  Let $\mathscr{P}(E)$ be the space of probability measures on a measurable
space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov
chain Monte Carlo (MCMC) methods for simulating from a probability measure
$\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman--Kac Formulae:
Genealogical and Interacting Particle Systems with Applications (2004)
Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be
constructed to, in some sense, improve over MCMC methods. However, such
nonlinear kernels cannot be simulated exactly, so approximations of the
nonlinear kernels are constructed using auxiliary or potentially
self-interacting chains. Several nonlinear kernels are presented and it is
demonstrated that, under some conditions, the associated approximations exhibit
a strong law of large numbers; our proof technique is via the Poisson equation
and Foster--Lyapunov conditions. We investigate the performance of our
approximations with some simulations.
