Preprints

• A Remark on Kac's Scattering Length Formula
• On General Perturbations of Symmetric Markov Processes
• Potential Theory of Moderate Makov Dual Processes
• Converse Jensen Inequality
• Martingale Functions of Brownian Motion and its Local Time
• Lévy Systems and Time Changes
• On Joint Fourier-Laplace Transforms
• Time Change Approach to Generalized Excursion Measures, and its Application to Limit Theorems
• Stochastic Calculus for Symmetric Markov Processes (Originally titled "Stochastic Calculus for Dirichlet Processes")
• Perturbation of Symmetric Markov Processes
• The Dirichlet Form of a Gradient-type Drift Transformation of a Symmetric Diffusion
• Drift Transformations of Symmetric Diffusions, and Duality
• On the Existence of Recurrent Extensions of Self-similar Markov Processes
• Excursion Theory Revisited
• A New Approach to the Martingale Representation Theorem
• Superposition Operators on Dirichlet Spaces
• Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process
• Absolute Continuity of Symmetric Markov Processes
• Non-symmetric Perturbations of Symmetric Dirichlet Forms
• Excursions Above the Minimum for Diffusions
• On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin
• Class of smooth functions in Dirichlet spaces

Short Notes

• I present a proof of A. Jakubowski's result that in the Chebyshev correlation inequality (cov[f(X)g(X)] >= 0 for increasing f and g) equality implies that either f(X) is a.s. constant or g(X) is a.s. constant. Available as a pdf (83k) file. [April 26, 2024]
  
  
• A proof based on excursions below the maximum is given of J. Lehoczky's theorem on the the joint distribution of the drawdown time and the maximun up to that time, for a one-dimensional diffusion. Available as a pdf (187k) file. [March 13, 2020]
  
  
• We provide an alternative proof of the fact that Kemeny's constant, for a positive recurrent discrete-time Markov chain with countably infinite state space, is infinite. Available as a pdf (123k) file. [February 13, 2020]
  
  
• A simple proof of the observation that the tail-sum formula from probability theory holds for arbitrary measures. Available as a pdf (124k) file. [November 6, 2019]
  
  
• A simple proof of a general form of a result of A. Adamou and O. Peters on the advantage of cooperation in certain multiplicative growth models. Available as a pdf (98K) file. [July 24, 2019]
  
  
• I provide a simple non-combinatorial proof of two integral identities of N. Kimura and O.G. Ruehr, the subject of a recent note in the American Mathematical Monthly by J.P. Allouche. Available as a pdf (106K) file. [June 26, 2019]
  
  
• A recent improvement on the discrete classic L^2 Hardy inequality, due to Keller, Pinchover, and Pogorzelski, is shown to follow from a general Hardy inequality for Dirichlet forms. Available as a pdf (107K) file. [April 10, 2018]
  
  
• A filtering formula of G. Nappo and B. Torti, for the conditional distribution of reflected Brownian motion at a fixed time given the history of its local time (at 0) up to that time, is shown to be a special case of a general result in the excursion theory of Markov processes. Available as a pdf (48K) file. [June 28, 2012]
  
  
• I give a new proof of J. Wesolowski's characterization of the Poisson process: If X(t) is an increasing cadlag process with X(0)=0, such that Y(t) := X(t) - t, Y(t)² - t, and Y(t)³ -3t Y(t) -t are all local martingales, then X(t) is a unit-rate Poisson process. Available as a pdf (40K) file. [June 28, 2012]
  
  
• I give a simple proof of the Dragomir-Jensen inequality: For a probability measure P with mean m, and a convex function f, define V(P) = int f dP - f(m); If Q is a second probability measure such that dQ/dP <= k (a constant), then V(Q) <= kV(P). Available as a pdf (40K) file. [March 29, 2012]
  
  
• I show how a method of D.W. Stroock can be adapted to show that if f is an additive mapping from a Banach space E to another Banach space, and if f is measurable with respect to the completion of the Borel sigma-field of E relative to any centered Gaussian meausre on E, then f is continuous and hence linear. Available as a pdf (56K) file. [February 21, 2012]
  
  
• I discuss a multi-player Gambler's Ruin problem of S. M. Ross, using martingale ideas familiar from the classical Gambler's Ruin problem. Available in dvi (8K) and pdf (36K) formats. [January 6, 2010]
  
  
• Let B be a standard one-dimensional Brownian motion and let S_t be the maximum to time t of |B|. For positive p define Y_t=S^p-2_t [B²_t-t]+cS^p_t. We give a simple proof of the following result of B. Davis and J. Suh [On Burkholder's supermartingales, Illinois J. Math. 50 (2006) 313--322]: Define c₀ = (2-p)/p. Then (i) for p in (0,2], Y is a submartingale iff c >= c₀, and (ii) for p in [0,infinity), Y is a supermartingale iff c < c₀. Available in dvi (4K) and pdf (32K) formats. [September 1, 2006]
  
