Stochastic Flows and Signed Measure Valued Stochastic Partial Differential Equations

Peter M. Kotelenez
Theory of Stochastic Processes
Vol.16 (32), no.2, 2010, pp.86-105
Let $N$ point particles be distributed over ${\mathbb{R}}^d,\ d \in {\mathbb{N}}$. The position of the $i$-th particle at time $t$ will be denoted $r (t,q^i)$ where $q^i$ is the position at $t=0$. $m_i$ is the mass of the $i$-th particle. Let $\delta_r$ be the point measure concentrated in $r$ and ${\mathcal X}_N(0) := \sum_{i =1}^N m_i \delta_{q^i}$ the initial mass distribution of the $ N$ point particles. The empirical mass distribution (also called the "empirical process") at time $t$ is then given by \footnote{We will not indicate the integration domain in what follows if it is $\mathbb{R}^d$.} $${\mathcal X}_N (t) := \sum_{i =1}^N m_i \delta_{r(t,q^i)} = \int \delta_{r(t,q)} {\mathcal X}_N(0,dq), $$ i.e., by the $N-$particle flow. In Kotelenez (2008) the masses are positive and the motion of the positions of the point particles is described by a stochastic ordinary differential equation (SODE). Further, the resulting empirical process is the solution of a stochastic partial differential equation (SPDE) which, by a continuum limit, can be extended to an SPDE in smooth positive measures. Some generalizations to the case of signed measures with applications in 2D fluid mechanics have been made.\footnote{Cf., e.g., Marchioro and Pulvirenti (1982), Kotelenez (1995a,b), Kurtz and Xiong (1999), Amirdjanova (2000), (2007), Amirdjanova and Xiong (2006).} We extend some of those results and results of Kotelenez (2008), showing that the signed measure valued solutions of the SPDEs preserve the Hahn-Jordan decomposition of the initial distributions which has been an open problem for some time.