100,711 research outputs found
Mean Field Games with Singular Controls
This paper establishes the existence of relaxed solutions to mean eld games (MFGs for short)
with singular controls. As a by-product, we obtain an existence of relaxed solutions results for
McKean-Vlasov stochastic singular control problems. Finally, we prove approximations of solutions results for a particular class of MFGs with singular controls by solutions, respectively control rules, for MFGs with purely regular controls. Our existence and approximation results strongly hinge on the use of the Skorokhod M1 topology on the space of cadlag functions
Quantum stochastic convolution cocycles III
Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is
shown to be equivalent to one governed by a quantum stochastic differential
equation, and the generating functionals of norm-continuous convolution
semigroups on a multiplier C*-bialgebra are then completely characterised.
These results are achieved by extending the theory of quantum Levy processes on
a compact quantum group, and more generally quantum stochastic convolution
cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier
C*-bialgebras. Strict extension results obtained by Kustermans, together with
automatic strictness properties developed here, are exploited to obtain
existence and uniqueness for coalgebraic quantum stochastic differential
equations in this setting. Then, working in the universal enveloping von
Neumann bialgebra, we characterise the stochastic generators of Markov-regular,
*-homomorphic (respectively completely positive and contractive), quantum
stochastic convolution cocycles.Comment: 20 pages; v2 corrects some typos and no longer contains a section on
quantum random walk approximations, which will now appear as a separate
submission. The article will appear in the Mathematische Annale
Basic properties of nonlinear stochastic Schr\"{o}dinger equations driven by Brownian motions
The paper is devoted to the study of nonlinear stochastic Schr\"{o}dinger
equations driven by standard cylindrical Brownian motions (NSSEs) arising from
the unraveling of quantum master equations. Under the Born--Markov
approximations, this class of stochastic evolutions equations on Hilbert spaces
provides characterizations of both continuous quantum measurement processes and
the evolution of quantum systems. First, we deal with the existence and
uniqueness of regular solutions to NSSEs. Second, we provide two general
criteria for the existence of regular invariant measures for NSSEs. We apply
our results to a forced and damped quantum oscillator.Comment: Published in at http://dx.doi.org/10.1214/105051607000000311 the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Global existence of weak solutions to viscoelastic phase separation: Part II Degenerate Case
The aim of this paper is to prove global in time existence of weak solutions
for a viscoelastic phase separation. We consider the case with singular
potentials and degenerate mobilities. Our model couples the diffusive interface
model with the Peterlin-Navier-Stokes equations for viscoelastic fluids. To
obtain the global in time existence of weak solutions we consider appropriate
approximations by solutions of the viscoelastic phase separation with a regular
potential and build on the corresponding energy and entropy estimates.Comment: 29 pages, 28 figure
Analysis of a quasi-variational contact problem arising in thermoelasticity
We formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions
Analysis of a regular black hole in Verlinde's gravity
This work focuses on the examination of a regular black hole within
Verlinde's emergent gravity, specifically investigating the Hayward-like
(modified) solution. The study reveals the existence of a single event horizon
under certain conditions. Our results indicate phase transitions and forbidden
regions based on the analysis of heat capacity and \textit{Hawking}
temperature. Geodesic trajectories and critical orbits (photon spheres) are
calculated, highlighting the presence of outer and inner light rings.
Additionally, we investigate the black hole shadows. Furthermore, the
\textit{quasinormal} modes are explored using third- and sixth-order WKB
approximations. In particular, we observe stable and unstable oscillations for
certain frequencies. Finally, in order to comprehend the phenomena of
time-dependent scattering in this scenario, we provide an investigation of the
time-domain solution.Comment: 33 pages, 8 figure
Mathematical and Numerical Analysis of a Pair of Coupled Cahn-Hilliard Equations with a Logarithmic Potential
Mathematical and numerical analysis has been undertaken for a pair of coupled Cahn-Hilliard equations with a logarithmic potential and with homogeneous Neumann boundary conditions. This pair of coupled equations arises in a phase separation model of thin film of binary liquid mixture. Global existence and uniqueness of a weak solution to the problem is proved using Faedo-Galerkin method. Higher regularity results of the weak solution are established under further regular requirements on the initial data. Further, continuous dependence on the initial data is presented.
Numerically, semi-discrete and fully-discrete piecewise linear finite element approximations to the continuous problem are proposed for which existence, uniqueness and various stability estimates of the approximate solutions are proved. Semi-discrete and fully-discrete error bounds are derived where the time discretisation error is optimal. An iterative method for solving the resulting nonlinear algebraic system is introduced and linear stability analysis in one space dimension is studied. Finally, numerical experiments illustrating some of the theoretical results are performed in one and two space dimensions
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