Superprocesses of stochastic flows. (English) Zbl 1015.60063

Measure-valued processes of a new type, arising as extensions of empirical processes of consistent \(k\)-point motions, are studied. More precisely, let \(b_{p}:\mathbb R^{d} \to\mathbb R^{d}\), \(1\leq p\leq d\), be bounded Borel functions, and let \(a\) be a bounded Borel mapping from \(\mathbb R^{d}\times\mathbb R^{d}\) to the set of all nonnegative definite \(d\times d\) symmetric matrices, satisfying \(a_{pq}(z_{i},z_{j}) = a_{pq}(z_{j}, z_{i})\) for all \(z_{i},z_{j}\in\mathbb R^{d}\), \(1\leq p,q\leq d\). For every \(k\geq 1\) define a differential operator \(A_{k}\) by \[ A_{k}f(z_1,\ldots,z_{k})= \frac 12 \sum^{k}_{i,j=1} \sum^{d}_{p,q=1} a_{pq}(z_{i},z_{j}){\partial^2 f\over \partial z_{ip}\partial z_{jq}}(z_1,\ldots,z_{k})+ \sum^{k}_{i=1}\sum^{d}_{p=1} b_{p}(z_{i}){\partial f \over \partial z_{ip}}(z_1,\ldots,z_{k}), \] where \(f\in C^2_{b}((\mathbb R^{d})^{k})\), \(z_{i} = (z_{i1}, \ldots,z_{id})\in\mathbb R^{d}\), \(1\leq i\leq k\). Suppose that the martingale problem for \((A_{k},C^2_{b}((\mathbb R^{d})^{k})\) is well posed and generates a unique continuous strong Markov Feller process on \((\mathbb R^{d})^{k}\). Let us denote this process, starting from \((y_1,\ldots,y_{k})\in (\mathbb R^{d})^{k}\) at time 0, by \(Y^{k}_{t} = (Y_{t}(y_1),\ldots,Y_{t}(y_{k}))\), and set \[ X^{k}_{t} = \frac 1{k}\sum^{k}_{i=1} \delta_{Y_{t} (y_{i})}. \tag{1} \] Let \(M(\mathbb R^{d})\) be the space of all finite Borel measures on \(\mathbb R^{d}\) equipped with the weak topology. It is proved that a unique continuous strong Markov Feller process \(X\) on \(M(\mathbb R^{d})\) with the following property exists: for all \(r>0\) and \(k\geq 1\), if \(X(0) = rk^{-1}\sum^{k}_{j=1}\delta _{y_{i}}\), then the law of \(X\) equals to the law of \(rX^{k}\), \(X^{k}\) being given by (1). Moreover, the generator of the process \(X\) is found and relations to stochastic coalescence and isotropic flows are discussed.


60J25 Continuous-time Markov processes on general state spaces
60G57 Random measures
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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