  
• Let X=(X_t)_{t >= 0} be a right Markov process with infinitesimal generator L and lifetime z. Let A_t = int₀^t a(X_s) ds be a continuous additive functional of X with A_z finite, and let f(x) be the expected value of exp(-A_z) as a function of the starting point x=X₀. We give a simple proof, based on the domination principle, that f is the minimal solution with values in [0,1] of the equation Lf=af. This extends a recent result of P.K. Pollett and V.T. Stefanov [Path integrals for continuous-time Markov chains. J. Appl. Prob. 39 (2002) 901-904.] Available in dvi (5K) and pdf (192K) formats. [June 13, 2003]
  
  
• Using Skorokhod stopping, we prove the converse of Jensen's inequality: If X and Y are integrable random variables with E[f(X)] greater than or equal to E[f(Y)] for every convex function f, then there are random variables X' and Y' equal in distribution to X and X respectively such that Y'=E[X'|Y']. Available in dvi (4K), postscript (29K), and pdf (179K) formats. [June 11, 2002]
  
  
• We give a simple proof of Grüss's inequality: If X is a bounded random variable with lower bound m and upper bound M, then the variance of X is bounded above by [(M-m)²]/4. Available in dvi (4K), postscript (29K), and pdf (193K) formats. [June 11, 2002]
  
  
• We provide a short proof of the fact that if X is a symmetric diffusion and u(X_t) is a Dirichlet process (basically, a process of finite quadratic variation) then u is locally in the Dirichlet space of X. Available in dvi (12K), postscript (76K), and pdf (76K) formats. [July 19, 2001]
  
  
• We present short proof that (aside from a deterministic non-atomic component) a completely random measure in the sense of Kingman [Completely random measures, Pacific J. Math., 21 (1967) 59-78. MR: 35#1079] is purely atomic, almost surely. Available in dvi (5K), postscript (49K), and pdf (160K) formats. [December 15, 2000]
  
  
• We present a simple approach to the approximation of a Markov process by pure jump processes, complementing recent work of Z.-M. Ma, M. Röckner, and T.-S. Zhang [Approximation of arbitrary Dirichlet processes by Markov chains. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 1--22. MR: 99c:60160] and Z.-M. Ma, M. Röckner, and W. Sun [Approximation of Hunt process by multivariate Poisson processes. Preprint, 1999]. Available in dvi (10K), postscript (72K), and pdf (86K) formats. [April 20, 2000]
  
  
• Using an integral representation theorem of the author [Skorokhod embedding by randomized hitting times, In: Seminar on Stochastic Processes 1990, Birkhäuser, Boston, pp. 183-191], we give a short proof of a result of Mokobodzki, characterizing the extreme points of the set of stopping distributions of a right Markov process. Available in dvi (4K), postscript (40K), and pdf (59K) formats. [November 18, 1999]
  
  
• We give a probabilistic proof of a recent result of A. Dunkels [A note on the equilibrium potential of certain Dirichlet spaces. Potential Anal. 6 (1997) 99--104], giving an explicit identification of the equilibrium measure of a Borel set on its (fine) interior. Dunkels dealt with symmetric Lévy processes in Euclidean space; our argument works for general symmetric Markov processes. Available in dvi (7K), postscript (54K), and pdf (102K) formats. [November 2, 1999]
  
  
• In a recent paper [Orthogonal measures and absorbing sets for Markov chains, Math. Proc. Camb. Phil. Soc. 121 (1997) 101--113], P.-D. Chen and R.L. Tweedie make use of the natural embedding of a separable and separated measurable space in the compact cube {0,1}^N. The point of this note is to show that a well-known representation of functions measurable with respect to the sigma-algebra generated by a sequence of functions can be substituted for the rather intricate arguments they employ. Available in dvi (8K), postscript (56K), and pdf (82K) formats. [March 24, 1999]
  
  
• If the sum of independent real-valued random variables is normally distributed, then so are the summands (Cramer-Lévy theorem). In this note an analogous result is proved for Brownian motion: If the sum of independent local martingales is Brownian motion, then the summands are (modulo deterministic time changes) also Brownian motion. Available in pdf form (116K). [September 29, 1995; corrected March 28, 2019]
  
  
• A new (?) proof of the fact that the Lebesgue measurable solutions of the functional equation f(x+y) = f(x) + f(y) are of the form f(x) = cx. In postscript form (45K) and in pdf form (3K). [September 29, 1995]