205 research outputs found
Ergodic Mean Field Games with H\"ormander diffusions
We prove existence of solutions for a class of systems of subelliptic PDEs
arising from Mean Field Game systems with H\"ormander diffusion. These results
are motivated by the feedback synthesis Mean Field Game solutions and the Nash
equilibria of a large class of -player differential games
Stochastic neural field equations: A rigorous footing
We extend the theory of neural fields which has been developed in a
deterministic framework by considering the influence spatio-temporal noise. The
outstanding problem that we here address is the development of a theory that
gives rigorous meaning to stochastic neural field equations, and conditions
ensuring that they are well-posed. Previous investigations in the field of
computational and mathematical neuroscience have been numerical for the most
part. Such questions have been considered for a long time in the theory of
stochastic partial differential equations, where at least two different
approaches have been developed, each having its advantages and disadvantages.
It turns out that both approaches have also been used in computational and
mathematical neuroscience, but with much less emphasis on the underlying
theory. We present a review of two existing theories and show how they can be
used to put the theory of stochastic neural fields on a rigorous footing. We
also provide general conditions on the parameters of the stochastic neural
field equations under which we guarantee that these equations are well-posed.
In so doing we relate each approach to previous work in computational and
mathematical neuroscience. We hope this will provide a reference that will pave
the way for future studies (both theoretical and applied) of these equations,
where basic questions of existence and uniqueness will no longer be a cause for
concern
Stochastic models associated to a Nonlocal Porous Medium Equation
The nonlocal porous medium equation considered in this paper is a degenerate
nonlinear evolution equation involving a space pseudo-differential operator of
fractional order. This space-fractional equation admits an explicit,
nonnegative, compactly supported weak solution representing a probability
density function. In this paper we analyze the link between isotropic transport
processes, or random flights, and the nonlocal porous medium equation. In
particular, we focus our attention on the interpretation of the weak solution
of the nonlinear diffusion equation by means of random flights.Comment: Published at https://doi.org/10.15559/18-VMSTA112 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
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