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1.
Luigi Manca 《随机分析与应用》2013,31(2):399-426
Abstract We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L 2(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup. 相似文献
2.
We consider stochastic equations in Hilbert spaces with singular drift in the framework of [G. Da Prato, M. Röckner, Singular dissipative stochastic equations in Hilbert spaces, Probab. Theory Related Fields 124 (2) (2002) 261-303]. We prove a Harnack inequality (in the sense of [F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields 109 (1997) 417-424]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure μ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure μ for noncontinuous drifts. 相似文献
3.
We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence
of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially
to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including
some where the noise does not act on every determining mode directly.
Received: 20 September 2001 / Revised version: 21 April 2002 / Published online: 10 September 2002 相似文献
4.
《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(2-3):159-188
We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process 相似文献
5.
We prove the existence of an invariant measure for the transition semigroup associated with a nonlinear damped stochastic wave equation in Rn of the Klein--Gordon type. The uniqueness of the invariant measure and the structure of the corresponding Kolmogorov operator are also studied. 相似文献
6.
In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with Beddington-DeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results.
相似文献7.
Vladimir Bogachev Giuseppe Da Prato Michael Röckner 《Journal of Evolution Equations》2010,10(3):487-509
We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence
and uniqueness results are known. We prove, under suitable assumptions, the existence and uniqueness of a measure valued solution,
for the corresponding Fokker–Planck equation. In particular, we verify the Chapman–Kolmogorov equations and get an evolution
system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation. 相似文献
8.
Marcelina Mocanu 《Complex Analysis and Operator Theory》2011,5(3):799-810
We prove a Poincaré inequality for Orlicz–Sobolev functions with zero boundary values in bounded open subsets of a metric
measure space. This result generalizes the (p, p)-Poincaré inequality for Newtonian functions with zero boundary values in metric measure spaces, as well as a Poincaré inequality
for Orlicz–Sobolev functions on a Euclidean space, proved by Fuchs and Osmolovski (J Anal Appl (Z.A.A.) 17(2):393–415, 1998).
Using the Poincaré inequality for Orlicz–Sobolev functions with zero boundary values we prove the existence and uniqueness
of a solution to an obstacle problem for a variational integral with nonstandard growth. 相似文献
9.
Under the nondegenerate condition as in the diffusion case, see [14, 21, 6], the linear stochastic jump-diffusion process
projected on the unit sphere is a strong Feller process and has a unique invariant measure which is also ergodic using the
relation between the transition probabilities of jump-diffusions and the corresponding diffusions due to [22]. The unique
deterministic Lyapunov exponent can be represented by the Furstenberg-Khas'minskii formula as an integral over the sphere
or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear
stochastic systems with jumps depends on its sign. The critical case of zero Lyapunov exponent is discussed and a large deviations
result for asymptotically stable systems is further investigated. Several examples are treated for illustration.
Received: 22 June 2000 / Revised version: 20 November 2001 / Published online: 13 May 2002 相似文献
10.
We prove existence and (in some special case) uniqueness of an invariant measure for the transition semigroup associated with the stochastic wave equations with nonlinear dissipative damping. 相似文献
11.
This paper is concerned with nonlinear partial differential equations of the calculus of variation (see [13]) perturbed
by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see [14]), using monotonicity methods.
The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup Pt. We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure ν. Moreover, we solve the
Kolmogorov equation and prove the so-called "identite du carre du champs". This will be used to study the Sobolev space W1,2(H,ν) and to obtain information
on the domain of the infinitesimal generator of Pt. 相似文献
12.
Andrea Cianchi 《Mathematische Zeitschrift》2010,266(2):431-449
We prove a sharp trace embedding theorem for Orlicz–Sobolev spaces into Orlicz spaces on the boundary. An optimal trace embedding
into more general rearrangement invariant spaces on the boundary is also established. 相似文献
13.
We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval.
As an immediate corollary we obtain a concentration inequality. We apply these results to the estimation of the rate of convergence
of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure.
Received: 14 August 2001 / Revised version: 13 February 2002 / Published online: 1 July 2002 相似文献
14.
Wei Liu 《Journal of Evolution Equations》2009,9(4):747-770
As a Generalization to Wang (Ann Probab 35:1333–1350, 2007) where the dimension-free Harnack inequality was established for
stochastic porous media equations, this paper presents analogous results for a large class of stochastic evolution equations
with general monotone drifts. Some ergodicity, compactness and contractivity properties are established for the associated
transition semigroups. Moreover, the exponential convergence of the transition semigroups to invariant measure and the existence
of a spectral gap are also derived. As examples, the main results are applied to many concrete SPDEs such as stochastic reaction-diffusion
equations, stochastic porous media equations and the stochastic p-Laplace equation in Hilbert space. 相似文献
15.
Tomasz Klimsiak 《Potential Analysis》2012,36(2):373-404
We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in
divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic
differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem
by means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic
homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic
analogue of the homographic approximation for solutions to the obstacle problem. 相似文献
16.
17.
We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes
the result of Hansson and Brezis-Wainger for W
n
k/k
as a special case. We deal with generalized Sobolev spaces W
A
k
, where instead of requiring the functions and their derivatives to be in Ln/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to Ln/k.
We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces
are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an,
earlier optimality result obtained by Hansson with respect to the Riesz potential operator.
In memory of Gene Fabes.
Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund. 相似文献
18.
Summary. We study the stationary measures of an infinite Hamiltonian system of interacting particles in ℝ
3
subject to a stochastic local perturbation conserving energy and momentum. We prove that the translation invariant measures
that are stationary for the deterministic Hamiltonian dynamics, reversible for the stochastic dynamics, and with finite entropy
density, are convex combination of “Gibbs” states. This result implies hydrodynamic behavior for the systems under consideration.
Received: 17 December 1994/In revised form: 12 April 1996 相似文献
19.
We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept
allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function
technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application,
we consider stochastic Burgers, reaction-diffusion, and Navier-Stokes equations and investigate the elliptic equations for
the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic
equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class
of models.
Received: 23 January 2000 / Revised version: 4 October 2000 / Published online: 5 June 2001 相似文献
20.
There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris?? theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such ??asymptotic couplings?? were central to (Mattingly and Sinai in Comm Math Phys 219(3):523?C565, 2001; Mattingly in Comm Math Phys 230(3):461?C462, 2002; Hairer in Prob Theory Relat Field 124:345?C380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553?C582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris?? celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are ??small?? (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a ??small set?? by the much weaker notion of a ??d-small set,?? which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris?? theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it. 相似文献