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A pure jump Markov process associated with Smoluchowski’s coagulation equation. (English) Zbl 1018.60067

The Smoluchowski’s coagulation equation is considered. Let \[ \begin{cases} \frac{d}{dt} n(k,t)=\tfrac 12 \sum^{k-1}_{j=1} K(j,k-j)n(j,t)n(k-j,t)-n(k,t)\sum^\infty_{j=1} K(j,k)n(j,t),\\ n(k,0)=n_0(k)\end{cases}\tag{SD} \] for \(k\in\mathbb{N}^*\) and \[ \begin{cases} \frac{\partial}{\partial t} n(x,t)=\tfrac 12 \int^x_0 K(y,x-y)n(y,t)n(x-y,t) dy-n(x,t)\int^\infty_0 K(x,y)n(y,t) dy,\\ n(x,0)=n_0(x)\end{cases}\tag{SC} \] for all \(x\in\mathbb{R}_+\), where the coagulation kernel \(K\) is nonnegative and symmetric, be the coagulation equations in the discrete and continuous case, respectively.
The authors construct a pure jump stochastic process \((X_t)\), \(t\geq 0\), whose law is the solution of the coagulation equation in the following sense: in the discrete case \(P(X_t= k) = k_n(k,t)\) for all \(t\geq 0\) and all \(k\in\mathbb{N}^*\), in the continuous case \(P(X_t\in dx)= xn(x,t)dx\) for all \(t\geq 0\). This jump process satisfies a nonlinear Poisson driven stochastic differential equation. Existence, uniqueness and pathwise regularity for the solution of this equation are studied. It is also proved that the nonlinear process \(X\) can be obtained as the limit of Marcus-Lushnikov procedure.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60H20 Stochastic integral equations
60J75 Jump processes (MSC2010)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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