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1.
Sharpe investigated the structure of full operator-stable measures μ on a vector group V and obtained decompositions, μ = μ1 1 μ2 and V = V1V2, in terms of the Gaussian component μ1 and the Poisson component μ2. The subspaces V1 and V2 are here identified in terms of an exponent B for μ. Sharpe also pointed out that the Lévy measure M of μ is a mixture of Lévy measures concentrated on single orbits of tB. Here, an explicit representation is obtained for M as such a mixture by constructing a measure on the unit sphere. Also, necessary and sufficient conditions are given that a Lévy measure be the Lévy measure of a full operator-stable measure. The final result deals with full Gaussian measures μ and establishes the connection between its covariance operator and the class of all exponents of μ.  相似文献   

2.
For Gaussian vector fields {X(t) ∈ Rn:tRd} we describe the covariance functions of all scaling limits Y(t) = Llimα↓0 B?1(α) Xt) which can occur when B(α) is a d × d matrix function with B(α) → 0. These matrix covariance functions r(t, s) = EY(t) Y1(s) are found to be homogeneous in the sense that for some matrix L and each α > 0, (1) r(αt, αs) = αL1r(t, s) αL. Processes with stationary increments satisfying (1) are further analysed and are found to be natural generalizations of Lévy's multiparameter Brownian motion.  相似文献   

3.
New and more elementary proofs are given of two results due to W. Littman: (1) Let n ? 2, p ? 2n(n ? 1). The estimate ∫∫ (¦▽u¦p + ¦ut¦p) dx dt ? C ∫∫ ¦□u¦p dx dt cannot hold for all u?C0(Q), Q a cube in Rn × R, some constant C. (2) Let n ? 2, p ≠ 2. The estimate ∫ (¦▽(t)¦p + ¦ut(t)¦p) dx ? C(t) ∫ (¦▽u(0)¦p + ¦ut(0)¦p) dx cannot hold for all C solutions of the wave equation □u = 0 in Rn x R; all t ?R; some function C: RR.  相似文献   

4.
The Lévy transform of a Brownian motion B is the Brownian motion B′t=∫0tsgnBsdBs; denote by Bn the Brownian motion obtained from B by iterating n times the Lévy transform. We establish that the set of all instants t such that Btn=0 for some n, is a.s. dense in the time-axis R+. To cite this article: M. Malric, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

5.
Let Ω denote a connected and open subset of Rn. The existence of n commuting self-adjoint operators H1,…, Hn on L2(Ω) such that each Hj is an extension of i∂∂xj (acting on Cc(Ω)) is shown to be equivalent to the existence of a measure μ on Rn such that f → \̂tf (the Fourier transform of f) is unitary from L2(Ω) onto Ω. It is shown that the support of μ can be chosen as a subgroup of Rn iff H1,…, Hn can be chosen such that the unitary groups generated by H1,…, Hn act multiplicatively on L2(Ω). This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of Rn by some subgroup, i.e., iff Ω is essentially a fundamental domain.  相似文献   

6.
We obtain, for a large class of measures μ, general inequalities of the form ∫Rn|u|p A(log1|u|) dμ ? K(6u : Wm,p(Rn,dμ)6p + 6 u 6p A(log1 6 u 6)), where 6u6 = 6 u: Lp(Rn,dμ)6p, log1 t = max{1, log t}, and the function A depends in an appropriate way on μ. Our results extend similar results obtained by Rosen for the case p = 2, A(t) = ts. We also investigate some implications of these inequalities for the imbedding of Sobolev spaces into Orlicz spaces.  相似文献   

7.
We provide conditions on a finite measure μ on Rn which insure that the imbeddings Wk, p(Rndμ)?Lp(Rndμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by dμ = e?¦x¦αdx, where α > 1.  相似文献   

8.
The iterative method of the first author makes possible a self-correcting scheme for the approximate solution obtained. This scheme accelerates the convergence and substantially decreases the number of terms necessary in computation for applications involving either linear or nonlinear stochastic systems. A “feedback” term compensates for the approximation of the system inverse operator by a partial sum. Further, errors are determined in calculating the mean solution 〈y〉 and the correlation Ry(t1,t2) = 〈y(t1)y1(t2)〉 by using the approximations 〈φn〉 and 〈φn(t1)φn(t2)〉 , where φn represents n terms of the solution by the iterative method for stochastic differential equations.  相似文献   

9.
Let {X(t) : t ∈ R+N} denote the N-parameter Wiener process on R+N = [0, ∞)n. For multiple sequences of certain independent random variables the authors find lower bounds for the distributions of maximum of partial sums of these random variables, and as a consequence a useful upper bound for the yet unknown function P{supt∈DnX(t) ≥ c}, c ≥ 0, is obtained where DN = Πk = 1N [0, Tk]. The latter bound is used to give three different varieties of N-parameter generalization of the classical law of iterated logarithm for the standard Brownian motion process.  相似文献   

10.
Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L2(Rk) are analyzed in terms of the elementary generator,
A = (?n)(n2 ? 1)(n!)?1kj = 1?n?xjn
, for n even. Integral operators are defined using the fundamental solutions pn(x, t) to ut = Au and using real polynomials ql,…, qk on Rm by the formula, for q = (ql,…, qk),
(F(t)?)(x) = ∫
Rm
?(x + q(z)) Pn(z, t)dz
. It is determined when, strongly on L2(Rk),
etQ = limj → ∞ Ftjj
. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.  相似文献   

