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1.
Sigurd Assing 《Probability Theory and Related Fields》2001,120(2):143-167
The paper deals with the infinite-dimensional stochastic equation dX= B(t, X) dt + dW driven by a Wiener process which may also cover stochastic partial differential equations. We study a certain finite dimensional
approximation of B(t, X) and give a qualitative bound for its rate of convergence to be high enough to ensure the weak uniqueness for solutions of
our equation. Examples are given demonstrating the force of the new condition.
Received: 6 November 1999 / Revised version: 21 August 2000 / Published online: 6 April 2001 相似文献
2.
Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise
Quantum stochastic differential equations of the form
govern stochastic flows on a C
*-algebra ?. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite
dimensional, and the coefficient matrix θ consists of bounded linear operators on ?. Weak and strong forms of solution are
distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and
uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution
k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when ? is a von Neumann algebra and the components of θ are normal, these
in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and *-homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy
equations:
in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes.
The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown
to have bounded generators, and to be necessarily strong solutions.
Received: 3 November 1998 / Published online: 30 March 2000 相似文献
3.
For a bounded C
1,α domain in ℝ
d
we show that there exists a strong solution to the multidimensional Skorokhod equation and that weak uniqueness holds for
this equation. These results imply that pathwise uniqueness and strong uniqueness hold for the Skorokhod equation.
Received: 3 February 1999 / Revised version: 2 September 1999 /?Published online: 11 April 2000 相似文献
4.
We consider the flow of a stochastic differential equation on d-dimensional Euclidean space. We show that if the Lie algebra generated by its diffusion vector fields is finite dimensional
and solvable, then the flow is conjugate to the flow of a non-autonomous random differential equation, i.e. one can be transformed
into the other via a random diffeomorphism of d-dimensional Euclidean space. Viewing a stochastic differential equation in this form which appears closer to the setting
of ergodic theory, can be an advantage when dealing with asymptotic properties of the system. To illustrate this, we give
sufficient criteria for the existence of global random attractors in terms of the random differential equation, which are
applied in the case of the Duffing-van der Pol oscillator with two independent sources of noise.
Received: 25 May 1999 / Revised version: 19 October 2000 / Published online: 26 April 2001 相似文献
5.
Jeffrey Bergen Piotr Grzeszczuk Małgorzata Hryniewicka 《Algebras and Representation Theory》2008,11(3):251-273
In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional
Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk
for finite dimensional restricted Lie algebras. Our main result is
Theorem 13
If R is a graded-reduced ring of characteristic p > 2 acted on by a finitely generated restricted K-Lie superalgebra L, then
.
We can then use Theorem 13 to prove
Corollary 15
Let R be a reduced algebra over a field K of characteristic p > 2 acted on by a finite dimensional restricted K-Lie superalgebra L and let H = u(L)#G, where G is the group of order 2 with the natural action on L. If R
H
satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN, where N is the dimension
of H.
Presented by Donald S. Passman. 相似文献
6.
Sylvie Méléard 《Probability Theory and Related Fields》2001,121(3):367-388
We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u
0 belonging to the Lorentz space L
2,∞ and such that curl u
0 is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation.
In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of
the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently
generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded
density initial data.
Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001 相似文献
7.
Sandra Cerrai 《Probability Theory and Related Fields》1999,113(1):85-114
In the present paper we consider the transition semigroup P
t
related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in
the Banach space of continuous functions , where ⊂ℝ
d
is a bounded open set. In L
2() the only result proved is the strong Feller property, for d=1. Here we are able to prove that if f∈C
∞(ℝ) and d≤3, then for any and t>0. An important application is to the study of the ergodic properties of the system. These results are also of interest for
some problem in stochastic control.
Received: 20 August 1997 / Revised version: 27 May 1998 相似文献
8.
Thomas P. Wakefield 《Algebras and Representation Theory》2011,14(4):609-623
Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H×A, where A is an abelian group. In this paper, we verify the conjecture for the twisted Ree groups 2
G
2(q
2) for q
2 = 32m + 1, m ≥ 1. The argument involves verifying five steps outlined by Huppert in his arguments establishing his conjecture for many
of the nonabelian simple groups. 相似文献
9.
A subgroup H of a finite group G is called a TI-subgroup if H ∩ H
x
= 1 or H for any x ∈ G. In this short note, the finite groups all of whose nonabelian subgroups are TI-subgroups are classified. 相似文献
10.
The main result in this paper states that if a one-parameter Gaussian process has C
2k
paths and satisfies a non-degeneracy condition, then the distribution of its maximum on a compact interval is of class C
k
. The methods leading to this theorem permit also to give bounds on the successive derivatives of the distribution of the
maximum and to study their asymptotic behaviour as the level tends to infinity.
Received: 14 May 1999 / Revised version: 18 October 1999 / Published online: 14 December 2000 相似文献
11.
For the signal in Gaussian white noise model we consider the problem of testing the hypothesis H
0 : f≡ 0, (the signal f is zero) against the nonparametric alternative H
1 : f∈Λɛ where Λɛ is a set of functions on R
1 of the form Λɛ = {f : f∈?, ϕ(f) ≥ Cψɛ}. Here ? is a H?lder or Sobolev class of functions, ϕ(f) is either the sup-norm of f or the value of f at a fixed point, C > 0 is a constant, ψɛ is the minimax rate of testing and ɛ→ 0 is the asymptotic parameter of the model. We find exact separation constants C
* > 0 such that a test with the given summarized asymptotic errors of first and second type is possible for C > C
* and is not possible for C < C
*. We propose asymptotically minimax test statistics.
