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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.

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.

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

One-dimensional temporally homogeneous strong Markov processes with continuous path functions are considered. No regularity conditions are assumed. Infinitesimal operators for all these processes are calculated. These calculations are based on a preliminary analysis of the local behaviour of path functions. The general results of sections 1—3 are used in sections 4—5 for conducting a more detailed investigation of the process in intervals of its regularity.Some results contained in this paper have been published in [3] without proofs.

Let $\mathcal{E}$ be a metric space, and suppose that $\mathfrak{B}$ is the Borel field generated by the open sets of $\mathcal{E}$. A stochastic process is defined on $\mathcal{E}$ if a function $x(t,\omega )$$(0 \leqq t < \infty ,\omega \in \Omega )$ and a system of probability measures ${\bf P}_x (x \in \mathcal{E})$ are given such that all ${\bf P}_x $ are defined on the Borel field generated by sets of type (1), and ${\bf P}_x \{ x(0,w) = x\} \equiv 1$. A random variable $\tau (\omega )$ is said to be independent of the future if for every s the $\omega $-set $\{ \tau (\omega ) \leqq s\} $ belongs to the Borel field generated by sets of type (1) with $t \leqq s$. A measurable stochastic process on $\mathcal{E}$ is called a strong Markov homogeneous process if for any $\Gamma _1 , \cdots ,\Gamma _n \in B,0 < t_1 < t_2 < \cdots < t_n $, and any $\tau $ independent of the future (5) is true for almost all $\omega $ such that $\tau (\omega ) < \infty $. Every strong Markov homogeneous process is a Markov...

In this paper the relations between various semigroups of operators and between various infinitesimal operators connected with a homogeneous in t Markov process are investigated. General conditions are established under which the Markov process is determined by its corresponding infinitesimal operator.Let $U_t $ be a semigroup of linear operators in the Banach space L such that $\|U_t \| \leqq 1$. Let $T_t = U_t $ be an adjoint semigroup in the conjugate space $B = L^ * $. More abstractly the main object of this paper can be characterized as the study of semigroups $T_t $ and its infinitesimal operators in strong and weak topologies of space B.

A Markov process $x(t,w),t \geqq 0,\omega \in \Omega $, on a measurable space $(\mathcal{E},\mathfrak{B})$ is called a discontinuous process, if for every $\omega \in \Omega $ and $t \geqq 0$ there exists an $\varepsilon > 0$ such that $x(t,\omega ) = x(t + h,\omega )$ for all $h \in (0,\varepsilon ]$. In this paper infinitesimal operators of all discontinuous processes are calculated. The results of these calculations imply the step-function processes described.

In 1931 A. Kolmogorov showed [9] that a wide class of one dimensional Markov processes can be described by the differential equation \[(1)\qquad \frac{{\partial u}}{{\partial t}} + a\frac{{\partial ^2 u}}{{\partial x^2 }} + b\frac{{\partial u}}{{\partial x}}.\]Are there one-dimensional Markov processes governed by equations of the type \[ (2)\qquad \frac{{\partial u}}{{\partial t}}\mathfrak{A}u, \] where $\mathfrak{A}$ is a differential operator of an order higher than 2? This question remained unsettled until 1954–1955 when it was completely solved by W. Feller. Feller showed that every one-dimensional Markov process with a continuous path function is described by (2), where $\mathfrak{A}$ is the generalized second derivative.The purely analytical method of Feller is essentially connected with the one-dimensional character of the problem. It is very difficult to extend this method to the case of two (and more) dimensions.In this paper, a method is developed which can be applied to n dimensions as well as...

E. B. Dynkin in [2] has defined a large class of transformations of Markov processes. He calls processes obtained by these transformations $\mathop X\limits^ \sim $ and $(\alpha ,\xi )$-subprocesses of the process X. In this paper we prove that if the process X is a strong Markov process, then, with some conditions imposed on $\alpha $ and $\xi $, the $(\alpha ,\xi )$-subprocess will also be a strong Markov process. One special case of a $(\alpha ,\xi )$ subprocess has been examined earlier by E. B. Dynkin in his book [1]. He proved in particular that the strong Markov property does not change in the formation of subprocesses. We show that the method he used for the special case can be carried over to the general case.

February 27, 2007, was the 75th birthday of the eminent mathematician, Professor Vladimir Mikhailovich Zolotarev. In 1949, Vladimir entered the faculty of mechanics and mathematics of Moscow State University. As his specialization field he chose probability theory and began his studies under the supervision of Eugene Borisovich Dynkin. After graduating from the university he was recommended to graduate studies, where his advisor was Andrei Nikolaevich Kolmogorov. Other distinguished mathematicians also have had a potent effect on Zolotarev's mathematical talent. Later, he mentions more than once not only his teachers E. B. Dynkin and A. N. Kolmogorov, but also B. V. Gnedenko and Yu. V. Linnik. In his graduate studies, Vladimir begins to study the properties of stable distributions. He continues to be interested in this theme even today. At first he was dealing with the stable distributions in the scheme of summation of independent identically distributed random variables. Later, he extended the concept of...

March 6, 2006, was the $75$th birthday of outstanding Russian mathematician, Academician, ALEXANDER ALEKSEEVICH BOROVKOV. Alexander Alekseevich Borovkov was born in Moscow into the family of an aircraft designer. After graduating from the Faculty of Mechanics and Mathematics of the Lomonosov Moscow State University, A. A. Borovkov worked for a number of years at an applied research institution. At the same time, Alexander Alekseevich continued to participate in research seminars on probability theory, which were convened by A. N. Kolmogorov and E. B. Dynkin, and after a while he was admitted as a postgraduate student into the Steklov Mathematical Institute of the Academy of Sciences of the USSR. His supervisor was Andrei Nikolaevich Kolmogorov, whose influence played a large role in forming the outlook and research interests of the young scientist. In 1959, A. A. Borovkov defended his Cand. Sci. (Ph.D.) thesis, and beginning in 1960 he worked at the Institute of Mathematics, Siberian Branch of the Academy...

The concept of an excessive function connected with a Markov process has been introduced by G. A. Hunt in [3]. Afterwards this class of functions was studied by E. B. Dynkin [2] and other authors.The purpose of this paper is to give a local criterion for excessive functions connected with a process. The class of processes we consider is the same as in [2] and [3].

In this paper a pair of random processes $X_t $, $Y_t $, which conjunctly form the Markov process $Z_t $ is considered. The conditional distribution of the process $Y_t $ for the condition of a known realization of the process $X_t $ during some time interval is examined. E. B. Dynkin has proposed that if $X_t $ is a Markov process, the conditional distribution of $Y_t $ will satisfy a functional equation similar to the known Kolmogorov-Chapman equation. The author has proved this proposition, but details of the proof are omitted here.