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1.
Summary A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupR
t are obtained. We show thatR
t is a compactC
0-semigroup in all Sobolev spacesW
n,p which are built on its invariant measure . Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inL
p() spaces and spacesW
1,p
. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to . It is shown also that the density of this measure with respect to is inL
p() for allp1.This work was done during the first author's stay at UNSW supported by ARC Grant 150.346 and the second author's stay at ód University supported by KBN Grant 2.1020.91.01 相似文献
2.
Some dimension results for super-Brownian motion 总被引:4,自引:0,他引:4
Laurent Serlet 《Probability Theory and Related Fields》1995,101(3):371-391
Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y
t)t0 be a super-Brownian motion on
d
(d2) andH be a Borel subset of
d
. We determine the Hausdorff Dimension of {t0; SuppY
tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of
SuppY
t
as a function of the dimension ofB. 相似文献
3.
P. Révész 《Probability Theory and Related Fields》1992,93(1):21-37
Summary We say that the discD()R
2, of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT. 相似文献
4.
Uwe Einmahl 《Probability Theory and Related Fields》1993,96(1):97-106
Summary A general integral test is established which refines the Jain-Pruitt Chung LIL for iid random variables. As a corollary we obtain that Chung's integral test for Brownian motion is valid for partial sums of iid random variables satisfyingEX
21{|X|t}=O((LLt)
–1) ast.Supported in part by NSF grant DMS 90-05804 相似文献
5.
Summary Let (X
t
n
) be a Poisson sequence of independent Brownian motions in
d
,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X
s
n
, 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in
d
. 相似文献
6.
Qi-Man Shao 《Probability Theory and Related Fields》1992,94(1):119-133
Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX
2(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t
–2) as t for some >0 and (iii) supts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)1/2}. The law of the iterated logarithm implies
a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e.,
a.s. if 0<<2 and
. Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati 相似文献
7.
El-Maati Ouhabaz 《Potential Analysis》1996,5(6):611-625
Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L
2(, ). Denote by T=T(t)
t0and S=S(t)
t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given. 相似文献
8.
Richard Sowers 《Probability Theory and Related Fields》1992,92(3):393-421
Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE)
t
v
=v
+f(x,v
)+(x,v
)
. Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH
xx-h, and the space variablex takes values on the unit circleS
1. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv
2 is aC
k
(S
1)-valued Markov process for each 0<1/2, whereC
(S
1) is the Banach space of real-valued continuous functions onS
1 which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC
(S
1) is a globally exponentially stable critical point of the unperturbed equation
t
0 = 0 +f(x,0), that has a unique stationary distributionv
K,
on (C
(S
1), (C
K
(S
1))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v
K,
tends tov
K,0, the point mass centered on the zero element ofC
(S
1). The main goal of this paper is to show that in factv
K,
is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv
K,
is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv
K,
based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC
(S
1) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA 相似文献
9.
B. Maisonneuve 《Probability Theory and Related Fields》1979,47(1):61-67
Summary Let X be a Markov process and M a homogeneous random set. For t0, we set G
t=Sup{st: sM}. The stochastic dependence between the past and the future of G
Tis investigated for certain stopping times T. This gives some insight to recent results of Getoor concerning the excursion straddling t and the first excursion exceeding a in length.This research was supported in part by NSF grant MCS76-8023 相似文献
10.
Summary Let be a centered Gaussian measure on a Hilbert spaceH and let
be the centered ball of radiusR>0. ForaH and
, we give the exact asymptotics of (B
R(t)+t·a) ast. Also, upper and lower bounds are given when is defined on an arbitrary separable Banach space. Our results range from small deviation estimates to large deviation estimates.Supported in part by NSF grant number DMS-9024961 相似文献
11.
LetE be a locally convex space endowed with a centered gaussian measure . We construct a continuousE-valued brownian motionW
t with covariance . The main goal is to solve the SDE of Langevin type dX
t=
dW
t–AX
t wherea andA are unbounded operators of the Cameron-Martin space of (E, ). It appears as the unique linear measurable extension of the solution of the classical Cauchy problemv(t)=
u–Av(t). 相似文献
12.
