Intégrales stochastiques par rapport aux processus de Wiener ou de Poisson
Integrals related to stationary processes and cylindrical martingales
Integration by parts and Cameron-Martin formulas for the free path space of a compact riemannian manifold
Integration by parts for jump processes
Integration in a dynamical stochastic geometric framework
Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary...
Integration in a dynamical stochastic geometric framework
Motivated by the well-posedness of birth-and-growth processes, a stochastic geometric differential equation and, hence, a stochastic geometric dynamical system are proposed. In fact, a birth-and-growth process can be rigorously modeled as a suitable combination, involving the Minkowski sum and the Aumann integral, of two very general set-valued processes representing nucleation and growth dynamics, respectively. The simplicity of the proposed geometric approach allows to avoid problems of boundary...
Intégration stochastique et géométrie des espaces de Banach
Intermittence and nonlinear parabolic stochastic partial differential equations.
Intermittency on catalysts: symmetric exclusion.
Intermittency on catalysts: three-dimensional simple symmetric exclusion.
Intermittency properties in a hyperbolic Anderson problem
We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension 3 with linear multiplicative spatially homogeneous gaussian noise that is white in time. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison,...
Interprétation probabiliste et extension des intégrales stochastiques non commutatives
Intrinsic location parameter of a diffusion process.
Introducing randomness into first-order and second-order deterministic differential equations.
Introduction au calcul stochastique
Introduction to the Theory of the It-type Stochastic Integrals and Stochastic Differential Equations
Invariance of Poisson measures under random transformations
We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method of Üstünel and Zakai (Probab. Theory Related Fields103 (1995) 409–429) on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples...
Invariant measure for some differential operators and unitarizing measure for the representation of a Lie group. Examples in finite dimension
Consider a Lie group with a unitary representation into a space of holomorphic functions defined on a domain 𝓓 of ℂ and in L²(μ), the measure μ being the unitarizing measure of the representation. On finite-dimensional examples, we show that this unitarizing measure is also the invariant measure for some differential operators on 𝓓. We calculate these operators and we develop the concepts of unitarizing measure and invariant measure for an OU operator (differential operator associated to...
Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations
We consider a stochastic delay differential equation with exponentially stable drift and diffusion driven by a general Lévy process. The diffusion coefficient is assumed to be locally Lipschitz and bounded. Under a mild condition on the large jumps of the Lévy process, we show existence of an invariant measure. Main tools in our proof are a variation-of-constants formula and a stability theorem in our context, which are of independent interest.
Invariant measures for nonlinear SPDE's: uniqueness and stability
The paper presents a review of some recent results on uniqueness of invariant measures for stochastic differential equations in infinite-dimensional state spaces, with particular attention paid to stochastic partial differential equations. Related results on asymptotic behaviour of solutions like ergodic theorems and convergence of probability laws of solutions in strong and weak topologies are also reviewed.