Summary1. Ski resorts increasingly affect alpine ecosystems through enlargement of ski pistes, machine-grading of ski piste areas and increasing use of artificial snow. 2. In 12 Swiss alpine ski resorts, we investigated the effects of ski piste management on vegetation structure and composition using a pairwise design of 38 plots on ski pistes and 38 adjacent plots off-piste. 3. Plots on ski pistes had lower species richness and productivity, and lower abundance and cover of woody plants and early flowering species, than reference plots. Plots on machine-graded pistes had higher indicator values for nutrients and light, and lower vegetation cover, productivity, species diversity and abundance of early flowering and woody plants. Time since machine-grading did not mitigate the impacts of machinegrading, even for those plots where revegetation had been attempted by sowing. 4. The longer artificial snow had been used on ski pistes (2-15 years), the higher the moisture and nutrient indicator values. Longer use also affected species composition by increasing the abundance of woody plants, snowbed species and late-flowering species, and decreasing wind-edge species. 5. Synthesis and applications. All types of ski piste management cause deviations from the natural structure and composition of alpine vegetation, and lead to lower plant species diversity. Machine-grading causes particularly severe and lasting impacts on alpine vegetation, which are mitigated neither by time nor by revegetation measures. The impacts of artificial snow increase with the period of time since it was first applied to ski piste vegetation. Extensive machine-grading and snow production should be avoided, especially in areas where nutrient and water input are a concern. Ski pistes should not be established in areas where the alpine vegetation has a high conservation value.

We study large deviation properties of systems of weakly interacting particles modeled by Itô stochastic differential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to infinity, to the weak solution of an associated McKean–Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of diffusion type. To illustrate this, we treat the case of SDEs with delay

Mean field games are limit models for symmetric N -player games with interaction of mean field type as N → ∞. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding N -player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? In this direction, rigorous results are mostly available for stationary problems with ergodic costs. Here, we identify limit points of sequences of certain approximate Nash equilibria as solutions to mean field games for problems with Itô-type dynamics and costs over a finite time horizon. Limits are studied through weak convergence of associated normalized occupation measures and identified using a probabilistic notion of solution for mean field games.2000 AMS subject classifications: 60B10, 60K35, 91A06, 93E20

Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for mean-field systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the W 1 Wasserstein distance between the empirical measures of the two systems on the space of trajectories D([0, T ], R d ).

The large deviation principle in the small noise limit is derived for solutions of possibly degenerate Itô stochastic differential equations with predictable coefficients, which may also depend on the large deviation parameter. The result is established under mild assumptions using the Dupuis-Ellis weak convergence approach. Applications to certain systems with memory and to positive diffusions with square-root-like dispersion coefficient are included.

We study mean field games and corresponding N -player games in continuous time over a finite time horizon where the position of each agent belongs to a finite state space. As opposed to previous works on finite state mean field games, we use a probabilistic representation of the system dynamics in terms of stochastic differential equations driven by Poisson random measures. Under mild assumptions, we prove existence of solutions to the mean field game in relaxed open-loop as well as relaxed feedback controls. Relying on the probabilistic representation and a coupling argument, we show that mean field game solutions provide symmetric ε N -Nash equilibria for the N -player game, both in open-loop and in feedback strategies (not relaxed), with ε N ≤ constant √ N . Under stronger assumptions, we also find solutions of the mean field game in ordinary feedback controls and prove uniqueness either in case of a small time horizon or under monotonicity. Date

We introduce a simple class of mean field games with absorbing boundary over a finite time horizon. In the corresponding N -player games, the evolution of players' states is described by a system of weakly interacting Itô equations with absorption on first exit from a bounded open set. Once a player exits, her/his contribution is removed from the empirical measure of the system. Players thus interact through a renormalized empirical measure. In the definition of solution to the mean field game, the renormalization appears in form of a conditional law. We justify our definition of solution in the usual way, that is, by showing that a solution of the mean field game induces approximate Nash equilibria for the N -player games with approximation error tending to zero as N tends to infinity. This convergence is established provided the diffusion coefficient is nondegenerate. The degenerate case is more delicate and gives rise to counter-examples.

We consider N -player and mean field games in continuous time over a finite horizon, where the position of each agent belongs to {−1, 1}. If there is uniqueness of mean field game solutions, e.g. under monotonicity assumptions, then the master equation possesses a smooth solution which can be used to prove convergence of the value functions and of the feedback Nash equilibria of the N -player game, as well as a propagation of chaos property for the associated optimal trajectories. We study here an example with anti-monotonous costs, and show that the mean field game has exactly three solutions. We prove that the value functions converge to the entropy solution of the master equation, which in this case can be written as a scalar conservation law in one space dimension, and that the optimal trajectories admit a limit: they select one mean field game soution, so there is propagation of chaos. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N -player game selects the optimizer of this problem. 1 2 ALEKOS CECCHIN, PAOLO DAI PRA, MARKUS FISCHER, AND GUGLIELMO PELINO are shown to be concentrated on weak solutions of the corresponding mean field game. This concept of solution is also used in another, more recent work by Lacker; see below.Here, we are interested in the convergence problem for Nash equilibria in Markov feedback strategies with full state information. A first result in this direction was given by Gomes, Mohr, and Souza [19] in the case of finite state dynamics. There, convergence of Markovian Nash equilibria to the mean field game limit is proved, but only if the time horizon is small enough. A breakthrough was achieved by Cardaliaguet, Delarue, Lasry, and Lions in [7]. In the setting of games with non-degenerate Brownian dynamics, possibly including common noise, those authors establish convergence to the mean field game limit, in the sense of convergence of value functions as well as propagation of chaos for the optimal state trajectories, for arbitrary time horizon provided the so-called master equation associated with the mean field game possesses a unique sufficiently regular solution. The master equation arises as the formal limit of the Hamilton-Jacobi-Bellman systems determining the Markov feedback Nash equilibria. It yields, if well-posed, the optimal value in the mean field game as a function of initial time, state and distribution. It thus also provides the optimal control action, again as a function of time, state, and measure variable. This allows, in particular, to compare the prelimit Nash equilibria to the solution of the limit model through coupling arguments.If the master equation possesses a unique regular solution, which is guaranteed under the Lasry-Lions monotonicity conditions, then the convergence analysis can be considerably refined. In this case, for games with finite state dynamics, Cecchin and Pelino [11] and, independently, Bayraktar and Cohen [3] obtain a central limit theorem and lar...