75 research outputs found
Ornstein-Uhlenbeck-Cauchy Process
We combine earlier investigations of linear systems with L\'{e}vy
fluctuations [Physica {\bf 113A}, 203, (1982)] with recent discussions of
L\'{e}vy flights in external force fields [Phys.Rev. {\bf E 59},2736, (1999)].
We give a complete construction of the Ornstein-Uhlenbeck-Cauchy process as a
fully computable model of an anomalous transport and a paradigm example of
Doob's stable noise-supported Ornstein-Uhlenbeck process. Despite the
nonexistence of all moments, we determine local characteristics (forward drift)
of the process, generators of forward and backward dynamics, relevant
(pseudodifferential) evolution equations. Finally we prove that this random
dynamics is not only mixing (hence ergodic) but also exact. The induced
nonstationary spatial process is proved to be Markovian and quite apart from
its inherent discontinuity defines an associated velocity process in a
probabilistic sense.Comment: Latex fil
Linear Control Systems on Unbounded Time Intervals and Invariant Measures of Ornstein--Uhlenbeck Processes in Hilbert Spaces
Coupling and Strong Feller for Jump Processes on Banach Spaces
By using lower bound conditions of the L\'evy measure w.r.t. a nice reference
measure, the coupling and strong Feller properties are investigated for the
Markov semigroup associated with a class of linear SDEs driven by
(non-cylindrical) L\'evy processes on a Banach space. Unlike in the
finite-dimensional case where these properties have also been confirmed for
L\'evy processes without drift, in the infinite-dimensional setting the
appearance of a drift term is essential to ensure the quasi-invariance of the
process by shifting the initial data. Gradient estimates and exponential
convergence are also investigated. The main results are illustrated by specific
models on the Wiener space and separable Hilbert spaces.Comment: 31 page
Functional and Banach Space Stochastic Calculi: Path-Dependent Kolmogorov Equations Associated with the Frame of a Brownian Motion
First, we revisit basic theory of functional It\uf4/path-dependent calculus, using the formulation of calculus via regularization. Relations with the corresponding Banach space valued calculus are explored. The second part of the paper is devoted to the study of the Kolmogorov type equation associated with the so called window Brownian motion, called path-dependent heat equation, for which well-posedness at the level of strict solutions is established. Then, a notion of strong approximating solution, called strong-viscosity solution, is introduced which is supposed to be a substitution tool to the viscosity solution. For that kind of solution, we also prove existence and uniqueness
Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy white noise
The paper is concerned with spatial and time regularity of solutions to
linear stochastic evolution equation perturbed by L\'evy white noise "obtained
by subordination of a Gaussian white noise". Sufficient conditions for spatial
continuity are derived. It is also shown that solutions do not have in general
\cadlag modifications. General results are applied to equations with fractional
Laplacian. Applications to Burgers stochastic equations are considered as well.Comment: This is an updated version of the same paper. In fact, it has already
been publishe
Controllability and Qualitative properties of the solutions to SPDEs driven by boundary L\'evy noise
Let be the solution to the following stochastic evolution equation (1)
du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking
values in an Hilbert space \HH, where is a \RR valued L\'evy process,
an infinitesimal generator of a strongly continuous semigroup,
\sigma:H\to \RR bounded from below and Lipschitz continuous, and B:\RR\to H
a possible unbounded operator. A typical example of such an equation is a
stochastic Partial differential equation with boundary L\'evy noise. Let
\CP=(\CP_t)_{t\ge 0} %{\CP_t:0\le t<\infty}T>0BAx\in H\CP_T^\star \delta_xH\HHLAB$ the solution of Equation [1] is
asymptotically strong Feller, respective, has a unique invariant measure. We
apply these results to the damped wave equation driven by L\'evy boundary
noise
O warunkowaniach dla zbieżnych do zera ciągów wektorów losowych w przestrzeniach Banacha
https://doi.org/10.26485/0459-6854/2018/68.3/1
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