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Let L be a second order elliptic differential operator in ${\bf R}^d$ and let E be a bounded domain in ${\bf R}^d$ with smooth boundary $\partial E$. A pair $(\Gamma,\nu)$ is associated with every positive solution of a semilinear differential equation $Lu=\psi(u)$ in E, where $\Gamma$ is a closed subset of~$\partial E$ and~$\nu$ is a Radon measure on $O=\partial E\setminus \Gamma$. We call this pair the {\it rough trace of u on~$\partial E$}. (In [E. B. Dynkin and S. E. Kuznetsov, Comm. Pure Appl. Math., 51 (1998), pp. 897--936], we introduced a fine trace allowing us to distinguish solutions with identical rough traces.) The case of $\psi(u)=u^\alpha$ with $\alpha>1$ was investigated using various methods by Le Gall, Dynkin, and Kuznetsov and by Marcus and V\'eron. In this paper we cover a wide class of functions~$\psi$ and simplify substantially the proofs contained in our earlier papers.

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.

A motivation for this paper comes from the role of Choquet capacities in the study of semilinear elliptic partial differential equations. In particular, the recent progress in the classification of all positive solutions of Lu=uα in a bounded smooth domain E⊂Rd was achieved by using, as a tool, capacities on a smooth manifold ∂E. Either the Poisson capacities (associated with the Poisson kernel in E) or the Bessel capacities (related to the Bessel kernel) have been used. In this and many other applications there is no advantage in choosing any special member in a class of equivalent capacities. (Two capacities are called equivalent if their ratio is bounded away from 0 and ∞.) In the literature Bessel capacities are considered mostly in the space Rd. We introduce two versions of Bessel capacities on a compact N-dimensional manifold. A class Capℓ,p of equivalent capacities is defined, for ℓp⩽N, on every compact Lipschitz manifold. Another class CBℓ,p is defined (for all ℓ>0, p>1) in terms of a diffusion process on a C2-manifold. These classes coincide when both are defined. If the manifold is the boundary of a bounded C2-domain E⊂Rd, then both versions of the Bessel capacities are equivalent to the Poisson capacities.

Our motivation is the following problem: to describe all positive solutions of a semilinear elliptic equation L u = u α L u=u^\alpha with α > 1 \alpha >1 in a bounded smooth domain E ⊂ R d E\subset \mathbb {R}^d . In 1998 Dynkin and Kuznetsov solved this problem for a class of solutions which they called σ \sigma -moderate. The question if all solutions belong to this class remained open. In 2002 Mselati proved that this is true for the equation Δ u = u 2 \Delta u=u^2 in a domain of class C 4 C^4 . His principal tool—the Brownian snake—is not applicable to the case α ≠ 2 \alpha \neq 2 . In 2003 Dynkin and Kuznetsov modified most of Mselati’s arguments by using superdiffusions instead of the snake. However a critical gap remained. A new inequality established in the present paper allows us to close this gap.

Suppose that E is a bounded domain of class C2,λ in Rd and L is a uniformly elliptic operator in E. The set U of all positive solutions of the equation Lu=ψ(u) in E was investigated by a number of authors for various classes of functions ψ. In Dynkin and Kuznetsov (Comm. Pure Appl. Math. 51 (1998) 897) we defined, for every Borel subset Γ of ∂E, two such solutions uΓ⩽wΓ. We also introduced a class of solutions uν in 1–1 correspondence with a certain class N0 of σ-finite measures ν on ∂E. With every u∈U we associated a pair (Γ,ν) where Γ is a Borel subset of ∂E and ν∈N0. We called this pair the fine boundary trace of u and we denoted in tr(u). Let u⊕v stand for the maximal solution dominated by u+v. We say that u belongs to the class EL,ψ if the condition tr(u)=(Γ,ν) implies that u⩽wΓ⊕uν and we say that u belongs to EL,ψ∗ if the condition tr(u)=(Γ,ν) implies that u⩾uΓ⊕uν. It was proved in Dynkin and Kuznetsov (1998) that, under minimal assumptions on L and ψ, the class EL,ψ∗ contains all bounded domains. It follows from results of Mselati (These de Doctorat de l'Universite Paris 6, 2002; C.R. Acad. Sci. Paris Ser. I 332 (2002); Mem. Amer. Math. Soc. (2003), to appear), that all E of the class C4 belong to EΔ,ψ where Δ is the Laplacian and ψ(u)=u2. [Mselati proved that, in his case, uΓ=wΓ and therefore the condition tr(u)=(Γ,ν) implies u=uΓ⊕uν=wΓ⊕uν.] By modifying Mselati's arguments, we extend his result to ψ(u)=uα with 1

