## 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.

### MSC:

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

### Keywords:

measure-valued processes; stochastic flows; coalescence
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### References:

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