1,722,018 research outputs found
On the form of the large deviation rate function for the empirical measures of weakly interacting systems
No full textA basic result of large deviations theory is Sanov’s theorem, which states that the sequence of em- pirical measures of independent and identically distributed samples satisfies the large deviation principle with rate function given by relative entropy with respect to the common distribution. Large deviation principles for the empirical measures are also known to hold for broad classes of weakly interacting systems. When the interaction through the empirical measure corresponds to an absolutely continuous change of measure, the rate function can be expressed as relative entropy of a distribution with respect to the law of the McKean-Vlasov limit with measure- variable frozen at that distribution. We discuss situations, beyond that of tilted distributions, in which a large deviation principle holds with rate function in relative entropy form
Ein Zweifelsfall: zweifeln im Deutschen
No full textTwo main types of sentences are traditionally distinguished in the context of semantic theories of questions and answers: declarative sentences, corresponding to statements, and interrogative sentences, corresponding to questions. The interrogative forms can be fur- ther subdivided into dialectical ones (yes-no-questions) and non- dialectical ones (constituent questions). These distinctions are made for both root and embedded sentences. The predicates that select sentential complements fall into three classes: predicates that license only declaratives, those that allow only for interrogatives, and those that embed both types of sentences. In this connection, verbs of doubt are interesting in that they allow for declaratives as well as dialectical interrogatives, while non-dialectical interrogatives do not seem to be appropriate complements.
In what follows, our main concern will be with the German verb of doubt zweifeln and its possible sentential complements. Speaker intuitions as to which constructions are grammatical or acceptable vary, particularly with respect to rare expressions like zweifeln. Interviews and corpus analysis were therefore applied as a means to acquire reliable linguistic data. These as well as data from historical sources and from some languages other than German (esp. English and Italian) are presented and analysed. In the last section, based on the notion of `subjective probability', an attempt is made at explaining the observations
On the connection between symmetric N-player games and mean field games
No full textMean 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
On the Moments of the Modulus of Continuity of Itô Processes
No full textThe modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm
Probabilistic Approach to Finite State Mean Field Games
Full text linkWe 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
An algorithm to construct subsolutions of convex optimal control problems
Full text linkWe propose an algorithm that produces a nondecreasing sequence of subsolutions for a class of optimal control problems distinguished by the property that the associated Bellman operators preserve convexity. In addition to a theoretical discussion and proofs of convergence, numerical experiments are presented to illustrate the feasibility of the method
Effects of structural heterogeneity on the diurnal temperature range in temperate forest ecosystems
No full textThe microclimate in forest ecosystems can be altered by modifications of stand structure due to forest management or natural forest development. Current forest management practices in Central Europe and North America aim to promote structural heterogeneity and maintain forest canopy cover, which is known to be a major driver of forest microclimate. Here, we investigated the impacts of forest management and structural heterogeneity on the diurnal temperature range (DTR) in 128 managed forest stands in three climatically different locations (Swabian Alb, Hainich-Dün and Schorfheide-Chorin) in Central Europe. Increasing structural heterogeneity by promoting tree size diversity and differentiation increased vertical stratification and resulted in an impaired DTR during the vegetation period. Linear regression models with geographic location, elevation above sea level, canopy openness and measures of structural heterogeneity as explanatory variables explained 79.4–80.9% of variance in DTR. However, the overall effect of structural heterogeneity on DTR was small. Differences in DTR between plots of different main tree species could be attributed to differences in canopy openness and light transmission, whereas tree species diversity had no significant effect on DTR. Unmanaged forests were characterized by a significantly lower DTR than managed, even-aged forests. DTR in uneven-aged stands managed under single tree selection was comparable to unmanaged stands. Terrestrial laser scanning (TLS) derived measures of canopy openness and vertical structure allowed to explain 79.4% of variance in DTR considering geographic location and elevation, which can also be assessed by TLS with integrated GPS and an altimeter. We conclude that structural characteristics of forest stands other than canopy openness contribute marginally to variation in forest microclimate. However, the analyses of structure-microclimate analyses indicate that effects of stand structure on DTR might be more pronounced in regions with low precipitation during the vegetation period
On large deviations for small noise Itô processes
No full textThe large deviation principle in the small noise limit is derived for solutions of possibly degenerate Itô stochastic differential equations with predictable coefficients, which may depend also 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
An iterative procedure for constructing subsolutions of discrete-time optimal control problems
Full text linkAn iterative procedure for constructing subsolutions of deterministic or stochastic optimal control problems in discrete time with continuous state space is introduced. The procedure generates a nondecreasing sequence of subsolutions, giving true lower bounds on the minimal costs. Convergence of the values at any fixed initial state is shown
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games
Full text linkThis paper presents a class of evolutive Mean Field Games with multiple solutions for all time horizons T and convex but non-smooth Hamiltonian H, as well as for smooth H and T large enough. The phenomenon is analysed in both the PDE and the probabilistic setting. The examples are compared with the current theory about uniqueness of solutions. In particular, a new result on uniqueness for the MFG PDEs with small data, e.g., small T , is proved. Some results are also extended to MFGs with two populations
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