This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Diffusions, Markov processes, and martingales

Introduction Parabolic equations and branching exit Markov systems: Linear parabolic equations and diffusions Branching exit Markov systems Superprocesses Semilinear parabolic equations and superdiffusions Elliptic equations and diffusions: Linear elliptic equations and diffusions Positive harmonic functions Moderate solutions of $Lu=\psi(u)$ Stochastic boundary values of solutions Rough trace Fine trace Martin capacity and classes $\mathcal{N}_1$ and $\mathcal{N}_0$ Null sets and polar sets Survey of related results Basic facts of Markov processes and Martingales Facts on elliptic differential equations Epilogue Bibliography Subject index Notation index.

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Diffusions, Superdiffusions and Partial Differential Equations

Finite state continuous time Markov decision processes with an infinite planning horizon

Gaussian and non-Gaussian random fields associated with Markov processes

We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous-time simple random walk on $\mathbb{Z}^d$, with jump rate $D_A$. These particles are called $A$-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with $N_A(x,0-)$ $A$-particles at $x$, and that the $N_A(x,0-),x\in\mathbb{Z}^d$, are i.i.d., mean-$\mu_A$ Poisson random variables. In addition, there are $B$-particles which perform continuous-time simple random walks with jump rate $D_B$. We start with a finite number of $B$-particles in the system at time 0. $B$-particles are interpreted as individuals who have heard a certain rumor or who are infected. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if $\widetilde{B}(t):=\{x\in\mathbb{Z}^d:$ a $B$-particle visits $x$ during $[0,t]\}$ and $B(t)=\widetilde{B}(t)+[-1/2,1/2]^d$, then we investigate the asymptotic behavior of $B(t)$. Our principal result states that if $D_A=D_B$ (so that the $A$- and $B$-particles perform the same random walk), then there exist constants $0<C_i<\infty$ such that almost surely $\mathcal{C}(C_2t)\subset B(t)\subset \mathcal{C}(C_1t)$ for all large $t$, where $\mathcal{C}(r)=[-r,r]^d$. In a further paper we shall use the results presented here to prove a full ``shape theorem,'' saying that $t^{-1}B(t)$ converges almost surely to a nonrandom set $B_0$, with the origin as an interior point, so that the true growth rate for $B(t)$ is linear in $t$. If $D_A\ne D_B$, then we can only prove the upper bound $B(t)\subset \mathcal{C}(C_1t)$ eventually.

The spread of a rumor or infection in a moving population

Theory of Markov Processes.

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Theory of Markov Processes.