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Abstract: Let D be a bounded domain in with smooth boundary ∂D. We give a probabilistic representation formula for the non-negative solution of the mixed Dirichlet non-linear Neumann boundary value problem (DNP) where is a non-trivial partition of ∂D, φ is a non-negative, bounded and continuous function defined on , and denotes the outward normal derivative on the boundary of D. To solve the DNP, we consider a catalytic super-Brownian motion with underlying motion a Brownian motion reflected on ∂D, killed when it reaches and catalysed by the set , i.e. the branching rate is given by the local time of the paths on . Then we prove that the log-Laplace transform of φ integrated with respect to the exit measure of the catalytic process on , is a non-negative weak solution of the DNP. In a second part we show that we still have a probabilistic representation formula if the Dirichlet condition on is replaced by a Neumann condition. [Copyright &y& Elsevier]

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Using an approximating scheme with the Brownian snake, we prove the existence of solution to a martingale problem for super Brownian motion with interactions. [Copyright &y& Elsevier]

We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ[sup d] . We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4]. [ABSTRACT FROM AUTHOR]

Consider the catalytic super-Brownian motion X[sup ϱ] (reactant) in ℝ[sup d] , d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L[sub [ϱ,X[sup ϱ] ]] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for X[sup ϱ] and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of X[sup ϱ] . In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds K[sub s] (dx). At fixed time s, the collision measures K[sub s] (dx) of ϱ[sub s] and X[sub s] [sup ϱ] have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts. [ABSTRACT FROM AUTHOR]

Abstract. 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 epsilon-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[sub t] is capacity-equivalent to [0, 1][sup 2] in R[sup d], d is greater than or equal to 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1][sup 4] in R[sup d] d less than or equal to 5. [ABSTRACT FROM AUTHOR]