Semiclassical analysis and a new result for Poisson-Lev́y excursion measures.
Sharp bounds for Green functions and jumping functions of subordinate killed Brownian motions in bounded domains.
Simulation and approximation of Lévy-driven stochastic differential equations
We consider the approximate Euler scheme for Lévy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Lévy measure of the driving process behaves like |z|−1−αdz near 0, for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the...
Simulation and approximation of Lévy-driven stochastic differential equations
We consider the approximate Euler scheme for Lévy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Lévy measure of the driving process behaves like |z|−1−αdz near 0, for some α∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα. For a similar error when neglecting the...
Skew-product representations of multidimensional Dunkl Markov processes
In this paper we obtain skew-product representations of the multidimensional Dunkl processes which generalize the skew-product decomposition in dimension 1 obtained in L. Gallardo and M. Yor. Some remarkable properties of the Dunkl martingales. Séminaire de Probabilités XXXIX, 2006. We also study the radial part of the Dunkl process, i.e. the projection of the Dunkl process on a Weyl chamber.
Small-time behavior of beta coalescents
For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1<α<2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time...
Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump.
Some informational properties of Markov pure-jump processes
Some remarks on the two-dimensional Dirac equation.
Spaces of generalized smoothness on h-sets and related Dirichlet forms
The paper is devoted to spaces of generalized smoothness on so-called h-sets. First we find quarkonial representations of isotropic spaces of generalized smoothness on ℝⁿ and on an h-set. Then we investigate representations of such spaces via differences, which are very helpful when we want to find an explicit representation of the domain of a Dirichlet form on h-sets. We prove that both representations are equivalent, and also find the domain of some time-changed Dirichlet form on an h-set.
Stability of impulsive hopfield neural networks with Markovian switching and time-varying delays
The paper is concerned with stability analysis for a class of impulsive Hopfield neural networks with Markovian jumping parameters and time-varying delays. The jumping parameters considered here are generated from a continuous-time discrete-state homogenous Markov process. By employing a Lyapunov functional approach, new delay-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities (LMIs). The proposed criteria can be easily checked by using some standard numerical...
Stability properties of constrained jump-diffusion processes.
Stationary distributions for jump processes with memory
We analyze a jump processes with a jump measure determined by a “memory” process . The state space of is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of is the product of the uniform probability measure and a Gaussian distribution.
Stochastic differential equations with jumps.
Stochastic FitzHugh-Nagumo equations on networks with impulsive noise.
Stochastic flow for SDEs with jumps and irregular drift term
We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes...
Stochastic stability of neural networks with both Markovian jump parameters and continuously distributed delays.
Sur le passage de certaines marches aléatoires planes au-dessus d'une hyperbole équilatère
Symmetric jump processes : localization, heat kernels and convergence
We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Hölder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.