Stochastic coalescence.
Stochastic differential equation driven by a pure-birth process
A generalization of the Poisson driven stochastic differential equation is considered. A sufficient condition for asymptotic stability of a discrete time-nonhomogeneous Markov process is proved.
Stochastic differential equations driven by processes generated by divergence form operators I: a Wong-Zakai theorem
We show in this article how the theory of “rough paths” allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.
Stochastic differential equations driven by processes generated by divergence form operators II: convergence results
We have seen in a previous article how the theory of “rough paths” allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals...
Stochastic differential equations with boundary conditions driven by a Poisson noise.
Stochastic differential equations with jumps.
Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings
In this continuation of the preceding paper (Part I), we consider a sequence of i.i.d. random Lipschitz mappings → , where is a proper metric space. We investigate existence and uniqueness of invariant measures, as well as recurrence and ergodicity of the induced stochastic dynamical system (SDS) starting at x ∈ . The main results concern the case when the associated Lipschitz constants are log-centered. Principal tools are local contractivity, as considered in detail in Part I, the Chacon-Ornstein...
Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity
Consider a proper metric space and a sequence of i.i.d. random continuous mappings → . It induces the stochastic dynamical system (SDS) starting at x ∈ . In this and the subsequent paper, we study existence and uniqueness of invariant measures, as well as recurrence and ergodicity of this process. In the present first part, we elaborate, improve and complete the unpublished work of Martin Benda on local contractivity, which merits publicity and provides an important tool for studying stochastic...
Stochastic equation of population dynamics of prey-predator type with diffusion in a territory.
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 flows associated to coalescent processes II : stochastic differential equations
Stochastic flows with interaction and measure-valued processes.
Stochastic harmonic morphisms : functions mapping the paths of one diffusion into the paths of another
We give several necessary and sufficient conditions that a function maps the paths of one diffusion into the paths of another. One of these conditions is that is a harmonic morphism between the associated harmonic spaces. Another condition constitutes an extension of a result of P. Lévy about conformal invariance of Brownian motion. The third condition implies that two diffusions with the same hitting distributions differ only by a chance of time scale. We also obtain a converse of the above...
Stochastic Holomorphy.
Stochastic integral representation of the modulus of Brownian local time and a central limit theorem.
Stochastic integrals and progressive measurability. An example
Stochastic integration with respect to fractional brownian motion
Stochastic methods and differential geometry
Stochastic modelling of the loss of telomeric sequences in cells with multiple chromosomes.