Symmetric Stable Process Of Index Research Articles

On solutions of equations with measurable coefficients driven by α- stable processes

We prove the existence of solutions for the stochastic differential equation with the measurable coefficients a and b satisfying the condition and for all , where and K are constants. The driving process Z is a symmetric stable process of index . This generalizes the result of Krylov [Controlled Diffusion Processes, Springer, New York, 1980] for the case of , that is, when Z is a Brownian motion. The proof is based on integral estimates of the Krylov type for the given equation, which are also derived in this note and are of independent interest. Moreover, unlike in Krylov [Controlled Diffusion Processes, Springer, New York, 1980], we use a different approach to derive the corresponding integral estimates.

A Law of the Iterated Logarithm for Some Additive Functionals of Symmetric Stable Process Via the Strong Approximation

In this note we give a strong approximation version for the first-order limit theorem of symmetric stable process of index 1 < α ⩽ 2. This generalizes a result of Csaki et al.[7] for Brownian motion. As an application, we give the law of iterated logarithm for some additive functionals of this process.

LIMIT THEOREMS FOR A CLASS OF ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESS AND FRACTIONAL BROWNIAN MOTION IN BESOV-ORLICZ SPACES

We use the Potter’s Theorem and the tightness criterion in Besov-Orlicz spaces, recently proved, to generalize some limit theorem for occupation times problem of certain self-similar process, namely symmetric stable process of index 1<α≤ and fractional Brownian motion of Hurst parameter 0

On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

We consider a one-dimensional stochastic equation , , with respect to a symmetric stable process of index . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation with respect to the semimartingale and corresponding matrix . In the case of we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The existence proofs are established using the method of Krylov's estimates for processes satisfying the 2-dimensional equation. On another hand, the Krylov's estimates are based on some analytical facts of independent interest that are also proved in the paper.

Large deviations and renormalization for Riesz potentials of stable intersection measures

We study the object formally defined as (0.1) γ ( [ 0 , t ] 2 ) = ∬ [ 0 , t ] 2 | X s − X r | − σ d r d s − E ∬ [ 0 , t ] 2 | X s − X r | − σ d r d s , where X t denotes the symmetric stable processes of index 0 < β ≤ 2 in R d . When β ≤ σ < min { 3 2 β , d } , this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for γ . This is applied to obtain results about stable processes in random potentials.

Large deviations for Riesz potentials of additive processes

Nous étudions les fonctionelles de la forme ζt=∫0t⋯∫0t|X1(s1)+⋯+Xp(sp)|−σ ds1 ⋯ dsp, où X1(t), …, Xp(t) sont des processus stables symétriques indépendants et identiquement distribués d’ordre 0<β≤2. Nous obtenons des résultats sur les grandes déviations et les lois du logarithme itéré.

On degenerate stochastic equations of Itô type with jumps

The one-dimensional stochastic equation d X t = b 1 ( X t ) d W t + b 2 ( X t − ) d Z t + a ( X t ) d t , t ≥ 0 , where b 1 , b 2 , a : R → R are Borel measurable functions, W is a Brownian motion, and Z is a symmetric stable process of index 0 < α < 2 , is considered. We prove the existence of (weak) solutions under some conditions of boundedness of coefficients when b 1 can be degenerate which improves the results of Lepeltier and Marchal [Lepeltier, J.P., Marchal, B., 1976. Probléme des martingales et équations différentielles stochastiques associées á un opérateur intégro-différentiel. Ann. IHP 12 (1), 43–103] for this case. Our approach is based on Krylov’s estimates for solutions X and the weak convergence arguments.

