868 research outputs found
The exit-time problem for a Markov jump process
The purpose of this paper is to consider the exit-time problem for a
finite-range Markov jump process, i.e, the distance the particle can jump is
bounded independent of its location. Such jump diffusions are expedient models
for anomalous transport exhibiting super-diffusion or nonstandard normal
diffusion. We refer to the associated deterministic equation as a
volume-constrained nonlocal diffusion equation. The volume constraint is the
nonlocal analogue of a boundary condition necessary to demonstrate that the
nonlocal diffusion equation is well-posed and is consistent with the jump
process. A critical aspect of the analysis is a variational formulation and a
recently developed nonlocal vector calculus. This calculus allows us to pose
nonlocal backward and forward Kolmogorov equations, the former equation
granting the various moments of the exit-time distribution.Comment: 15 pages, 7 figure
Reflected Spectrally Negative Stable Processes and their Governing Equations
This paper explicitly computes the transition densities of a spectrally
negative stable process with index greater than one, reflected at its infimum.
First we derive the forward equation using the theory of sun-dual semigroups.
The resulting forward equation is a boundary value problem on the positive
half-line that involves a negative Riemann-Liouville fractional derivative in
space, and a fractional reflecting boundary condition at the origin. Then we
apply numerical methods to explicitly compute the transition density of this
space-inhomogeneous Markov process, for any starting point, to any desired
degree of accuracy. Finally, we discuss an application to fractional Cauchy
problems, which involve a positive Caputo fractional derivative in time
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
A time-fractional mean field game
We consider a Mean Field Games model where the dynamics of the agents is
subdiffusive. According to the optimal control interpretation of the problem,
we get a system involving fractional time-derivatives for the
Hamilton-Jacobi-Bellman and the Fokker-Planck equations. We discuss separately
the well-posedness for each of the two equations and then we prove existence
and uniqueness of the solution to the Mean Field Games syste
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