Convergence of Markov chain approximations to stochastic reaction-diffusion equations
The authors consider a stochastic partial differential equation arising from the modelization of the transport of contaminants through a sheet of water [based on a model of \textit{G. Kallianpur} and \textit{J. Xiong}, Adv. Appl. Probab. 26, 377-403 (1994; Zbl 0804.60041)]: \[ du(t,x)=[\mathcal{A}u(t,x)+R(u(t,x))]dt+d\Theta(t,x), \] with initial condition \(u(0)=u_0\), where \(t\in [0,T]\), \(x\) belongs to some region \([0,L_1]\times [0,L_2]\), the operator \(\mathcal{A}\) is of the form \(D\Delta- V\cdot \nabla\), with von Neumann boundary conditions, \(D>0\) and \(V\) stands for the water velocity, \(R\) is assumed to be non-negative at the origin, with linear growth on the positive real line, locally Lipschitz and with polynomial growth, and finally the random perturbation \(d\Theta(t,x)\) has a Poisson source. Using the mild formulation, a process solution with values in some Hilbert space is considered. The aim is to prove existence and uniqueness of mild solution; for this pathwise uniqueness using Markov chain approximations via stochastic particle method is checked, obtaining a law of large numbers. This model is motivated by \textit{L. Arnold} and \textit{M. Theodosopulu} [Adv. Appl. Probab. 12, 367-379 (1980; Zbl 0429.60025)], \textit{P. Kotelenez} [Ann. Probab. 14, 173-193 (1986; Zbl 0661.60053) and [Probab. Theory Relat. Fields 78, 11-37 (1988; Zbl 0628.60108)], \textit{D. Blount} [Ann. Probab. 19, 1440-1462 (1991; Zbl 0741.92022); ibid. 22, 2040-2070 (1994; Zbl 0843.60057) and ibid. 24, 639-659 (1996; Zbl 0868.60078)].
scientific article; zbMATH DE number 1906176