143 research outputs found
Differentiability of solutions of stationary Fokker–Planck–Kolmogorov equations with respect to a parameter
We obtain sufficient conditions for the differentiability of solutions to stationary Fokker-Planck-Kolmogorov equations with respect to a parameter. In particular, this gives conditions for the differentiability of stationary distributions of diffusion processes with respect to a parameter
Construction of -strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions
We provide a general construction scheme for -strong Feller
processes on locally compact separable metric spaces. Starting from a regular
Dirichlet form and specified regularity assumptions, we construct an associated
semigroup and resolvents of kernels having the -strong Feller
property. They allow us to construct a process which solves the corresponding
martingale problem for all starting points from a known set, namely the set
where the regularity assumptions hold. We apply this result to construct
elliptic diffusions having locally Lipschitz matrix coefficients and singular
drifts on general open sets with absorption at the boundary. In this
application elliptic regularity results imply the desired regularity
assumptions
Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation
We study a quasilinear parabolic Cauchy problem with a cumulative
distribution function on the real line as an initial condition. We call
'probabilistic solution' a weak solution which remains a cumulative
distribution function at all times. We prove the uniqueness of such a solution
and we deduce the existence from a propagation of chaos result on a system of
scalar diffusion processes, the interactions of which only depend on their
ranking. We then investigate the long time behaviour of the solution. Using a
probabilistic argument and under weak assumptions, we show that the flow of the
Wasserstein distance between two solutions is contractive. Under more stringent
conditions ensuring the regularity of the probabilistic solutions, we finally
derive an explicit formula for the time derivative of the flow and we deduce
the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations
(2013) http://dx.doi.org/10.1007/s40072-013-0014-
The Formation of Crystal Defects in a Fe-Mn-Si Alloy Under Cyclic Martensitic Transformations
Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below
We prove existence and uniqueness of optimal maps on spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation. \ua9 2015, Mathematica Josephina, Inc
Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces
We find necessary and sufficient conditions for a finite K–bi–invariant
measure on a compact Gelfand pair (G, K) to have a square–integrable
density. For convolution semigroups, this is equivalent to having a
continuous density in positive time. When (G, K) is a compact Riemannian
symmetric pair, we study the induced transition density for
G–invariant Feller processes on the symmetric space X = G/K. These
are obtained as projections of K–bi–invariant L´evy processes on G,
whose laws form a convolution semigroup. We obtain a Fourier series
expansion for the density, in terms of spherical functions, where the
spectrum is described by Gangolli’s L´evy–Khintchine formula. The
density of returns to any given point on X is given by the trace of
the transition semigroup, and for subordinated Brownian motion, we
can calculate the short time asymptotics of this quantity using recent
work of Ba˜nuelos and Baudoin. In the case of the sphere, there is an
interesting connection with the Funk–Hecke theorem
- …