A note on 𝐿₂-estimates for stable integrals with drift

Let X X be of the form X t = ∫ 0 t b s d Z s + ∫ 0 t a s d s , t ≥ 0 , X_t=\int _0^tb_sdZ_s+\int _0^ta_sds, t\ge 0, where Z Z is a symmetric stable process of index α ∈ ( 1 , 2 ) \alpha \in (1,2) with Z 0 = 0 Z_0=0 . We obtain various L 2 L_2 -estimates for the process X X . In particular, for m ∈ N , t ≥ 0 , m\in \mathbb N, t\ge 0, and any measurable, nonnegative function f f we derive the inequality \[ E ∫ 0 t ∧ τ m ( X ) | b s | α f ( X s ) d s ≤ N ‖ f ‖ 2 , m . {\mathbf E}\int _0^{t\land \tau _m(X)}|b_s|^{\alpha }f(X_s)ds\le N\|f\|_{2,m}. \] As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation d X t = b ( X t − ) d Z t + a ( X t ) d t dX_t=b(X_{t-})dZ_t+a(X_t)dt for any initial value x 0 ∈ R x_0\in \mathbb R .

The Exact Hausdorff Measure Function of the Level Sets of Multi–parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N–parameter Rd–valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N–parameter Rd–valued stable processes. We show that the exact Hausdorff measure function of level sets of N–parameter Rd–valued symmetric stable processes of index α is ϕ(r) = rN−d/α(log log 1/r)d/α when Nα > d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general (N, d,α) strictly stable processes.

Convergence in law for certain additive functionals of symmetric stable processes under strong topology

We give some limit theorems of certain additive functionals for symmetric stable process of index 1 < α ⩽ 2 in anisotropic Besov space. These results generalize those obtained by Eisenbaum (1997) and by Csaki et al. (2002). To cite this article: M. Ait Ouahra, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Systems of equations driven by stable processes

Let Zjt, j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations

Exponential asymptotics for intersection local times of stable processes and random walks

We study large deviations for intersection local times of p independent d-dimensional symmetric stable processes of index β, under the condition p ( d − β ) < d . Our approach is based on Feynman–Kac type large deviations, moment computations and some techniques from probability in Banach spaces.

Théorèmes limites pour certaines fonctionnelles associées aux processus stables sur l’espace de Hölder

In this paper we study the Holder regularity property of the local time of a symmetric stable process of index 1 alt; [alpha] [less than or equal] 2 and of its fractional derivative as a doubly indexed process with respect to the space and the time variables. As an application we establish some limit theorems for occupation times of one--dimensional symmetric stable processes in the space of Holder continuous functions. Our results generalize those obtained by Fitzsimmons and Getoor for stable processes in the space on continuous functions. The limiting processes are fractional derivatives and Hilbert transforms of local times.

The most visited sites of symmetric stable processes

Let X be a symmetric stable process of index α∈ (1,2] and let L x t denote the local time at time t and position x. Let V(t) be such that L t V(t) = sup x∈ ℝ L t x . We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim t →∞ |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and further to the winding problem for planar Brownian motion.

Régularité du temps local du processus symétrique stable en norme besov

Let be a real-valued symmetric stable process of index β,1 < β ≤ 2, i.e., a Lévy process with characteristic function Let be the local time of the process X. The existence of an a.s continuous (as a function of two variables) version of L(t, x) has been proved in [1,6]The mean zero Gaussian process with covariance is said to be associated with Markov process X if: where is the symmetric transition density of X Marcus and Rosen used in [11] the Dynkin isomorphism theorem (cf. Lemma 3.2), which relates sample path properties of the local time L(t, x) to those of the associated Gaussian process G of X, to study the almost sure variation in the spatial variable of the local time of X. In the special case the process X is a Brownian motion and it is known (cf. [4,9]) that for each fixed t > 0 and p < ∞, the random function satisfies a.s a Hölder condition in the Lp norm, with the modulus of smoothness In this paper the Holder condition for the local time of the symmetric stable process of index 1 < β ≤ 2 in the Lp norm is studied. The main result – Theorem 3.1 – states that for each t > 0 and 1 ≤ p < ∞-the modulus of smoothness of the local time in the Lp norm-satisfies a. s Our theorem can be regarded as an extension of the result of [4]. The main tool of the proof is the Dynkin isomorphism theorem and some regularity results for Gaussian processes presented in [8]

