A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

Relative entropy and mixing properties of infinite dimensional diffusions

Summary. Let η be a diffusion process taking values on the infinite dimensional space T ^Z , where T is the circle, and with components satisfying the equations dη _i =σ _i (η) dW _i +b _i (η) dt for some coefficients σ _i and b _i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t _n →∞ such that lim _{n →∞}μS _tn =ν exists, where S _t is the semi-group of the process. We prove that if σ _i and b _i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf _{i ,η}σ _i (η)>0, then ν is invariant. Received: 12 September 1996 / In revised form: 10 November 1997     

A family of diffusion processes on Sierpinski carpets

We construct a family of diffusions P ^α = {P ^x} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d _H < α < ∞, where d _H = log(3 ^d − 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions P ^α are reversible with invariant measures μ = μ^[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3^α·μ(A) for all A∈ℬ(F^). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet. Received: 30 September 1999 / Revised version: 15 June 2000 / Published online: 24 January 2000     

Maximal dissipativity for degenerate Kolmogorov operators

In this paper we are interested in studying the properties of an elliptic degenerate operator N₀ in the space L^p of with respect to an invariant measure μ. The existence of μ is proven under suitable conditions on coefficients of the operator. We prove that the closure of N₀ is m-dissipative in      

A new class of unbalanced haar wavelets that form an unconditional basis for Lp on general measure spaces

Given a complete separable σ-finite measure space (X,Σ, μ) and nested partitions of X, we construct unbalanced Haar-like wavelets on X that form an unconditional basis for L_p (X,Σ, μ) where1<p<∞. Our construction and proofs build upon ideas of Burkholder and Mitrea. We show that if(X,Σ, μ) is not purely atomic, then the unconditional basis constant of our basis is (max(p, q) −1). We derive a fast algorithm to compute the coefficients.     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

On the Einstein relation for a mechanical system

We consider a mechanical model in the plane, consisting of a vertical rod, subject to a constant horizontal force f and to elastic collisions with the particles of a free gas which is “horizontally” in equilibrium at some inverse temperature β. In a previous paper we proved that, in the appropriate space and time scaling, the motion of the rod is described as a drift term plus a diffusion term. In this paper we prove that the drift d(f) and the diffusivity σ ² (f) are continuous functions of f, and moreover that the Einstein relation holds, i.e., lim _{f → 0}  d(f)f = β2 σ ² (0) . Received: 26 January 1996 / In revised form: 2 October 1996     

Skorokhod embeddings, minimality and non-centred target distributions

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality. We then apply these results, firstly to the problem of embedding non-centred target distributions in Brownian motion, and secondly to embedding general target laws in a diffusion. We construct an embedding (which reduces to the Azema-Yor embedding in the zero-target mean case) which maximises the law of sup_s≤TB_s among the class of minimal embeddings of a general target distribution μ in Brownian motion. We then construct a minimal embedding of μ in a diffusion X which maximises the law of sup_s≤Th(X_s) for a general function h.     

Characterizations of some classes ofL 1-preduals by the Alfsen-Effros structure topology

Using the Alfsen-Effros structure topology on the extreme boundary of the dual unit ball of a complex Banach space, we give characterizations ofL ¹-preduals (i.e., Banach spaces whose duals are isometrically isomorphic toL ¹ (μ) for a non-negative measure μ) and some of its subclasses viz.G-spaces,C _σ-spaces and _c ⁰(Г) spaces.     

Spectral mapping theorems for exponentially bounded c-semigroups in Banach spaces

LetZ be a generator of an exponentially boundedC-semigroup {S _t } _t≥0 in a Banach space and letT _t =C ⁻¹ S _t . We show that the spectral mapping theorems such as exp(tσ(Z)) ⊂ σ(T _t ) and exp(tσ _p (Z)) ⊂ tσ _p (T _t ) ⊂ exp(tσ _p (Z)) ⋃ {0} for everyt≥0 hold. The present studies were supported by the Basic Science Research Institute Program, Ministry of Education, 1987.     

