A pure jump Markov process associated with Smoluchowski's coagulation equation
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.
nonlinear stochastic differential equation
scientific article; zbMATH DE number 1906099
scientific article