## 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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### References:

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