3,641 research outputs found
Long-term and blow-up behaviors of exponential moments in multi-dimensional affine diffusions
This paper considers multi-dimensional affine processes with continuous
sample paths. By analyzing the Riccati system, which is associated with affine
processes via the transform formula, we fully characterize the regions of
exponents in which exponential moments of a given process do not explode at any
time or explode at a given time. In these two cases, we also compute the
long-term growth rate and the explosion rate for exponential moments. These
results provide a handle to study implied volatility asymptotics in models
where returns of stock prices are described by affine processes whose
exponential moments do not have an explicit formula.Comment: 36 pages, 5 figure
Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian
We study the long-time behavior of the unique viscosity solution of the
viscous Hamilton-Jacobi Equation with inhomogeneous Dirichlet boundary conditions,
where is a bounded domain of . We mainly focus on the
superquadratic case () and consider the Dirichlet conditions in the
generalized viscosity sense. Under rather natural assumptions on the
initial and boundary data, we connect the problem studied to its associated
stationary generalized Dirichlet problem on one hand and to a stationary
problem with a state constraint boundary condition on the other hand
Post-collapse dynamics of self-gravitating Brownian particles in D dimensions
We address the post-collapse dynamics of a self-gravitating gas of Brownian
particles in D dimensions, in both canonical and microcanonical ensembles. In
the canonical ensemble, the post-collapse evolution is marked by the formation
of a Dirac peak with increasing mass. The density profile outside the peak
evolves self-similarly with decreasing central density and increasing core
radius. In the microcanonical ensemble, the post-collapse regime is marked by
the formation of a ``binary''-like structure surrounded by an almost uniform
halo with high temperature. These results are consistent with thermodynamical
predictions
The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation
We consider the diffusive Hamilton-Jacobi equation with Dirichlet boundary conditions in two space dimensions, which
arises in the KPZ model of growing interfaces. For , solutions may develop
gradient singularities on the boundary in finite time, and examples of
single-point gradient blowup on the boundary are known, but the space-profile
in the tangential direction has remained a completely open problem. In the
parameter range , for the case of a flat boundary and an isolated
singularity at the origin, we give an answer to this question, obtaining the
precise final asymptotic profile, under the form Interestingly, this result displays a new phenomenon of strong
anisotropy of the profile, quite different to what is observed in other blowup
problems for nonlinear parabolic equations, with the exponents in the
normal direction and in the tangential direction .
Furthermore, the tangential profile violates the (self-similar) scale
invariance of the equation, whereas the normal profile remains self-similar.Comment: Int. Math. Res. Not. IMRN, to appea
- …