155 research outputs found
Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations
The purpose is to study the Cauchy problem for non-linear in time and space
pseudo-differential equations. These include the fractional in time versions of
HJB equations governing the controlled scaled CTRW. As a preliminary step which
is of independent interest we analyse the corresponding linear equation proving
its well-posedness and smoothing properties
Generalised Fractional Evolution Equations of Caputo Type
This paper is devoted to the study of generalised time-fractional evolution
equations involving Caputo type derivatives. Using analytical methods and
probabilistic arguments we obtain well-posedness results and stochastic
representations for the solutions. These results encompass known linear and
non-linear equations from classical fractional partial differential equations
such as the time-space-fractional diffusion equation, as well as their far
reaching extensions. \\ Meaning is given to a probabilistic generalisation of
Mittag-Leffler functions.Comment: To be published in 'Chaos, Solitons & Fractals
The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis
We introduce a max-plus analogue of the Petrov-Galerkin finite element method
to solve finite horizon deterministic optimal control problems. The method
relies on a max-plus variational formulation. We show that the error in the sup
norm can be bounded from the difference between the value function and its
projections on max-plus and min-plus semimodules, when the max-plus analogue of
the stiffness matrix is exactly known. In general, the stiffness matrix must be
approximated: this requires approximating the operation of the Lax-Oleinik
semigroup on finite elements. We consider two approximations relying on the
Hamiltonian. We derive a convergence result, in arbitrary dimension, showing
that for a class of problems, the error estimate is of order or , depending on the
choice of the approximation, where and are respectively the
time and space discretization steps. We compare our method with another
max-plus based discretization method previously introduced by Fleming and
McEneaney. We give numerical examples in dimension 1 and 2.Comment: 31 pages, 11 figure
Evolutionary game of coalition building under external pressure
We study the fragmentation-coagulation (or merging and splitting)
evolutionary control model as introduced recently by one of the authors, where
small players can form coalitions to resist to the pressure exerted by the
principal. It is a Markov chain in continuous time and the players have a
common reward to optimize. We study the behavior as grows and show that the
problem converges to a (one player) deterministic optimization problem in
continuous time, in the infinite dimensional state space
REGULARITY AND SENSITIVITY FOR MCKEAN-VLASOV TYPE SPDEs GENERATED BY STABLE-LIKE PROCESSES
In this paper we study the sensitivity of nonlinear stochastic differential equations of McKeanâVlasov type generated by
stable-like processes. By using the method of stochastic characteristics, we transfer these equations to non-stochastic equations
with random coefficients, thus making it possible to use results obtained for nonlinear PDEs of McKeanâVlasov type generated by
stable-like processes in previous works. The motivation for studying sensitivity of nonlinear McKeanâVlasov SPDEs arises naturally
from the analysis of the mean-field games with common noise
Semiring and semimodule issues in MV-algebras
In this paper we propose a semiring-theoretic approach to MV-algebras based
on the connection between such algebras and idempotent semirings - such an
approach naturally imposing the introduction and study of a suitable
corresponding class of semimodules, called MV-semimodules.
We present several results addressed toward a semiring theory for
MV-algebras. In particular we show a representation of MV-algebras as a
subsemiring of the endomorphism semiring of a semilattice, the construction of
the Grothendieck group of a semiring and its functorial nature, and the effect
of Mundici categorical equivalence between MV-algebras and lattice-ordered
Abelian groups with a distinguished strong order unit upon the relationship
between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of
Section
The influence of fractional diffusion in Fisher-KPP equations
We study the Fisher-KPP equation where the Laplacian is replaced by the
generator of a Feller semigroup with power decaying kernel, an important
example being the fractional Laplacian. In contrast with the case of the stan-
dard Laplacian where the stable state invades the unstable one at constant
speed, we prove that with fractional diffusion, generated for instance by a
stable L\'evy process, the front position is exponential in time. Our results
provide a mathe- matically rigorous justification of numerous heuristics about
this model
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