Convergence of moderately interacting particle systems to a diffusion-convection equation
The diffusion-convection equation \[ \frac {\partial u}{\partial t}+|u|^{q-1}\frac {\partial u}{\partial x}=\frac 12\frac {\partial ^2u}{\partial x^2}\tag{1} \] in the domain \((t,x)\in(0,\infty)\times R \) with initial condition \(\delta _0\), where \(q\geq 2\), admits a unique positive solution \(v_q\) in \(C((0,\infty),L^1(R))\cap C^{\infty }((0,\infty)\times R)\). By interpreting the equation (1) as a Fokker-Planck equation a probabilistic interpretation to the solution \(v_q\) is given. To this end it is shown that an appropriate martingale problem has a unique solution. Consequently, the solution is obtained as the propagation of chaos limit of a sequence of moderately interacting particle systems.
scientific article; zbMATH DE number 1409173
scientific article