Let X=(XD,Pμ) be a superdiffusion in a domain E⊂Rd. We introduce a germ σ-algebra FE− at the boundary of E and we prove that, on this σ-algebra, Pμ1 is absolutely continuous with respect to Pμ2 if μ1 and μ2 are concentrated on compact subsets of E. In combination with previous results of Dynkin, Kuznetsov and Mselati, this leads to a complete classification of positive solutions of equation Δu=uα in a bounded domain E of class C4 for the case 1

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.

By using super-Brownian motion, all positive solutions of the non-linear differential equation with in a bounded smooth domain are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case was treated by B. Mselati in 2002.

All positive harmonic functions in an arbitrary domain of a Euclidean space can be described in terms of the so-called exit boundary. This was established in 1941 by R.S. Martin. A probabilistic approach to the Martin theory is due to Doob and Hunt. It was extended later to harmonic functions associated with a wide class of Markov processes. The subject of this paper are harmonic functions associated with a superdiffusion X (we call them X -harmonic). The results of Evans and Perkins imply the existence and uniqueness of an integral representation of positive X -harmonic functions through extreme functions. An outstanding problem is to find all extremal functions (they are in 1–1 correspondence with the points of the exit boundary). An interest in X -harmonic functions is motivated, in part, by the fact that each of them provides a way of conditioning a superprocess. Path properties of conditioned superprocesses (corresponding to various special X -harmonic functions) were investigated by a number of authors. Important classes of X -harmonic functions are related to positive solutions of semilinear partial differential equations. Almost nothing is known about their decomposition into extreme elements. A progress in this direction may create new tools for investigating solutions of the equations. The goal of this paper is to summarize all known facts about X -harmonic functions, to present the results of various authors in a more general setting in a unified form and to outline a program of further work.

Let L be a second order elliptic differential operator and let D be an arbitrary open subset of Rd. In [1] we introduced a class U1(D) of positive solutions of the equation Lu=−u2 which is in 1–1 correspondence with a convex class H1(D) of positive solutions of the equation Lu=0. In the present paper, we give a probabilistic characterization of U1(D) and a probabilistic representation of u∈U1(D) in terms of a superdiffusion. Similar results are obtained also for a parabolic equation u+Lu=−u2.

Let $\xi$ be a Markov process with transition function $p(r,x;\,t,\,dy)$ and let X be the corresponding Dawson--Watanabe superprocess (i.e., the superprocess with the branching characteristic $\psi(u)=\gamma u^2$). Denote by $\cal P$ the transition function of X and put $$ p_n(r,x;\,t,\,dy)=\prod_{i=1}^n p(r,x_i;\,t,\,dy_i). $$ To every $p_n$-exit law $\ell$ there corresponds a $\cal P$-exit law $L_\ell$ such that, for every t, $L_\ell^t(\mu)$ is a polynomial of degree n in $\mu$ with the leading term $\langle \ell^t,\mu^n\rangle $. Every polynomial $\cal P$-exit law has a unique representation of the form $L_{\ell_1}+\cdots+L_{\ell_n}$, where $\ell_k$ is a $p_k$-exit law.

