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1.
In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007. 相似文献
2.
Michel Talagrand 《Probability Theory and Related Fields》1998,112(4):545-563
Consider 0<α<1 and the Gaussian process Y(t) on ℝ
N
with covariance E(Y(s)Y(t))=|t|2α+|s|2α−|t−s|2α, where |t| is the Euclidean norm of t. Consider independent copies X
1,…,X
d
of Y and␣the process X(t)=(X
1(t),…,X
d
(t)) valued in ℝ
d
. When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ
k
N
/α−(
k
−1)
d
(loglog(1/ɛ))
k
. If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛ
d
(log(1/ɛ) logloglog 1/ɛ)
k
. (This includes the case k=1.)
Received: 20 May 1997 / Revised version: 15 May 1998 相似文献
3.
Piotr Haj?asz 《Mathematische Annalen》2009,343(4):801-823
We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N
1,p
(X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π
1(Y) = π
2(Y) = · · · = π
[p](Y) = 0, where [p] is the largest integer less than or equal to p.
This work was supported by the NSF grant DMS-0500966. 相似文献
4.
Rong-mao ZHANG & Zheng-yan LIN Department of Mathematics Zhejiang University Hangzhou China 《中国科学A辑(英文版)》2007,50(1):35-46
Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established. 相似文献
5.
Let X be a Banach space, A : D(A) X → X the generator of a compact C0- semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b) × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t) = Au(t) + f(t, u(t - q)), t ∈ [to, to + T], with initial condition uto = φ ∈C([-q, 0]; X), is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t, v(-q)); D) = 0 for almost every t ∈ (a, b) and every v ∈ C([-q, 0]; X) with v(0), v(-q)∈ D. 相似文献
6.
Clustering of linearly interacting diffusions and universality of their long-time limit distribution
J. M. Swart 《Probability Theory and Related Fields》2000,118(4):574-594
Let K⊂ℝ
d
(d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K
Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX
i
(t) = ∑
j
a(j−i) (X
j
(t) −X
i
(t))dt + σ (X
i
(t))dB
i
(t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a
S
(i) = a(i) + a(−i) is recurrent, then each component X
i
(∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a
S
is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of
the components. This distribution turns out to be universal in all recurrent kernels a
S
on Abelian groups Λ.
Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 相似文献
7.
For ν(dθ), a σ-finite Borel measure on R
d
, we consider L
2(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫
t
0
e
−λ(θ)(
t
−
s
)
dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫
t
0
g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions
of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process
t
(τ)≗Y(τt∧t).
Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001 相似文献
8.
We investigate the completeness and completions of the normed algebras (D
(1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D
(1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of
connected, compact plane sets X for which the completeness of (D
(1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for
all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. 相似文献
9.
We will study the following problem.Let X_t,t∈[0,T],be an R~d-valued process defined on atime interval t∈[0,T].Let Y be a random value depending on the trajectory of X.Assume that,at each fixedtime t≤T,the information available to an agent(an individual,a firm,or even a market)is the trajectory ofX before t.Thus at time T,the random value of Y(ω) will become known to this agent.The question is:howwill this agent evaluate Y at the time t?We will introduce an evaluation operator ε_t[Y] to define the value of Y given by this agent at time t.Thisoperator ε_t[·] assigns an (X_s)0(?)s(?)T-dependent random variable Y to an (X_s)0(?)s(?)t-dependent random variableε_t[Y].We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) withthe drift coefficient b and diffusion coefficient σcontaining an unknown parameter θ=θ_t.We then consider theso called super evaluation when the agent is a seller of the asset Y.We will prove that such super evaluation is afiltration consistent nonlinear expectation.In some typical situations,we will prove that a filtration consistentnonlinear evaluation dominated by this super evaluation is a g-evaluation.We also consider the correspondingnonlinear Markovian situation. 相似文献
10.
Let X(t) be an N parameter generalized Lévy sheet taking values in ℝd with a lower index α, ℜ = {(s, t] = ∏
i=1
N
(s
i, t
i], s
i < t
i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established. 相似文献
11.
Let C1,···,Cd be Mumford curves defined over a finite extension of and let X=C1×···×Cd. We shall show the following: (1) The cycle map CH0(X)/n → H2d(X, μn⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH0(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH0(X). 相似文献
12.
