Analytic and Geometric Stochastics: Papers in Honour of G. E. H. Reuter || Harry Reuter: An Appreciation

Harry Reuter: An AppreciationAuthor(s): David KendallSource: Advances in Applied Probability, Vol. 18, Analytic and Geometric Stochastics: Papersin Honour of G. E. H. Reuter (Dec., 1986), pp. 1-7Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/20528771 .

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Alm' ^^^H

IS^'i^^H

Professor G. E. H. Reuter



Harry Reuter: An Appreciation

Harry Reuter was born in Berlin on 21 November 1921, the son of Ernst

Reuter and his first wife Charlotte. This was the Ernst Reuter who was to

inspire, guide and comfort the people of West Berlin in the difficult years after

the Second World War, and to whom Europe as a community owes much.

Harry arrived in England as a refugee in 1935, when he was 13, to be received

into the extended family of Charles and Greta Burkill of Cambridge. Rumour

has it that Greta wanted to be foster mother to many such young people and

that Charles added the stipulation that they must show promise in mathe

matics. Harry was educated at the Leys School and later at Trinity College,

Cambridge. He took Part II of the Mathematical Tripos in 1941, and then

joined the Admiralty for the war years as a Scientific Officer. In 1945 he was

back at Trinity as a research student, interacting with J. E. Little wood and

M. L. Cartwright, but with Frank Smithies as his official supervisor. From Frank

he learned functional analysis (which he was later to hand on to me). A year later ('did not complete Ph.D.') he joined the famous team of young

mathematics lecturers built up at that time in Manchester by Max Newman, and he stayed there till 1959 (finishing as Senior Lecturer).

When I was in Princeton in 1952, along with Brian Murdoch and James

Taylor, Harry had started work in probability with Walter Ledermann, and

they sent me a copy of their famous paper [11] which I showed to Will Feller, whose reaction was 'delight mingled with fury'. For they presented a complete and elegant account of the existence theory for countable-state Markov

processes with a stable conservative generator by a novel truncation technique which Feller had tried earlier, and had abandoned because 'I convinced myself that it could not work'. This paper was quickly followed by one [12] which

explored the family of Markov processes of birth-and-death type with an

arbitrary tri-diagonal q-matrix, using spectral techniques exploiting the sym metrisable character of the q-matrix, and showing how various modifications in

the truncation method can lead on occasion to distinct Markov processes. This

work, enormously in advance of its time, was published in the Royal Society's

Philosophical Transactions. Its full significance was seen later in the context of

the similar definitive work by Karlin and McGregor. I met Harry first at the BMC in Durham in 1953, and we immediately began

to work together on q-processes, as we called them. The theme of those early

papers was to put the Hille-Yosida theory of semigroups of bounded linear

1



2 HARRY REUTER

operators to work in the elucidation of q -processes for which the generator was

either not stable, or not conservative, or both. Kolmogorov had recently constructed two specific examples of such pathological q -processes which

became known as Kl and K2, and in [13] we presented a detailed analysis of

these from the new point of view in the volumes published after the

Amsterdam ICM in 1954. From this we went on to ergodic theorems, and

rounded off our joint work by solving the problem of determining the limits

Pij( ) directly from a knowledge of the generator. All this work was done by correspondence between Manchester and Oxford,

supplemented by conferences which we both attended (and neglected) in order to

be able to work together without GPO intervention. During these years our

two families twice enjoyed joint seaside holidays at Rhosneigr in the Isle of

Anglesey. After these events our interests swerved apart, but while I sowed wild oats,

writing about comets and pre-dynastic egyptian pots, Harry kept on working at

more and more difficult problems concerning existence theorems for anoma

lous #-processes. As this (at least at first) was before the development of

'boundary theory', his results had a very great influence on the evolution of the

subject, and even after Feller's and Chung's boundary-theoretic investigations his papers from this period still in specific contexts often provide the best

approach. This is well exemplified in recent work by the Changsha school on

the 'germ' problem, and in the latest contributions to that topic by Harry himself [32].

In 1959 Harry became Professor of Pure Mathematics (and later Head of

Department) at Durham. Though not so directly working together we still kept in very close touch. Thus we both took part with John Hammersley in an

operation conceived to cheer up mathematical schoolteachers by introducing them to (for example) Markov chains and their applications, we were both

involved in the LMS Instructional Conference on Probability, and we both

attended the 'Probability Methods in Analysis' symposium organised by D. A.