11.
It is shown, for n ? m ? 1, that there exist inner maps Φ: BnBm with boundary values Φ1: Bn → Bm such that σm(A) = σn1?1(A)). where σn and σm are the Haar measures on ?Bn and ?Bm, respectively, and A ? Bn is an arbitrary Borel set.  相似文献   

12.
For a given pair (A,b)∈Rn×n×Rn×1 such that A is cyclic and b is a cyclic generator (with respect to A) of Rn×1, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence {fj}tj=0,fjR1×n,so that a the zeros of the rational function det P(z), where P(z) = zI ? A ? ∑tj=0z-(m+j)b?f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action.  相似文献   

13.
Let p be an infinitely divisible n-variate probability having μ for Poisson measure. We give here some sufficient conditions for p to belong to the class Iαn of n-variate probabilities having only infinitely divisible α-factors. These results are interesting since they are concerned with the case when μ is a continuous measure.  相似文献   

14.
Let L = ∑j = 1mXj2 be sum of squares of vector fields in Rn satisfying a Hörmander condition of order 2: span{Xj, [Xi, Xj]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X1,…, Xm are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator ?2?x2 + x2?2?t2 in R2. (b) The real part of the Kohn Laplacian on the Heisenberg group j ? 1n (??xj + 2yj??t)2 + (??yj ? 2xj??t)2 in R2n + 1. In contrast to non-characteristic points, C regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.  相似文献   

15.
This paper deals with asymptotic behavior for (weak) solutions of the equation utt ? Δu + β(ut) ? ?(t, x), on R+ × Ω; u(t, x) = 0, on R+ × ?Ω. If ?∈L∞(R+,L2(Ω)) and β is coercive, we prove that the solutions are bounded in the energy space, under weaker assumptions than those used by G. Prouse in a previous work. If in addition ?t∈S2(R+,L2(Ω)) and ? is srongly almost-periodic, we prove for strongly monotone β that all solutions are asymptotically almost-periodic in the energy space. The assumptions made on β are much less restrictive than those made by G. Prouse: mainly, we allow β to be multivalued, and in the one-dimensional case β need not be defined everywhere.  相似文献   

16.
Let F1(Rn) denote the Fourier algebra on Rn, and D(Rn) the space of test functions on Rn. A closed subset E of Rn is said to be of spectral synthesis if the only closed ideal J in F1(Rn) which has E as its hull
h(J)={x ? Rn:f(x)=0 for all f ? J}
is the ideal
k(E)={f?F1(Rn):f(E)=0}
. We consider sufficiently regular compact subsets of smooth submanifolds of Rn with constant relative nullity. For such sets E we give an estimate of the degree of nilpotency of the algebra (k(E)∩D(Rn))?j(E), where j(E) denotes the smallest closed ideal in F1(Rn) with hull E. Especially in the case of hypersurfaces this estimate turns out to be exact. Moreover for this case we prove that k(E)∩D(Rn) is dense in k(E). Together this solves the synthesis problem for such sets.  相似文献   

17.
Variational problems for the multiple integral IΩ(u) = ∝Ω g(▽u(x))dx, where Ω?Rm and u:Ω→Rn are studied. A new condition on g, called W1,p-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of IΩ in W1,p(Ω;Rn) and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in W1,p(Ω;Rn), p ? n = m. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.  相似文献   

18.
Let u∈C([0,T1[;Ln(Rn)n) be a maximal solution of the Navier–Stokes equations. We prove that u is C on ]0,T1Rn and there exists a constant ε1>0, which depends only on n, such that if T1 is finite then, for all ω∈S(Rn)n, we have limt→T16u(t)?ω6B?1,∞1.To cite this article: R. May, C. R. Acad. Sci. Paris, Ser. I 336 (2003).  相似文献   

19.
The compactness method to weighted spaces is extended to prove the following theorem:Let H2,s1(B1) be the weighted Sobolev space on the unit ball in Rn with norm
6ν612,s=B1 (1rs)|ν|2 dx + ∫B1 (1rs)|Dν|2 dx.
Let n ? 2 ? s < n. Let u? [H2,s1(B1) ∩ L(B1)]N be a solution of the nonlinear elliptic system
B11rs, i,j=1n, h,K=1N AhKij(x,u) DiuhDK dx=0
, ψ ? ¦C01(B1N, where ¦Aijhk¦ ? L, Aijhk are uniformly continuous functions of their arguments and satisfy:
|η|2 = i=1n, j=1Nij|2 ? i,j=1n, 1rs, h,K=1N AhKijηihηik,?η?RNn
. Then there exists an R1, 0 < R1 < 1, and an α, 0 < α < 1, along with a set Ω ? B1 such that (1) Hn ? 2(Ω) = 0, (2) Ω does not contain the origin; Ω does not contain BR1, (3) B1 ? Ω is open, (4) u is Lipα(B1 ? Ω); u is LipαBR1.  相似文献   

20.
Let ψ be convex with respect to ?, B a convex body in Rn and f a positive concave function on B. A well-known result by Berwald states that 1¦B¦B ψ(f(x)) dx ? n ∝01 ψ(ξt)(1 ? t)n ? 1) dt (1) if ξ is chosen such that 1¦B¦B ?(f(x)) dx = n ∝01 ?(ξt)(1 ? t)n ? 1) dt.The main purpose in this paper is to characterize those functions f : BR+ such that (1) holds.  相似文献   

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