A superdiffusion is a measure-valued branching process associated with a nonlinear operator Lu − ψ ( u ) where L is a second order elliptic differential operator and ψ is a function from R d × R + to R + . In the case L 1⩽0 (the so-called subcritical case), the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1. Most results on connections between the superdiffusion and differential equations involving Lu − ψ ( u ) were obtained for the subcritical case. In the present paper, we assume only that L 1 is a bounded function. In this more general setting, the probability of extinction can be smaller than 1, and we show that his happens if and only if there exists a strictly positive solution of the equation Lu − ψ ( u )=0 in R d . We also establish a relation between the probability of extinction in a domain D and strictly positive solutions of equation Lu − ψ ( u )=0 in D that are equal to 0 on the boundary of D . We call such solutions special. For the equation ( Δ + c ) u = u α , where Δ is the Beltrami–Laplace operator on a complete Riemannian manifold E , strictly positive solutions in E were studied in connection with a geometrical problem: Which two functions represent scalar curvatures of two Riemannian metrics related by a conformal mapping (see [1] and references there)?

We investigate positive solutions of a nonlinear equationLu=uαwhereLis a second order elliptic differential operator in a Riemannian manifoldEand 12. We establish a 1-1 correspondence between the set U of positive solutions and a class Z of functionals ofXwhich we call linear boundary functionals (they depend only on the behavior ofXnear the Martin boundaryE′). The class Z is a closed convex cone andu∈U is a subadditive function ofZ∈Z. Special roles belong to moderate solutions corresponding toZwith finite mathematical expectations and to a family of solutions determined by the range ofX. A new formula is deduced connectingu,ZandL-diffusions conditioned to hit the boundaryE′ at a given pointy. A concept of a singular boundary point foruis introduced in terms of the conditioned diffusion.

Let L be a second order elliptic differential operator in Rd with no zero order terms and let E be a bounded domain in Rd with smooth boundary 'E. We say that a function h is L-harmonic if Lh = 0 in E. Every positive L-harmonic function has a unique representation h(x) J(x, y)i(dy), E where k is the Poisson kernel for L and z' is a finite measure on &E. We call z the trace of h on ME. Our objective is to investigate positive solutions of a nonlinear equation Lu = u' in E for 1 2]. We associate with every solution u a pair (IF, i), where F is a closed subset of &E and zis a Radon measure on 0 -E \ F. We call (F, zi) the trace of u on ME. F is empty if and only if u is dominated by an L-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair (F, z') to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.

We investigate the minimal solution of the problem $$ \eqalign{ Lu&=u^\ap\q\hbox{in}\ D, \cr u&=f\q\hbox{on}\ O, \cr} $$ where $1

Let L be a second order elliptic differential operator on a Riemannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh 0. Every positive L-harmonic function has a unique representation

We establish a relation between a diffusion ξ and the superdiffusion X corresponding to the spatial motion ξ and branching parameter α. It involves, on one hand, the range R and linear additive functionals of X and, on the other hand, h -transforms of process ξ. By using this relation, we investigate boundary singularities of solutions of the semilinear partial differential equation Lu=u α .

We describe some directions of research in probability theory and related problems of analysis to which S. E. Kuznetsov has made fundamental contributions.A Markov process (understood as a random path X t , 0 ≤ t < ∞ such that past before t and future after t are independent given X t ) is determined by a probability measure P on a path space. This measure can be constructed starting from a transition function and probability distribution of X 0. For a number of applications, it is also important to consider a path in both, forward and backward directions which leads to a concept of dual processes. In 1973, Kuznetsov constructed, as a substitute for such a pair of processes, a single random process (X t , ℙ) determined on a random time interval (α, β). The corresponding forward and backward transition functions define a dual pair of processes. A σ-finite measure ℙ became, under the name “Kuznetsov measure,” an important tool for research on Markov processes and their applications.

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

The Martin boundary theory allows to describe all positive solutions of a linear elliptic equation in an arbitrary domain E of a Euclidean space R. Our goal is to describe all positive solutions of a semilinear equation Lu = ψ(u). As a result of efforts of probabilists and analysts since early 1990s, now we have a solution of this problem for the equation ∆u = u with 1 < α ≤ 2 in a bounded smooth domain E. The present article contains an exposition of the theory developed to obtain this solution. 1 The central role is played by the boundary trace theory. A survey of this theory is given in Part One. In Part Two we outline the principal steps needed to construct an arbitrary positive solution starting from its trace. Our main probabilistic tool is (L,ψ)-superdiffusions.