Stochastic flows and signed measure valued stochastic partial differential equations

Peter M. Kotelenez

Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue Cleveland, OH 44106

Abstract: 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 (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 (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.

Keywords: Stochastic partial differential equations, signed measures, Hahn–Jordan decomposition, stochastic flows, stochastic ordinary differential equations, correlation Brownian motions.

MSC: Primary 60G57; Secondary 60H15

Language: English

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