Kappos in Loutraki in the Gulf of Corinth. This included a free weekend in

which we made our way to Diakofton despite the train strike, and there (aided now by the train strike) walked up the sleepers of the airy (and almost aerial)

trackway threading the Diakofton gorge to the Megaspelion Monastery, where

we spent the night. We had intended climbing Mount Chelmos, and plumbing the sources of the Styx, but fortunately the fates were against us and we

returned in safety. At about this time Harry and family spent a year in Yale, and this made the

opportunity for the first meeting between Feller and Reuter. Long afterwards I

asked Feller how they got on. 'It was love at first sight', he replied. I am sure



An appreciation 3

Harry would want to join me in saying how much we both owe to Will Feller

for his generous encouragement and appreciation.

Throughout this whole period we toyed from time to time with trying to get to the same university, but it never seemed to work out; at last, in retirement,

we have happily succeeded. So we had the idea of starting the very informal

Stochastic Analysis Group in order to promote probabilistic activity within

both the London Mathematical Society and the Royal Statistical Society; it

quietly expired when the task was done, and amazingly this did not take long. We were aided greatly in this by the LMS President, Dame Mary Cartwright,

by the Treasurers of the LMS and of the Medical Research Council (both of

whom were the mathematician Sir Edward Collingwood), and by energetic

support from James Taylor and John Kingman. We were also helped by Joe

Gani who conceived the idea of the Applied Probability journals at about this

time. Harry has been associated with these as Editor and/or Trustee ever

since.

Three dei ex machina must be mentioned here, who as friends and

colleagues of us both enhanced our own friendship and collaboration. One of

these was David Williams whom I came to know as a crab-fishing schoolboy on

our family holidays in Gower, who became first my research pupil and then

Harry's, and who was eventually to succeed me in Cambridge. Another was

Rollo Davidson, a dazzling light too soon extinguished. After Rollo's death

Harry became a Foundation Trustee of the Rollo Davidson Memorial Trust, and his wisdom and perception has guided a long series of awards. Some

people do not readily grasp the criteria governing the award of Davidson

prizes; the chief consideration, always in our minds, is 'what sort of thing would Rollo have become interested in by nowV. The third close friend and

colleague I wish to mention is John Kingman. It was not part of the STAG

program to produce probabilists who would head the SERC and become

Vice-Chancellors, but if we had thought of this we might well have added it to

our list of aims. This is the place to thank both David and John for their help in producing this volume. I also wish to thank Joe Gani, the Applied

Probability Trust, and the London Mathematical Society for making it

possible, and Mavis Hitchcock for translating that possibility into actuality. In 1965 Harry moved to Imperial College London, first as Professor and

then as Head of its prestigious Department of Mathematics. This appointment can now be seen as a culminating success of the STAG program. Young people who slip easily today into university posts in probability don't know (and often

won't believe) that 'twas not always thus. They largely have Harry to thank for

the change. As the list of publications on pp. 5-6 will show, Harry throughout his career



4 HARRY REUTER

has kept in close touch with practical problems, writing lucid papers which had

a profound effect through their clarity and insight. An especially important

example is the article [3] written with Sondheimer; this is still regarded as a

classic in the field and is in addition often referred to as a particularly happy collaboration between a mathematical physicist and a pure mathematician.

A little earlier I mentioned the encouragement we both received from

Collingwood who, wearing his MRC hat, brought about a conference in 1965

on Mathematics and Computer Science in Biology and Medicine. This included

a discussion, based on a not very realistic approximation, of the reaction

diffusion equation controlling the spatial spread of epidemics. Harry became

very interested in this topic and later, with Atkinson and Ridler-Rowe, wrote

two startling papers [29] and [30] which gave the exact solution.

About ten years ago Robert and Mary Rankin very kindly lent a cottage in

the Isle of Arran to Harry, my son Wilfrid, and myself, and here, in addition

to admiring eider ducks, we pored over a manuscript recently sent to Harry by Professor Hou Zhen-ting of Changsha in the People's Republic of China.

Harry saw at once that this represented a breakthrough in ^-process theory on

the unicity front, and a significant chain of events then began to unfold. First, Hou was awarded a Davidson Prize for this remarkable paper. Then he came

to London to collect the prize (in the form of books), and there met us both, and broached the possibility of visits to China. A little after this David

Williams and I went to China at Hou's invitation, lecturing in Guangzhou,

Changsha, Xiangtan, and Xi'an, and met enthusiastic audiences armed with

photocopies of all our works and asking us 'what happens next'. The emotional

impact was terrific, and we returned to England (and Wales) in love with the

Chinese people, and especially with her young mathematicians. Now one of

these is in Cambridge and is a contributor to this volume. I am convinced that

this story of Anglo-Chinese collaboration in probability theory will in its

continuation prove worthy of its romantic beginning. This little book was composed as a birthday greeting by a random small

sample of Harry's admirers. As it travels out into the world it will certainly reach many readers who would have wished to be authors. Write and tell him

so!

