The stochastic transport equation driven by the standard Brownian motion was first addressed in Kunita's books (see [10], [11]). More recently it has been studied by several authors; in [1] the linear additive case is considered, existence and uniqueness of weak L p -solutions and a representation for the general solution were shown. The non-blow-up problem is addressed for the multiplicative case with Stratonovich form in [5]. In [7] the authors have shown that the introduction of a multiplicative noise in the PDE allows some improvements in the traditional hypothesis needed to prove that the problem is well-posed, this is extended later to a non-linear case in [20]. A new uniqueness result is obtained in [14] by means of Wiener-chaos decomposition, and working on the associated Kolmogorov equation. The extension of the model to the fractional Brownian noise has been done in [21], where the existence of density of the solution and Gaussian estimates of the density were proven.Our purpose is to solve the equation (2) and to analyze the properties of its solution in the case when the noise is a more general stochastic process, possibly non-Gaussian. We will focus on the situation when the noise Z in (2) is a stochastic process with zero quadratic variation, this is well defined in the next section of the paper.The reason why we chose such a noise is that the stochastic integration theory in the sense of Russo-Vallois (see [24], [25]) can be applied to it. In fact, the stochastic integral in (2) will be understood as a symmetric integral in the Russo-Vallois sense with respect to the noise Z. Besides, in most of the papers cited in the previous paragraph, the Itô-Wentzell formula plays a crucial role in the characterization of the solution, consider a zero quadratic variation process is as far as one can go in order to prove that characterization, avoiding the presence of second order terms (see Proposition 9 in [8]). Among the zero covariance processes lies the fractional Brownian motion (for H ≥ 1/2), a self-similar process that find some of their applications in various kind of phenomena, going from hydrology and surface modelling to network traffic analysis and mathematical finance, to name a few...