Jean Bertoin 《Probability Theory and Related Fields》1995,103(3):317-327
Summary LetG=(G(t),t0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA
+=(A
+
(t),t0), the time spent in [0, ) by a real-valued diffusion process, in connection with Watanabe's recent extension of Lévy's second arcsine law. 相似文献
13.
Youngmee Kwon 《Probability Theory and Related Fields》1995,101(2):211-226
Summary In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone. The basic question considered here is When is this process a semimartingale?. Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter depending on the cone and the direction of reflection. It is shown that starting from any point of the cone, the process is a semimartingale if < 1, + 0 and not a semimartingale if < < 2.This research is supported by KOSEF grant 941-0100-011-1 相似文献
14.
Noboru Hamada 《Graphs and Combinatorics》1989,5(1):137-147
Recently, Hamada [5] characterized all {v
2 + 2v
1,v
1 + 2v
0;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev
l
= (q
l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v
+ 1 + 2v
,v
+ 2v
– 1;t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q
k-1 – (2q
-1 +q
) andq 5 using the results in Hamada [3, 4, 5]. 相似文献
15.
Summary LetC be the symmetric cusp {(x, y)2:–x
yx
,x0} where >1. In this paper we decide whether or not reflecting Brownian motion inC has a semimartingale representation. Here the reflecting Brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary. Our results carry through for a wide class of asymmetric cusps too. 相似文献
16.
Barbara Priwitzer 《Geometriae Dedicata》1995,58(3):245-258
LetP=(P, L) be a compact projective plane with 0P< and let be a compact connected subgroup of Aut(P). If dim dimE – dimP, whereE is the elliptic motion group of the corresponding classical plane, then E or is isomorphic to a point stabilizerE
0 inE, cf. [31]. Here we consider the case E
0. It is shown that the action of on the point spaceP is equivalent to the classical action ofE
0. For dimP {8, 16} the planeP is uniquely determined by a 2-dimensional subplane with SO2 Aut().Für H. Reiner Salzmann zum 65. Geburtstag 相似文献
17.
Summary LetB=(B
t,t0) be a planar Brownian motion and let >0. For anyt0, the pointz=B
t is called a one-sided cone point with angle if there exist >0 and a wedgeW(,z) with vertexz and angle such thatB
sW(,z) for everys[t, t+]. Burdzy and Shimura have shown independently that one-sided cone points with angle exist when >/2 but not when 2. The present paper deals with the critical case =/2. We show that cone points with angle /2 do not exist. 相似文献
18.
We prove that if the indicator-function1
E
of a measurable setE is a Fourier multiplier in the spaceE
p
() for somep2 thenE is an open set (up to a set of measure zero). 相似文献
19.
Jürg Hüsler 《Probability Theory and Related Fields》1979,46(2):159-164
Summary Let {X
k
, k0} be i.i.d. random variables with EX
+< and define t
=max{k0: X
k
> k} if such a k exists and =0 else, the last exit time of the sequence X
k
for fixed >0. We discuss weak limit laws for t
as 0; in particular the limit distributions, the stability and the relative stability.This work is partially supported by a grant of the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung, while the author was at the University of Pittsburgh, USAHerrn Prof. L. Schmetterer zu seinem 60. Geburtstag gewidmet 相似文献
20.
István Gyöngy 《Acta Appl Math》1994,35(1-2):165-177
We coasider a partially observable diffusion process (x
t,yt)t0 whose unobservable componentx
t lives on a submanifold M ofR
n
. We present some general conditions under which the conditional law ofx
t, given the observationsy
s
,s [0,t], admits a density w.r.t. a given measure on M. We characterize the analytical properties of this density by using appropriate Sobolev spaces.Research supported by the Hungarian National Foundation of Scientific Research No. 2290. 相似文献