共查询到20条相似文献,搜索用时 78 毫秒
1.
The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained
paths to paths that are confined to live within a given domain G⊂ℝ
n
. Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force
acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic
processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space,
the study of many problems involving these constrained processes is greatly simplified.
We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems
of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why
such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered
in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach,
Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined
in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using
convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative
theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions
to the Skorokhod Problem.
These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems.
Received: 17 April 1998 / Revised version: 8 January 1999 相似文献
2.
Damir Filipović 《Probability Theory and Related Fields》2000,118(3):323-341
Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C
2 submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: any weak solution, which is viable in
a finite dimensional C
2 submanifold, is a strong solution.
These results are related to finding finite dimensional realizations for stochastic equations. There has recently been increased
interest in connection with a model for the stochastic evolution of forward rate curves.
Received: 15 April 1999 / Revised version: 4 February 2000 / Published online: 18 September 2000 相似文献
3.
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
4.
We study nonlinear wave and heat equations on ℝ
d
driven by a spatially homogeneous Wiener process. For the wave equation we consider the cases of d = 1, 2, 3. The heat equation is considered on an arbitrary ℝ
d
-space. We give necessary and sufficient conditions for the existence of a function-valued solution in terms of the covariance
kernel of the noise.
Received: 1 April 1998 / Revised version: 23 June 1999 / Published online: 7 February 2000 相似文献
5.
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
6.
Sergio Albeverio Zbignew Haba Francesco Russo 《Probability Theory and Related Fields》2001,121(3):319-366
A two-space dimensional heat equation perturbed by a white noise in a bounded volume is considered. The equation is perturbed
by a non-linearity of the type λ : f(AU) :, where :: means Wick (re)ordering with respect to the free solution;λ, A are small parameters, U denotes a solution, f is the Fourier transform of a complex measure with compact support.
Existence and uniqueness of the solution in a class of Colombeau-Oberguggenberger generalized functions is proven. An explicit
construction of the solution is given and it is shown that each term of the expansion in a power series in λ is associated
with an L
2-valued measure when A is a small enough.
Received: 20 July 1997 / Revised version: 1 February 2001 / Published online: 9 October 2001 相似文献
7.
Lorenzo Zambotti 《Probability Theory and Related Fields》2000,118(2):147-168
We prove existence and uniqueness for a class of martingale problems in a Hilbert space. We solve the associated Kolmogorov
equation and prove that the corresponding semigroup is determined by a kernel of measures if a Schauder-type regularity is
satisfied.
Received: 18 May 1998 / Revised version: 27 September 1999 / Published online: 5 September 2000 相似文献
8.
M. van den Berg 《Probability Theory and Related Fields》2000,118(1):17-36
We investigate the asymptotic behaviour of the heat content as the time t→ 0 for an s-adic von Koch snowflake generated by a square. We show that the heat content satisfies a functional equation which, after
appropriate transformations, takes the form of an inhomogeneous renewal equation. We obtain the structure of the solution
of this equation in the arithmetic case up to an exponentially small remainder in t.
<!-ID="Mathematics Subject Classification (2000): 35K05, 60J65, 28A80-->
<!-ID="Key words: Heat equation – Arithmetic – Snowflake-->
Received: 24 March 1999 / Revised version: 14 October 1999 / Published online : 8 August 2000 相似文献
9.
Emmanuel Rio 《Probability Theory and Related Fields》2000,118(3):342-348
The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L
1(ℝ/ℤ), then the averages
converge to the mean value of ƒ over [0, 1] for almost every x in [0, 1]. In this paper we prove that, for ƒ in L
1(ℝ/ℤ), the averages A
n
a
ƒ(x) converge a.e. to the integral of ƒ over [0, 1] for almost every a > 1. Furthermore we obtain convergence rates in this strong law of large numbers.
Received: 1 March 1999 / Revised version: 20 October 1999 / Published online: 12 October 2000 相似文献
Lois fortes des grands nombres presque s?res pour les sommes de Riesz–Raikov English title: Almost sure versions of the Riesz–Raikov strong law of large numbers
Received: 1 March 1999 / Revised version: 20 October 1999 / Published online: 12 October 2000 相似文献
10.
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
11.
We present an upper bound O(n
2
) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant,
and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion
process on a circle indeed mixes more rapidly than the corresponding symmetric process.
Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000 相似文献
12.
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
13.
M. H. Hooshmand 《Ukrainian Mathematical Journal》2011,63(2):328-336
We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied earlier by the author
and present two classes of special functions, namely, ultraexponential and infralogarithm f -type functions. As a result of this investigation, we obtain a general solution of the Abel equation α(f(x)) = α (x) + 1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that an infralogarithm f -type function is its unique solution. We also show that an infralogarithm f -type function is an essentially unique solution of the Abel equation. Similar theorems are proved for ultraexponential f -type functions and their functional equation β(x) = f(β(x − 1)), which can be considered as dual to the Abel equation. We also solve a certain problem unsolved before and study some
properties of two considered functional equations and some relations between them. 相似文献
14.
E. P. Golubeva 《Journal of Mathematical Sciences》2009,157(4):543-552
The solvability of the equation n = x
2 + y
2 + 6pz
2 (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm
12 > p
21. As an implication, the solvability of the equation n = x
2 + y
2 + u
3 + v
3 + z
4 + w
16 + t
4k+1 for all sufficiently large n is established. Bibliography: 13 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21. 相似文献
15.
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
16.
Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the
nonautonomous ordinary differential equation u
(2m)
=f(t,u,...,u
(m-1)
), where the function f:ℝ×ℝ
m
→ℝ is periodic with respect to the first argument with period ω.
Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001 相似文献
17.
P. Maremonti 《Journal of Mathematical Sciences》2009,159(4):486-523
The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are
studied. The existence and uniqueness of classical solutions (u, π) (considered at least C
2 × C
1 smooth with respect to the space variable and C
1 × C
0 smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the
Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that,
for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x|
γ
). Bibliography: 20 titles.
To V. A. Solonnikov on his 75th birthday
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240. 相似文献
18.
Kurt Johansson 《Probability Theory and Related Fields》2000,116(4):445-456
Consider a realization of a Poisson process in ℝ2 with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number
of points in such a path has fluctuations of order N
χ, where χ = 1/3, [BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order N
ξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1 which is believed to hold in many random growth models.
Received: 16 April 1999 / Revised version: 5 July 1999 / Published online: 14 February 2000 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.
Assuming GCH, we prove that for every successor cardinal μ > ω1, there is a superatomic Boolean algebra B such that |B| = 2μ and |Aut B| = μ. Under ◊ω1, the same holds for μ = ω1. This answers Monk's Question 80 in [Mo].
Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000 相似文献