LetT(t) be the translation group onY=C
0(ℝ×K)=C
0(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY
⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y
⊙=L
1(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT
0(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT
0(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ
⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C
0(Ω;X) andL
p
(μ;X).
This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful
to the CWI and the Dutch National Science Foundation NWO for financial support. 相似文献
13.
Joanna Janczewska 《Central European Journal of Mathematics》2004,2(4):561-572
In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R
k
→Y is a C
2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R
1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R
k
and we describe the solution set of the studied equation in a small neighbourhood of this point. 相似文献
14.
Yan Xia Ren 《数学学报(英文版)》2002,18(1):69-78
Suppose X is a superdiffusion in R
d
with general branching mechanism ψ, and Y
r
D
denotes the total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions on ψ to guarantee that Y
r
D
has absolutely continuous states. And for particular ψ(
z) = z
1+β, 0 < β≤ 1, we prove that, in the case d < 2 + 2/β, Y
r
D
is absolutely continuous with respect to the Lebesgue measure in , whereas in the case d > 2 + 2/β, it is singular. As we know the absolute continuity and singularity of Y
r
D
have not been discussed before.
Received February 1, 1999, Revised February 25, 2000, Accepted March 9, 2000 相似文献
15.
Let M^n be a smooth, compact manifold without boundary, and F0 : M^n→ R^n+1 a smooth immersion which is convex. The one-parameter families F(·, t) : M^n× [0, T) → R^n+1 of hypersurfaces Mt^n= F(·,t)(M^n) satisfy an initial value problem dF/dt (·,t) = -H^k(· ,t)v(· ,t), F(· ,0) = F0(· ), where H is the mean curvature and u(·,t) is the outer unit normal at F(·, t), such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton. 相似文献
16.
Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (d
t
)
t∈ℤ of real numbers; if the parameter sequence is constant d
t
≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d
. They also studied partial sums limits of filtered white noise nonstationary processes A(d)ε
t
and B(d)ε
t
for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes X
t
A
= A(d)ε
t
and X
t
B
= B(d)ε
t
by assuming that d = (d
t
, t ∈ ℤ) is a random iid sequence, independent of the noise (ε
t
). In the case where the mean
, we show that large sample properties of X
A
and X
B
are similar to FARIMA(0,
, 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter
. The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear
functions h(X
t
A
) of a randomly fractionally integrated process X
t
A
with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence d
t
, this reduces to the standard Hermite rank used in Dobrushin and Major [2].
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 3–28, January–March, 2007. 相似文献
17.
Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ+×ℤ
d
associated with the Anderson Hamiltonian H=κΔ+ξ(·) for i.i.d. random potentials ξ(·). For the Cauchy problem with nonnegative
homogeneous initial condition we study the second order asymptotics of the statistical moments <u(t,0)
p
> and the almost sure growth of u(t,0) as t→∞. We point out the crucial role of double exponential tails of ξ(0) for the formation of high intermittent peaks of the
solution u(t,·) with asymptotically finite size. The challenging motivation is to achieve a better understanding of the geometric structure
of such high exceedances which in one or another sense provide the essential contribution to the solution.
Received: 10 December 1996 / In revised form: 30 September 1997 相似文献
18.
The bicompletion of an asymmetric normed linear space 总被引:5,自引:0,他引:5
A biBanach space is an asymmetric normed linear space (X,‖·‖) such that the normed linear space (X,‖·‖s) is a Banach space, where ‖x‖s= max {‖x‖,‖-x‖} for all x∈X. We prove that each asymmetric normed linear space (X,‖·‖) is isometrically isomorphic to a dense subspace of a biBanach space (Y,‖·‖Y). Furthermore the space (Y,‖·‖Y) is unique (up to isometric isomorphism).
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
19.
Endre Csáki Miklós Csörgő Antónia Földes Zhan Shi 《Probability Theory and Related Fields》2000,117(4):515-531
Let W be a standard Brownian motion, and define Y(t)= ∫0
t
ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y.
Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000 相似文献
20.
We establish an estimate for the rate of convergence of a solution of an ordinary stochastic differential equation of order
p ≥ 2 with a small parameter in the coefficient of the leading derivative to a solution of a stochastic equation of order p − 1 in the metric ρ(X, Y) = (sup0≤t≤T
M|X(t) − Y(t)|2)1/2
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 12, pp. 1587–1601, December, 2006. 相似文献