Thick Points for Transient Symmetric Stable Processes

Let $T(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for a transient symmetric stable processes of index $b \lt d$ in $R^d$ and $K(b,d)$ denote the norm of the convolution with its 0-potential density, considered as an operator on $L^2(B(0,1),dx)$. We prove that as $r$ approaches 0, almost surely $\sup_{|x| \leq 1} T(x,r)/(r^b|\log r|) \to b K(b,d)$. Furthermore, for any $a \in (0,b/K(b,d))$, the Hausdorff dimension of the set of ``thick points'' $x$ for which $\limsup_{r \to 0} T(x,r)/(r^b |\log r|)=a$, is almost surely $b-a/K(b,d)$; this is the correct scaling to obtain a nondegenerate ``multifractal spectrum'' for transient stable occupation measure. The liminf scaling of $T(x,r)$ is quite different: we exhibit positive, finite, non-random $c(b,d), C(b,d)$, such that almost surely $c(b,d) \lt \sup_x \liminf_{r \to 0} T(x,r)/r^b \lt C(b,d)$.

Occupation time large deviations for critical branching Brownian motion, super-Brownian motion and related processes

We derive a large deviation principle for the occupation time func-tional, acting on functions with zero Lebesgue integral, for both super-Brownian motion and critical branching Brownian motion in three dimensions. Our technique, based on a moment formula of Dynkin, allows us to compute the exact rate functions, which differ for the two processes. Obtaining the exact rate function for the super-Brownian motion solves a conjecture of Lee and Remillard. We also show the corresponding CLT and obtain similar results for the superprocesses and critical branching process built over the symmetric stable process of index $\beta$ in $R^d$, with $d < 2\beta < 2 + d$ .

Feynman-Kac semigroup with discontinuous additive functionals

LetX be a symmetric stable process of index α, 0<α<2, inRd, let μ be a (signed) Radon measure onRd belonging to the Kato classKd, α and letF be a Borel function onRd×Rd satisfying certain conditions. Suppose thatA t μ is the continuous additive functional with μ as its Revuz measure and $$\begin{gathered} A_{t } = A_t^\mu + \Sigma F(X_{s - } , X_s ) \hfill \\ 0< S \leqslant t \hfill \\ \end{gathered} $$ Then the defined semigroup $$T_t f(x) = E^x \{ e^{A_1 } f(X_1 )\} $$ is called the Feynman-Kac semigroup. In this paper we study the Feynman-Kac semigroup (T t ) t>0 and identify the bilinear form corresponding to it.

Probabilistic approach to the Dirichlet problem of perturbed stable processes

LetXt be a symmetric stable process of index α, 0<α<2, inRd (d≧2). In this paper we deal with the perturbation ofXt by a multiplicative functional of the following form: $$M_t = \exp \left\{ {\sum\limits_{s\mathop \leqslant \limits_ - t} {F(X_{s - } ,X_s )} } \right\}$$ withF being a function onRd×Rd satisfying certain conditions. First we prove the following gauge theorem: IfD is a bounded open domain ofRd, then the functiong(x)=Ex{M(τD)} is either identically infinite onD or bounded onD, where τD is the first exit time fromD. Then we formulate the Dirichlet problem associated with the perturbed symmetric stable process by using Dirichlet form theory. Finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the Dirichlet problem mentioned above.

Polar Sets and Multiple Points for Super-Brownian Motion

We study the closed support of the measure-valued diffusions of Watanabe and Dawson. When the spatial motion is Brownian, sufficient conditions involving capacity are given for a fixed set to be hit by the $k$-multiple points of the support process. The conditions are close to the necessary conditions found by Dawson, Iscoe and Perkins and lead to necessary and sufficient conditions for the existence of $k$-multiple points. When the spatial motion is a symmetric stable process of index $\alpha < 2$, the closed support is shown to be $\mathbb{R}^d$ or $\varnothing$.