A study on Markovian maximality,change of probability and regularity

LetE be a rigid separable Banach space andm a bounded Borel measure onE. Let Ext denote the family of all gradient type Dirichlet forms onL ²(E, m) such that the domain of their extended generators (cf. Definition 1.1) contain the smooth functions. We prove three results. First, we prove the existence of the maximum element in Ext whenever Ext is not empty. Secondly, let be the maximum element in Ext (when Ext Ø) and let be a positive function in D(). We define a new measure =²·m and we consider the family Ext associated with the measure . We prove that if is associated with a diffusion process, Ext is not empty and its maximum element is also associated with a diffusion process. Finally, whenm is a centered Gaussian measure onE, we can prove that Ext contains exactly one element.     

On a semigroup generated by a convex combination of two Feller generators

Let be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in with related Feller processes {X _A (t), t ≥ 0} and {X _B (t), t ≥ 0} and let α and β be two non-negative continuous functions on with α + β = 1. Assume that the closure C of C ₀ = αA + βB with generates a Feller semigroup {T _C (t), t ≥ 0} in . It is natural to think of a related Feller process {X _C (t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point , with probability α(p) the process behaves like {X _A (t), t ≥ 0} and with probability β(p) it behaves like {X _B (t), t ≥ 0}. We provide an approximation of {T _C (t), t ≥ 0} via a sequence of semigroups acting in that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].     

On the completeness of algebraic polynomials in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(ℝ, <Emphasis Type="Italic">d</Emphasis>μ)

We prove that the theorem on the incompleteness of polynomials in the space C ₀ ^w established by de Branges in 1959 is not true for the space L _p (ℝ, dμ) if the support of the measure μ is sufficiently dense.     

Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces

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     

Operator-semistable,operator semi-selfdecomposable probability measures and related nested classes on <Emphasis Type="Italic">p</Emphasis>-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

The uniform random tree in a Brownian excursion

Summary To any Brownian excursione with duration (e) and anyt ₁, ...,t _p [0,(e)], we associate a branching tree withp branches denoted byT _p (e, t ₁,...,t _p ), which is closely related to the structure of the minima ofe. Our main theorem states that, ife is chosen according to the Itô measure and (t ₁, ...,t _p ) according to Lebesgue measure on [0,(e)] ^p , the treeT _p (e, t ₁, ...,t _p ) is distributed according to the uniform measure on the set of trees withp branches. The proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree, thus generalizing a well-known representation theorem of Bismut. If we replace the Itô measure by the law of the normalized excursion, a simple conditioning argument leads to another remarkable result originally proved by Aldous with a very different method.     

Some equivalence relations which are borel reducible to isomorphism between separable banach spaces

We prove that the relationE _K σ is Borel reducible to isomorphism and complemented biembeddability between subspaces ofc ₀ orl _p with 1≤p<2. We also show that the relationE _K σ ⊗=⁺ is Borel reducible to isomorphism, complemented biembeddability, and Lipschitz isomorphism between subspaces ofL _p for 1≤p<2. This author was supported by FAPESP Grant 2002/09662-1.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

Processus d'Ornstein-Uhlenbeck et ensemblesW 2,2—polaires

Résumé Nous démontrons que, dans un espace de Fréchet gaussien (E,), les ensembles dec _2,2-capacité nulle sont les ensembles que le processus d'Ornstein-Uhlenbeck à deux indices à valeurs dansE ne rencontres pas. Afin de prouver ce résultat, nous établissons une inégalité entre la probabilité de rencontre d'un ensemble et la capacité de celui-ci. Nous établissons également une représentation de lac _2,2-capacité d'un ensemble par une mesure aléatoire à deux indices.

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We show that, on a Fréchet gaussian space (E,) thec _2,2-capacity null sets are those which cannot be hit by the two-parameterE-valued Ornstein-Uhlenbeck process. In order to prove this result, an inequality between the hitting probability of a set and itsc _2,2-capacity, as well as a representation of thec _2,2-capacity of a set by a two parameter random measure, is established.