Path-valued Process Research Articles

New large deviation results for some super-Brownian processes

We give large deviation results for the super-Brownian excursion conditioned to have unit mass or unit extinction time and for super-Brownian motion with constant non-positive drift. We use a representation of these processes by a path-valued process, the so-called Brownian snake for which we state large deviation principles.

Representations of the Brownian snake with drift

We consider a path-valued process which is a generalization of the classical Brownian snake introduced by Le Gall. More precisely we add a drift term b to the lifetime process, which may depends on the spatial process. Consequently, this introduces a coupling between the lifetime process and the spatial motion. This process can be obtained from the standard Brownian snake by Girsanov's theorem or by killing of the spatial motion. It can also be viewed as the limit of discrete snakes or, in some special cases, as conditioned Brownian snakes. We also use this process to describe the solutions of the non-linear partial differential equation j u =4 u 2 +4 bu .

Stochastic volterra equations in the plane: smoothness of the law

We give sufficient conditions ensuring the smoothness of the density for the law of the solution of Volterra equations in the plane. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof is based on Malliavin Calculus.

Reflecting Brownian snake and a Neumann–Dirichlet problem

The paper deals with a path-valued Markov process: the reflecting Brownian snake. It is a particular case of the path-valued process previously introduced by Le Gall. Here the spatial motion is a reflecting Brownian motion in a domain D of R d . Using this probabilistic tool, we construct an explicit function v solution of an integral equation which is, under some hypotheses on the regularity of v, equivalent to a semi-linear partial differential equation in D with some mixed Neumann–Dirichlet conditions on the boundary. When the hypotheses on v are not satisfied, we prove that v is still solution of a weak formulation of the Neumann–Dirichlet problem.

Large Deviations for Stochastic Volterra Equations in the Plane

We prove a Large Deviation Principle for the family of solutions of Volterra equations in the plane obtained by perturbation of the driving white noise. One of the motivations for the study of such class of equations is provided by non-linear hyperbolic stochastic partial differential equations appearing in the construction of some path-valued processes on manifolds. The proof uses the method developped by Azencott for diffusion processes. The main ingredients are exponential inequalities for different classes of two-parameter stochastic integrals; these integrals are related to the representation of the stochastic term in the differential equation as a representable semimatringale.

Some properties of the range of super-Brownian motion

We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the ɛ-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of X t is capacity-equivalent to [0, 1]2 in ℝd, d≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]4 in ℝd, d≥ 5.

Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses

We use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism. This construction depends on a path-valued process called the Lévy snake, which is of independent interest. Our method of proof involves a calculation of the Laplace functional of the occupation field of the Lévy snake. This calculation relies on an evaluation of the corresponding moment functionals, which requires precise information about the underlying genealogical structure.

Cone paths for the planar Brownian snake

For the Brownian path-valued process of Le Gall (or Brownian snake) in \({\Bbb R}^2\), the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard Brownian motion.

A large deviation principle for the Brownian snake

We consider the path-valued process called the Brownian snake, conditioned so that its lifetime process is a normalised Brownian excursion. This process denoted by (( W s , ξ s ); s ∈ [0, 1]) is closely related to the integrated super-Brownian excursion studied recently by several authors. We prove a large deviation principle for the law of ((εW s(ζ s), ε 2 3 ζ s); s ϵ [0, 1]) as ε↓0. In particular, we give an explicit formula for the rate function of this large deviation principle. As an application we recover a result of Dembo and Zeitouni.

Spatial Branching Processes and Subordination

AbstractWe present a subordination theory for spatial branching processes. This theory is developed in three different settings, first for branching Markov processes, then for superprocesses and finally for the path-valued process called the Brownian snake. As a common feature of these three situations, subordination can be used to generate new branching mechanisms. As an application, we investigate the compact support property for superprocesses with a general branching mechanism.

A probabilistic approach to blow-up of a semilinear heat equation

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.

Twisted Sheets

We give a simplified presentation of two-parameter stochastic calculus in terms of a symbolic calculus of differentials. We establish a new existence, uniqueness, and non-explosion theorem for certain hyperbolic two-parameter stochastic differential equations driven by the Brownian sheet. The theorem is applied to construct some natural path-valued processes in a Riemannian manifold, where the twisting effect of a connection makes the existing two-parameter theory inapplicable. An analysis of these path-valued processes is made using the two-parameter stochastic calculus.

The Brownian snake and solutions of Δu=u 2 in a domain

We investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation Δu=u2 in a domain of ℝ d . In particular, we prove two conjectures recently formulated by Dynkin. The first one gives a complete characterization of the boundary polar sets, which correspond to boundary removable singularities for the equation Δu=u2. The second one establishes a one-to-one correspondence between nonnegative solutions that are bounded above by a harmonic function, and finite measures on the boundary that do not charge polar sets. This correspondence can be made explicit by a probabilistic formula involving a special class of additive functionals of the Brownian snake. Our proofs combine probabilistic and analytic arguments. An important role is played by a new version of the special Markov property, which is of independent interest.

Sur la mesure de sortie du super mouvement brownien

We study some properties of the exit measure of super Brownian motion from a smooth domainD inR d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of Δu=u 2 inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of Δu=u 2 inD that tends to ∞ on an open subsetO of ϖD and to 0 on the complement in ϖD of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

Hitting probabilities and potential theory for the brownian path-valued process

Hitting probabilities and potential theory for the brownian path-valued process

A class of path-valued Markov processes and its applications to superprocesses

Let (ξ s ) be a continuous Markov process satisfying certain regularity assumptions. We introduce a path-valued strong Markov process associated with (ξ s ), which is closely related to the so-called superprocess with spatial motion (ξ s ). In particular, a subsetH of the state space of (ξ s ) intersects the range of the superprocess if and only if the set of paths that hitH is not polar for the path-valued process. The latter property can be investigated using the tools of the potential theory of symmetric Markov processes: A set is not polar if and only if it supports a measure of finite energy. The same approach can be applied to study sets that are polar for the graph of the superprocess. In the special case when (ξ s ) is a diffusion process, we recover certain results recently obtained by Dynkin.