Dynkin & Yushkevich — Markov Processes: Theorems and Problems (1969)

Evgenii B. Dynkin and Aleksandr A. Yushkevich, both then at Moscow University. Plenum Press, New York, 1969, translated from the Russian by James S. Wood; the authors' foreword is dated 22 January 1966. 245 pages, four chapters plus an appendix, literature list and index.

Arrived by Telegram 2026-09-05, from the sources directory of a harvard-university course page — the same site that supplied mcmullen-probability-theory. The scan carries no text layer (a 2020 tiff2pdf job), so it was read page by page as images rather than extracted.

Held at raw/dynkin-yushkevich-markov-processes.pdf.

Not a course, and that is the point

The spoke's stochastic-processes cluster is three teaching sources that disagree about the order (wu-introduction-stochastic-processes, lawler-stochastic-calculus, gamarnik-advanced-stochastic-processes — see synthesis.md, Three entrances, one summit). This is a fourth entrance and it is a different kind of object. It grew out of Dynkin's 1962–63 Moscow lectures, recorded by M. B. Malyutov, S. A. Molchanov and M. I. Freidlin, then "radically revised" with problems added. It is organised by problem area, not by curriculum:

It runs the other way

The three courses in this spoke build analytic machinery and point it at probability, ending at Itô calculus and (for two of them) option pricing. This book goes probability → analysis. Chapter II proves the existence of a solution to the Dirichlet problem for the Laplace equation "in a broad class of domains" by probabilistic argument. Chapter IV derives boundary conditions for differential equations from the paths of a birth-and-death process.

The foreword's own footnote is precise about who did which direction, and is worth keeping because it dates the idea: the link between probability and the Dirichlet problem predates the general theory of Markov processes (Phillips and Wiener 1923; Courant, Friedrichs and Lévy 1928), was developed by Khinchin (1933) and Petrovskii (1934), and Doob derived a formula for the Dirichlet solution from Wiener paths in 1954 — but "Doob applied it in a direction converse to our own, namely, for the derivation of the properties of paths from theorems of analysis."

They also state their disagreement with Feller in method rather than in result. Feller is named as "the pioneer in the application of the probabilistic approach to boundary problems", and the objection is that "even though Feller is guided by probabilistic intuition, he still introduces all his constructions in purely analytical form." Dynkin and Yushkevich work from the properties of paths and from the characteristic operator. This matters here because mcmullen-probability-theory is explicitly a companion to Feller — the spoke now holds a text that follows Feller and a text that names what it will not take from him.

The measure-theory question, dodged a fourth way

synthesis.md promoted measure theory to growth edge 17 on 2026-08-16 because all three stochastic sources avoid it differently: indented as optional (Lawler), taught in passing from metric spaces (Gamarnik), or excluded from the prerequisites (Wu). This book gives the fourth way, and it is the only one stated as a decision:

"We have consciously avoided references in the main text to measure theory and measurability. The reader who has these concepts at his command will have no difficulty in grasping the presentation at a more rigorous set-theoretic level."

Not an omission and not a prerequisite list. An editorial choice, announced, with the claim that the rigorous reading is recoverable by a reader who already has the machinery. Four sources, four avoidances, zero constructions — which is the edge getting sharper rather than closer.

The problems carry content

"At the end of each chapter is a set of problems which go beyond the simple function of providing material for exercises in that they supplement the text proper and present certain new information." The example given: the Chapter III problems "include a discussion of the Martin boundary for a denumerable Markov chain." Chapter IV's problem set runs eleven pages (210–220). A reader who skips the exercises here loses results, not practice — a different contract from the courses in this spoke, where problem sets are assessment.

Ancillary analytical arguments are pushed to the Appendix "so as not to impede the main flow of the probabilistic discussions", which is the same instinct as Lawler's font-switch, solved by relocation instead of typography.

Chapter III is the cross-spoke one

The optimal stopping problem opens on "The Problem of Optimal Choice" — the secretary problem — and develops it into optimal stopping for a Markov chain via excessive functions, the value of a game and the optimal strategy, then applies it to a random walk with absorption and to the Wiener process. The authors flag the connection as unexpected: "The connections between Markov processes and potentials find an unexpected application in Chapt. III." ../optimization-algorithms-wiki owns optimisation method; this is the probabilistic side of stopping rules, and it stays here with the process theory.

Prerequisites, as stated

"All that is expected of the reader is an acquaintance with the basic tenets of probability theory and classical analysis. Some of the problems, however, require greater erudition."

Related

stochastic-processes · markov-chain · brownian-motion · martingale · probability-theory · mcmullen-probability-theory · wu-introduction-stochastic-processes · lawler-stochastic-calculus · gamarnik-advanced-stochastic-processes · harvard-university · mathematics · synthesis