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The root system Σ of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of Σ is, in general, not large enough to contain the diagrams of all subsystems of Σ, the answer to this question is negative. In this paper we introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow us to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in Σ. In this paper we consider onlyADEroot systems (i.e., systems represented by simply laced Dynkin diagrams). The general case will be the subject of the next publication.

Semilinear equations Lu=ψ(u) where L is an elliptic differential operator and ψ is a positive function can be investigated by using (L,ψ)-superdiffusions. In a special case Δu=u2 a powerful probabilistic tool – the Brownian snake – introduced by Le Gall was successfully applied by him and his school to get deep results on solutions of this equation. Some of these results (but not all of them) were extended by Dynkin and Kuznetsov to general equations by applying superprocesses. An important role in the theory of the Brownian snake and its applications is played by measures Open image in new windowx on the space of continuous paths. Our goal is to introduce analogous measures related to superprocesses (and to general branching exit Markov systems). They are defined on the space of measures and we call them Open image in new window-measures. Using Open image in new window-measures allows to combine some advantages of Brownian snakes and of superprocesses as tools for a study of semilinear PDEs.

LetL be a second order elliptic differential operator on a differentiable manifoldM and let 1

We suggest the name Markov snakes for a class of path-valued Markov processes introduced recently by J.-F. Le Gall in connection with the theory of branching measure-valued processes. Le Gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator Δv−v2. We establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of Markov snakes and vice versa. By using this theorem, we get limit theorems for discrete Markov snakes.

The objective of this paper is to investigate the structure of a general subcritical branching measure-valued processX subject to the usual regularity conditions. We prove that, if the second moments of the total massXt(E) are finite, thenX is a superprocess and we give an explicit expression of the branching characteristicsQ andl in terms of the continuous martingale component of the total massXt(E) and the Levy measure (jumps compensator) ofX.

We consider Markov processes with a fixed transition functionp(r, x; t, B) and with random birth times. We show that a process $$(\tilde X_t ,\tilde P)$$ can be obtained from (X t ,P) by birth delay if and only if $$\tilde P\left\{ {\tilde X_t \in B} \right\} \leqq P\left\{ {X_t \in B} \right\}$$ for allt andB. As an application, we give a new version and a new proof of the results of Rost [R] and Fitzsimmons [F2] on stopping distributions of Markov processes. The key Lemma 1.1 replaces the “filling scheme” used by the previous authors. Birth delay was considered from a different prospective in [F1].

Three theorems on regularity of measure-valued processesX with branching property are established which improve earlier results of Fitzsimmons [F1] and the author [D5]. The main difference is that we treatX as a family of random measures associated with finely open setsQ in time-space. Heuristically,X describes an evolution of a cloud of infinitesimal particles. To everyQ there corresponds a random measureX τ which arises if each particle is observed at its first exit time fromQ. (The stateX t at a fixed timet is a particular case.) We consider a monotone increasing familyQ t of finely open sets and we establish regularity properties of $$\bar X_t = X_{\tau _t } $$ as a function oft. The results are used in [D6], [D7] and [D10] for investigating the relations between superprocesses and non-linear partial differential equations. Basic definitions on Markov processes and superprocesses are introduced in Sect. 1. The next three sections are devoted to proving the regularity theorems. They are applied in Sect. 5 to study parts of superprocess. The relation to the previous work is discussed in more detail in the concluding section. It may be helpful to look briefly through this section before reading Sects. 2–5.

We establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes. As an application, we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations.

A superprocessX over a Markov process ξ can be obtained by a passage to the limit from a branching particle system for which ξ describes the motion of individual particles.The historical process\(\hat \xi \) for ξ is the process whose state at timet is the path of ξ over time interval [0,t]. The superprocess\(\hat X\) over\(\hat \xi \)the historical superprocess over ξ—reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process\(\hat \xi \) is that\(\hat \xi _s \) is a function of\(\hat \xi _t \) for alls