University of Cambridge March 1986 David Kendall



Publications of G. E. H. Reuter

[1] An inequality for integrals of subharmonic functions. /. London Math. Soc. 23 (1948), 56-8.

[2] (with A. B. Pippard and E. H. Sondheimer) The conductivity of metals at microwave

frequencies. Phys. Rev. 73 (1948), 920-921.

[3] (with E. H. Sondheimer) The theory of the anomalous skin effect. Proc. R. Soc. London A

195 (1948), 336-364.

[4] (with I. J. Good) Bounded integral transforms. Quart. J. Math. 19 (1948), 224-234.

[5] The boundedness of the Hermite orthogonal system. J. London Math. Soc. 24 (1949), 159-160.

[6] Subharmonics in a non-linear system. Quart. J. Mech. Appl. Math. 2 (1949), 198-207.

[7] Note on the preceding paper (by F. W. J. Olver). Quart. J. Mech. Appl. Math. 2 (1949), 457-9.

[8-9] Boundedness theorems for non-linear differential equations. I: Proc. Camb. Phil. Soc. 47

(1951), 49-54; II: J. London Math. Soc. 27 (1952), 48-58.

[10] Some non-linear differential equations. /. London Math. Soc. 26 (1951), 215-221.

[11] (with W. Ledermann) On the differential equations for the transition probabilities of Markov processes. Proc. Camb. Phil Soc. 49 (1953), 247-262.

[12] (with W. Ledermann) Spectral theory for the differential equations of simple birth and death

processes. Phil. Trans. R. Soc. London A 246 (1954), 321-369.

[13] (with D. G. Kendall) Some pathological Markov processes. Proc. Internat. Congr. Math.

(1954), Vol. Ill, 377-415.

[14] ?ber eine Volterrasche Integralgleichung. Arch. Math. 7 (1956), 59-66.

[15] A note on contraction semigroups. Math. Scand. 3 (1956), 275-280.

[16] (with D. G. Kendall) Some ergodic theorems. Phil. Trans. R. Soc. London A 249 (1956), 151-177.

[17] (with F. F. Bonsall) A fixed point theorem for transition operators. Quart. J. Math. 7

(1956), 244-8.

[18] Denumerable Markov processes and the associated contraction semigroups on /. Acta Math.

97 (1957), 1-46.

[19] (with D. G. Kendall) The calculation of the ergodic projection for Markov chains and

processes. Acta Math. 97 (1957), 103-144.

[20] Denumerable Markov processes (II). /. London Math. Soc. 34 (1959), 81-91.

[21] (with K. Stewartson) A non-existence theorem in magneto-fluid dynamics. Phys. Fluids 4

(1961), 276-7.

[22] Competition processes. Proc. 4th Berkeley Symp. Math. Statistic. Prob. (1961) Vol. II, 421-430.

[23] Denumerable markov processes (III). /. London Math. Soc. 37 (1962), 63-73.

[24] Null solutions of the Kolmogorov differential equations. Mathematika 14 (1967), 56-61.

[25] Note on resolvents of denumerable submarkovian processes. Z. Wahrscheinlichkeitsth. 9

(1967), 16-19.

[26] Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitsth. 13 (1969), 315-320.

[27] (with P. W. Riley) The Feller property for Markov semigroups on a countable state space. /.

London Math. Soc. (2) 5 (1972), 267-275.

[28] Denumerable Markov processes (IV): On C. T. Hou's uniqueness theorem for Q

semigroups. Z. Wahrscheinlichkeitsth. 33 (1976), 309-315.

[29] (with C. Atkinson) Deterministic epidemic waves. Math. Proc. Camb. Phil. Soc. 80 (1976), 315-330.



6 G. E. H. REUTER

[30] (with C. Atkinson and C. J. Ridler-Rowe) Traveling wave solutions for some non-linear

diffusion equations. SIAMJ. Math. Anal. M (1981), 880-892.

[31] (with T. J. Lyons) On exponential bounds for solutions of second-order differential

equations. Bull. London Math. Soc. 17 (1985), 139-143.

[32] On Kendall's conjecture concerning 0+ equivalence of Markov transition functions. Bull.

London Math. Soc. (submitted).



Research Students of G. E. H. Reuter

E. J. R. Archinard

A. M. Ben Lashiher

J. R. Choksi

A. G. Cornish

R. A. Doney D. J. Emery J. Hawkes (shared with S. J. Taylor) J. Lane

D. Mannion (shared with D. G. Kendall) J. Ortega-Sanchez C. J. Ridler-Rowe

P. W. Riley K. S. Slack

R. Trottnow (shared with Y. N. Dowker) D. Williams (shared with D. G. Kendall)



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Analytic and Geometric Stochastics: Papers in Honour of G. E. H. Reuter || Harry Reuter: An Appreciation