Received: 23 February 1998 / Revised version: 6 April 1999 / Published online: 30 March 2000 相似文献
12.
Model selection for regression on a fixed design 总被引:1,自引:0,他引:1
Yannick Baraud 《Probability Theory and Related Fields》2000,117(4):467-493
We deal with the problem of estimating some unknown regression function involved in a regression framework with deterministic
design points. For this end, we consider some collection of finite dimensional linear spaces (models) and the least-squares
estimator built on a data driven selected model among this collection. This data driven choice is performed via the minimization
of some penalized model selection criterion that generalizes on Mallows' C
p
. We provide non asymptotic risk bounds for the so-defined estimator from which we deduce adaptivity properties. Our results
hold under mild moment conditions on the errors. The statement and the use of a new moment inequality for empirical processes
is at the heart of the techniques involved in our approach.
Received: 2 July 1997 / Revised version: 20 September 1999 / Published online: 6 July 2000 相似文献
13.
A new probabilistic method, based on the Girsanov theorem, for establishing the strong Feller property of diffusion processes
in both finite and infinite dimensional spaces is proposed. Applications to second order stochastic differential equations,
stochastic delay equations and stochastic partial differential equations of parabolic type are discussed, with a twofold aim:
both to extend some older results, usually by weakening the assumptions on the drift term, and to obtain simpler proofs of
them.
Received: 13 December 1999 / Published online: 5 September 2000 相似文献
14.
Nathanaël Enriquez Jacques Franchi Yves Le Jan 《Probability Theory and Related Fields》2001,119(2):213-255
Let ℳ be a geometrically finite hyperbolic surface with infinite volume, having at least one cusp. We obtain the limit law
under the Patterson-Sullivan measure on T
1ℳ of the windings of the geodesics of ℳ around the cusps. This limit law is stable with parameter 2δ− 1, where δ is the Hausdorff
dimension of the limit set of the subgroup Γ of M?bius isometries associated with ℳ. The normalization is t
−1/(2δ−1), for geodesics of length t. Our method relies on a precise comparison between geodesics and diffusion paths, for which we need to approach the Patterson-Sullivan
measure mentioned above by measures that are regular along the stable leaves.
Received: 8 October 1999 / Revised version: 2 June 2000 / Published online: 21 December 2000 相似文献
15.
A. de Acosta 《Probability Theory and Related Fields》2000,118(4):483-521
We prove an abstract large deviation result for a sequence of random elements of a vector space satisfying an “abstract exponential
martingale condition”. The framework naturally generates non-convex rate functions. We apply the result to solutions of It?
stochastic equations in R
d
driven by Brownian motion and a Poisson random measure.
Received: 23 June 1999 / Revised version: 17 February 2000 / Published online: 22 November 2000 相似文献
16.
We consider the stochastic differential equation dX
t
= a(X
t
)dW
t
+ b(X
t
)dt, where W is a one-dimensional Brownian motion. We formulate the notion of solution and prove strong existence and pathwise uniqueness
results when a is in C
1/2 and b is only a generalized function, for example,the distributional derivative of a H?lder function or of a function of bounded
variation. When b = aa′, that is, when the generator of the SDE is the divergence form operator ℒ = , a result on non-existence of a strong solution and non-pathwise uniqueness is given as well as a result which characterizes
when a solution is a semimartingale or not. We also consider extensions of the notion of Stratonovich integral.
Received: 23 February 2000 / Revised version: 22 January 2001 / Published online: 23 August 2001 相似文献
17.
Donald I. Cartwright 《Monatshefte für Mathematik》2001,133(2):93-109
To any locally finite thick building of type there is naturally associated a commutative algebra of operators. When is constructed from a local field F with local ring , and , then is isomorphic to the convolution algebra of compactly supported bi-K-invariant functions on PGL(n+1,F). We give a proof, valid for any , that the multiplicative functionals on may all be expressed in terms of Hall–Littlewood polynomials. Regarding as a subalgebra of the C
*-algebra of bounded operators on the space of square summable functions on the vertex set of , we find the spectrum of the C
*-algebra , the closure of . This generalizes results obtained in [3] when n = 1 and in [5] when n = 2.
(Received 26 June 2000; in revised form 21 February 2001) 相似文献
18.
We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the
bond dilute Ising model on ℤ
d
at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap
of the generator of the dyamics in a box of side L centered at the origin scales like L
−2. Such an estimate is then used to prove a decay to equilibrium for local functions of the form where ε is positive and arbitrarily small and α = ? for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition
when β crosses the critical value β
c
of the pure system.
Received: 10 April 2000 / Revised version: 23 October 2000 / Published online: 5 June 2001 相似文献
19.
Given a frame F = {f
j
} for a separable Hilbert space H, we introduce the linear subspace HpFH^{p}_{F} of H consisting of elements whose frame coefficient sequences belong to the ℓ
p
-space, where 1 ≤ p < 2. Our focus is on the general theory of these spaces, and we investigate different aspects of these spaces in relation
to reconstructions, p-frames, realizations and dilations. In particular we show that for closed linear subspaces of H, only finite dimensional ones can be realized as HpFH^{p}_{F}-spaces for some frame F. We also prove that with a mild decay condition on the frame F the frame expansion of any element in HFpH_{F}^{p} converges in both the Hilbert space norm and the ||·||
F, p
-norm which is induced by the ℓ
p
-norm. 相似文献
20.
Raluca M. Balan 《Potential Analysis》2012,36(1):1-34
We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the
solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of
any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation. 相似文献