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We consider a version of random motion of hard core particles on thesemi-lattice $ 1, 2, 3,...$, where in each time instant one of three possibleevents occurs, viz., (a) a randomly chosen particle hops to a free neighboringsite, (b) a particle is created at the origin (namely, at site 1) provided thatsite 1 is free and (c) a particle is eliminated at the origin (provided thatthe site 1 is occupied). Relations to the BKP equation are explained. Namely,the tau functions of two different BKP hierarchies provide generating functionsrespectively (I) for transition weights between different particleconfigurations and (II) for an important object: a normalization function whichplays the role of the statistical sum for our non-equilibrium system. As anexample we study a model where the hopping rate depends on two parameters ($r$and $\beta$). For time $\time\to\infty$ we obtain the asymptotic configurationof particles obtained from the initial empty state (the state withoutparticles) and find an analog of the first order transition at $\beta=1$.

Random turn walk on a half line with creation of particles at the origin