Lenya Ryzhik

We discuss the occurrence of some notions and results from contact topology in non-equilibrium thermodynamics. This includes Reeb chords and the partial order on the space of Legendrian submanifolds.

Mean Field Game (MFG) models implicitly assume “rational expectations”, meaning that the heterogeneous agents being modeled correctly know all relevant transition probabilities for the complex system they inhabit. When there is common noise, it becomes necessary to solve the “Master equation”, in which the infinite-dimensional density of agents is a state variable. The rational expectations assumption and the implication that agents solve Master equations are unrealistic in many applications. We show how to instead formulate MFGs with non-rational expectations. Departing from rational expectations is particularly relevant in “MFGs with a low-dimensional coupling”, i.e. MFGs in which agents’ running reward function depends on the density only through low-dimensional functionals of this density. This happens, for example, in most macroeconomics MFGs in which these low-dimensional functionals have the interpretation of “equilibrium prices”. In MFGs with a low-dimensional coupling, departing from rational expectations allows for completely sidestepping the Master equation and for instead solving much simpler finite-dimensional HJB equations. We introduce an adaptive learning model as a particular example of non-rational expectations and discuss its properties.

We consider recursion equations of the form u _{n +1} ( x ) = Q [ u _n ]( x ), n ≥ 1, x ∈ ℝ, with a non-local operator Q [ u ] = g ( u * q ), where g is a polynomial, satisfying g (0) = 0, g (1) = 1, g ((0, 1)) ⊆ (0, 1), and q is a (compactly supported) probability density with * denoting convolution. Motivated by a line of works for nonlinear PDEs initiated by Etheridge, Freeman and Penington (2017), we show that for general g , a probabilistic model based on branching random walk can be given to the solution of the recursion, while in case g is also strictly monotone, a probabilistic threshold-based model can be given. In the latter case, we provide a conditional tightness result. We analyze in detail the bistable case and prove for it convergence of the solution shifted around a linear in n centering.

The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. Their cumulative distribution function satisfies a forward in time nonlinear non-local reaction-diffusion type equation. On the other hand, the learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation for the agents density. Together, these equations form a system of the mean-field game type. When the learning rate is sufficiently large, existence of balanced growth path solutions to the Lucas-Moll system was proved in~\cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the balanced growth paths do not exist. The main result is a long time convergence theorem. Namely, the solution to the initial-terminal value problem behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength \beta\in\mathbb{R} . This equation exhibits a transition from pulled to pushed front behavior at \beta_{c}=2 . We prove convergence of the solutions to a traveling wave in a reference frame centered at a position m_{\beta}(t) and study the asymptotics of the front location m_{\beta}(t) . When \beta < 2 , it has the same form as for the standard Fisher-KPP equation established by Bramson: m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1) as t\to\infty . This form is typical of pulled fronts. When \beta > 2 , the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1) with c_{*}(\beta)=\beta/2+2/\beta , which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2 , the expansion changes to m_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1) , reflecting the “pushmi-pullyu” nature of the front. The arguments for \beta2 relies on standard pushed front techniques. The proof in the case \beta=\beta_{c} is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at \beta_{c}=2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.