Kai Lai Chung

Markov processes, Brownian motion, and time symmetry

Markov Process.- Basic Properties.- Hunt Process.- Brownian Motion.- Potential Developments.- Generalities.- Markov Chains: a Fireside Chat.- Ray Processes.- Application to Markov Chains.- Time Reversal.- h-Transforms.- Death and Transfiguration: A Fireside Chat.- Processes in Duality.- The Martin Boundary.- The Basis of Duality: A Fireside Chat.

Introduction to random time and quantum randomness

Introduction to Random Time: Prologue Stopping Martingale Stopped Random Past and Future Other Times From First to Last Gapless Time Markov Chain in Continuum Time The Trouble with Infinite Introduction to Quantum Randomness: Classical Prologue Standard Quantum Mechanics Probabilities in Standard Quantum Mechanics Feynmans Approach to Quantum Probabilities Schrodingers Euclidean Quantum Mechanics Beyond Feynmans Approach Time for a Dialogue.

Doubly-Feller Process with Multiplicative Functional

Despite the common use of the term “Feller property”, there are variations in its definition. In the early literature on Markov processes, there are discussions of this and related properties, often under sets of bewildering assumptions. The coast should now be clear, but certain neat formulations may have been overlooked. In §1 of this note, some old results are reviewed in more general forms and an apparently new one is derived. In §2, the results are extended to include a multiplicative functional, of which the prime example is that of Feynman-Kac, properly generalized.

Contributions to the theory of Markov chains. II

Introduction. This is a sequel to my paper [1]. The present developments are largely independent of the previous results except in so far as given in the Appendix. Theorem 1 shows a kind of solidarity among the states of a recurrent class; it generalizes a classical result due to Kolmogorov and permits a classification of recurrent states and classes. In ?2 some relations involving the mean recurrence and first passage times are given. In ??3-5 sequences of random variables associated in a natural way with a Markov chain are studied. Theorem 2 is a generalized ergodic theorem which applies to any recurrent class, positive or null. It turns out that in a null class there is a set of numbers which plays the role of stationary absolute probabilities. In the case of a recurrent random walk with independent, stationary steps these numbers are all equal to one and the result is particularly simple. Theorem 3 shows that the kind of solidarity exhibited in Theorem 1 persists in such a sequence; it leads to the clarification of certain conditions stated by Doblin(2) in connection with his central limit theorem. Using a fundamental idea due to Doblin, the weak and strong laws of large numbers, the central limit theorem, the law of the iterated logarithm, and the limit theorems for the maxima of the associated sequence are proved very simply. Owing to the great simplicity of the method it is the conditions of validity of these limit theorems that should deserve attention. Among other things, we shall show by an example that a certain set of conditions, attributed to Kolmogorov, is in reality not sufficient for the validity of the central limit theorem. Furthermore, conditions of validity for the strong limit theorems and the limit theorems for the maxima are obtained by a rather natural strengthening of corresponding conditions for the weak limit theorems. A word about the connection of these conditions with martingale theory closes the paper. 1. The sequence of random variables {Xn}, n=0, 1, 2, * , forms a denumerable Markov chain with stationary transition probabilities. The states will be denoted by the non-negative integers(3) 0, 1, 2, * * . The n-step transition probability from the state i to the state j will be denoted by P(n) (P(l) = Pij). Thus we have

On the recurrence of sums of random variables

1. S. Banach, Theorie des operations lineaires, Warsaw, Monogr. Mat., Tom 1,1932. 2. R. V. Kadison, Isometries of operator algebras, Ann. of Math. vol. 54 (1951) pp. 325-338. 3. M. A. Krasnoselski and Ya. Ruticki, Convex functions and Orlicz spaces (in Russian), Moscow, Gosudarstv. Izdat. Fiz.-Mat. Lit., 1958. 4. J. Lamperti, On the isometries of certain f unction spaces, Pacific J. Math. vol. 8 (1958) pp. 459-466. 5. G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. vol. 100 (1961) pp. 29-43. 6. , Semi-inner-product spaces and isometries, (to be published). 7. W. Orlicz, fiber eine gewisse Klasse von Raumen von Typus B, Bull. Inst. Acad. Polon. Sci. Ser. A (1932) pp. 207-220. 8. M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. vol. 41 (1937) pp. 375-481.

Seminar on Stochastic Processes, 1992

Categorical combinatorics sequential algorithms CDSO - the kernel of a functional language the full abstraction problem conclusion mathematical prerequisites.

Downcrossings and local time

and computed the second moment of the difference of the expressions in (1) and (2). In this paper, by examining the excursions in Brownian motion and using a new formula for the distribution of their maxima, we obtain a direct identification of the limit in (1) without using (2). Let Tx=inf{t: W(t )=x} , Tx=inf{t: Y( t )=x} . For a > 0 let R~=0, R~= T/, + T/~ o Or,~, and for n> 2 let R~= R~,_ 1 + R~ o OR~_ 1 . Here {0~, t>0} is the usual collection of shift operators: W(s, Ot co) = W(s + t, co) and if S is a random variable, Os=O t on {S=t}. If S is a random variable, let d , (S )=sup{n: R~, 0 for ~ < s < fl; { Y(s, co), c~ < s < fl} is called an excursion if

Maxima in Brownian excursions

2. S. I. Grossman and R. K. Miller, Nonlinear Volterra integro-differential systems with L kernels, J. Differential Equations 13 (1973), 551-566. 3. M. J. Leitman and V. J. Mizel, Hereaitary laws and nonlinear integral equations on the line, J. Differential Equations (to appear). 4. J. L. Levin and J. A. Nohel, Perturbations o f a nonlinear Volterra equation, Michigan Math. J. 12 (1965), 431-447. MR 32 #336. 5. R. C. MacCamy, Remarks on frequency domain methods for Volterra integral equations, J. Math. Anal. Appl. (to appear) 6. R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164 (1972), 1-37. MR 45 #2432. 7. R. K. Miller, Nonlinear Volterra integral equations, Benjamin, New York, 1971. 8. G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341-346. MR 29 #6319.

An extension of renewal theory

Renewal theory can be and has been viewed in several different ways. One way is to reduce the problems to those concerning the addition of independent, non-negative random variables having a common distribution. Accordingly we introduce the random variables X1, X2, * * *, independent, non-negative, and all possessing the same distribution function F(x) with mean m-=f.r.xdF(x) (O<m =< X), and their successive sums S. = Et=1 X. Either F(x) is purely discontinuous with all its discontinuities located at the multiples of a fixed real number or it is not so; we call the first case the lattice case and the second the nonlattice case. In the lattice case there is no real loss of generality by assuming (as we shall do in the following) the said discontinuities are all located at integers whose greatest common divisor is one. One of the main results of renewal theory can then be stated as follows: (i) In the lattice case, if x runs through integers:

Greenian bounds for Markov processes

In Section 1, a temporal bound is estimated by a spatial bound, for a Markov process whose transition density satisfies a simple condition. This includes the Brownian motion, for which comparison with a more special method is made. In Section 2, the result is related to the Green operator and examples are given. In Section 3, the result is applied to an old problem of eigenvalues of the Laplacian. In Section 4, recent extensions from the Laplacian to the Schrödinger circle-of-ideas are briefly described. In this case, time is measured by an exponential functional of the process, commonly known under the names Feynman-Kac.This article was written as script for two talks, one general and one technical, given in Taiwan in January 1991. In the spirit of the occasion, the style of exposition is deliberately paced and discursive.