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European Mathematical Society Publishing House
2024-03-29 08:39:35
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=6&iss=4&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2004
4
Metastability in Reversible Diffusion Processes I: Sharp Asymptotics for Capacities and Exit Times
Anton
Bovier
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Michael
Eckhoff
Universität Zürich, ZÜRICH, SWITZERLAND
Véronique
Gayrard
CNRS Luminy, MARSEILLE CEDEX 9, FRANCE
Markus
Klein
Universität Potsdam, POTSDAM, GERMANY
Metastability, diffusion processes, potential theory, capacity, exit times potential theory, capacity, exit times
We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form $-\e \Delta +\nabla F(\cdot)\nabla$ on $\R^d$ or subsets of $\R^d$, where $F$ is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that {\it metastable exit times} from the attractive domains of the minima of $F$ can be related, up to multiplicative errors that tend to one as $\e\downarrow 0$, to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of $F$ at the starting minimum and the relevant {\it saddle points}. As a result, we are able to give the first rigorous proof of the classical {\it Eyring-Kramers formula} in dimension larger than $1$. The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.
Statistical mechanics, structure of matter
Probability theory and stochastic processes
General
399
424
10.4171/JEMS/14
http://www.ems-ph.org/doi/10.4171/JEMS/14
The regular inverse Galois problem over non-large fields
Jochen
Koenigsmann
University of Oxford, OXFORD, GREAT BRITAIN
Inverse Galois problem, embedding problems, large fields, Mordell conjecture for function fields, diophantine theory of fields
By a celebrated theorem of Harbater and Pop, the regular inverse Galois problem is solvable over any field containing a large field. Using this and the Mordell conjecture for function fields, we construct the first example of a field $K$ over which the regular inverse Galois problem can be shown to be solvable, but such that $K$ does not contain a large field. The paper is complemented by model-theoretic observations on the diophantine nature of the regular inverse Galois problem.
Field theory and polynomials
General
425
434
10.4171/JEMS/15
http://www.ems-ph.org/doi/10.4171/JEMS/15
Representation of equilibrium solutions to the table problem of growing sandpiles
Piermarco
Cannarsa
Università di Roma, ROMA, ITALY
Pierre
Cardaliaguet
Université de Bretagne Occidentale, BREST, FRANCE
Granular matter, eikonal equation, singularities, semiconcave functions, viscosity solutions, optimal mass tranfer
In the dynamical theory of granular matter the so-called table problem consists instudying the evolution of a heap of matter poured continuously onto a bounded domain $\Omega\subset \R^2$. The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge-Kantorovich problem, is the object of this paper. Our main result is an integral representation formula for the solution, in terms of the boundary curvature and of the normal distance to the cut locus of $\Omega$.
Partial differential equations
Operations research, mathematical programming
General
435
464
10.4171/JEMS/16
http://www.ems-ph.org/doi/10.4171/JEMS/16
Intersection cohomology of reductive varieties
Michel
Brion
Université Grenoble I, SAINT MARTIN D'HERES CEDEX, FRANCE
Roy
Joshua
Ohio State University, COLUMBUS, UNITED STATES
We extend the methods developed in our earlier work to algorithmically compute the intersection cohomology Betti numbers of reductive varieties. These form a class of highly symmetric varieties that includes equivariant compactifications of reductive groups. Thereby, we extend a well-known algorithm for toric varieties.
Algebraic geometry
General
465
481
10.4171/JEMS/17
http://www.ems-ph.org/doi/10.4171/JEMS/17
Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion
Moshe
Marcus
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Laurent
Véron
Université François Rabelais, TOURS, FRANCE
Bessel capacities, maximal solutions, rate of blow-up
Let $\Gw$ be a bounded domain of class $C^2$ in $\mathbb R^N$ and let $K$ be a compact subset of $\prt\Gw$. Assume that $q\geq (N+1)/(N-1)$ and denote by $U_K$ the maximal solution of $ -\Gd u+u^q=0$ in $\Gw$ which vanishes on $\prt\Gw\setminus K$. We obtain sharp upper and lower estimates for $U_K$ in terms of the Bessel capacity $C_{2/q,q'}$ and prove that $U_K$ is $\gs$-moderate. In addition we describe the precise asymptotic behavior of $U_K$ at points $\gs\in K$, which depends on the 'density' of $K$ at $\gs$, measured in terms of the capacity $C_{2/q,q'}$.
Partial differential equations
Potential theory
General
483
527
10.4171/JEMS/18
http://www.ems-ph.org/doi/10.4171/JEMS/18