36 research outputs found
On the time evolution of Bernstein processes associated with a class of parabolic equations
In this article dedicated to the memory of Igor D. Chueshov, I first
summarize in a few words the joint results that we obtained over a period of
six years regarding the long-time behavior of solutions to a class of
semilinear stochastic parabolic partial differential equations. Then, as the
beautiful interplay between partial differential equations and probability
theory always was close to Igor's heart, I present some new results concerning
the time evolution of certain Markovian Bernstein processes naturally
associated with a class of deterministic linear parabolic partial differential
equations. Particular instances of such processes are certain conditioned
Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and
Bernstein loops, whose laws may evolve in space in a non trivial way.
Specifically, I examine in detail the time development of the probability of
finding such processes within two-dimensional geometric shapes exhibiting
spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal
is to allow approximate computations with controlled error terms of the various
probability distributions involved
Mild Solutions for a Class of Fractional SPDEs and Their Sample Paths
In this article we introduce and analyze a notion of mild solution for a
class of non-autonomous parabolic stochastic partial differential equations
defined on a bounded open subset and driven by an
infinite-dimensional fractional noise. The noise is derived from an
-valued fractional Wiener process whose covariance operator
satisfies appropriate restrictions; moreover, the Hurst parameter is
subjected to constraints formulated in terms of and the H\"{o}lder exponent
of the derivative of the noise nonlinearity in the equations. We
prove the existence of such solution, establish its relation with the
variational solution introduced in \cite{nuavu} and also prove the H\"{o}lder
continuity of its sample paths when we consider it as an --valued
stochastic processes. When is an affine function, we also prove uniqueness.
The proofs are based on a relation between the notions of mild and variational
solution established in Sanz-Sol\'e and Vuillermot 2003, and adapted to our
problem, and on a fine analysis of the singularities of Green's function
associated with the class of parabolic problems we investigate. An immediate
consequence of our results is the indistinguishability of mild and variational
solutions in the case of uniqueness.Comment: 37 page
On some Gaussian Bernstein processes in RN and the periodic Ornstein-Uhlenbeck process
In this article we prove new results regarding the existence of Bernstein
processes associated with the Cauchy problem of certain forward-backward
systems of decoupled linear deterministic parabolic equations defined in
Euclidean space of arbitrary dimension N, whose initial and final conditions
are positive measures. We concentrate primarily on the case where the elliptic
part of the parabolic operator is related to the Hamiltonian of an isotropic
system of quantum harmonic oscillators. In this situation there are many
Gaussian processes of interest whose existence follows from our analysis,
including N-dimensional stationary and non-stationary Ornstein-Uhlenbeck
processes, as well as a Bernstein bridge which may be interpreted as a
Markovian loop in a particular case. We also introduce a new class of
stationary non-Markovian processes which we eventually relate to the
N-dimensional periodic Ornstein-Uhlenbeck process, and which is generated by a
one-parameter family of non-Markovian probability measures. In this case our
construction requires an infinite hierarchy of pairs of forward-backward heat
equations associated with the pure point spectrum of the elliptic part, rather
than just one pair in the Markovian case. We finally stress the potential
relevance of these new processes to statistical mechanics, the random evolution
of loops and general pattern theory.Comment: Research articl
One-parameter families of Sturm-Liouville eigenfunctions and of periodic solutions for a class of nonlinear Hill's equations
On the asymptotic behavior of solutions to a class of grand canonical master equations
In this article, we investigate the long-time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential.We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitably weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly but not all. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature
Product Approximations for Solutions to a Class of Evolution Equations in Hilbert Space
International audienceIn this article we prove approximation formulae for a class of unitary evolution operators associated with linear non-autonomous evolution equations of Schrödinger type defined in a Hilbert space . An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for involve weak operator limits of products of suitable approximating functions taking values in , the algebra of all linear bounded operators on . Our results may be relevant to the numerical analysis of and we illustrate them by considering two evolution problems in quantum mechanics