"Multidimensional Diffusion Processes" is a book by Daniel W. Stroock and S.R. Srinivasa Varadhan, first published in 1997. It is a reprint of the original edition and is part of the "Grundlehren der mathematischen Wissenschaften" series. The book is a comprehensive study of multidimensional diffusion processes, which are mathematical models used to describe the movement of particles or other entities in a continuous space. The book is divided into 12 chapters, each covering different aspects of diffusion processes. The first chapter introduces the necessary preliminary material, including extension theorems, martingales, and compactness. The second chapter discusses Markov processes, the regularity of their sample paths, and the Wiener measure. The third chapter covers parabolic partial differential equations, which are used to model diffusion processes. The fourth chapter introduces the stochastic calculus of diffusion theory, including Brownian motion, equivalence of certain martingales, Itô processes, and Itô's formula. The fifth chapter discusses stochastic differential equations, including existence and uniqueness, the Lipschitz condition, and the equivalence of different choices of the square root. The sixth chapter presents the martingale formulation of diffusion processes, including existence, uniqueness, the Cameron-Martin-Girsanov formula, and random time change. The seventh chapter discusses uniqueness in more detail, including local and global cases. The eighth chapter covers Itô's uniqueness and uniqueness to the martingale problem. The ninth chapter provides some estimates on the transition probability functions, while the tenth chapter discusses explosion, which refers to the possibility of a diffusion process reaching infinity in finite time. The eleventh chapter covers limit theorems, including the convergence of diffusion processes and Markov chains to diffusions. The twelfth chapter discusses the non-unique case, including the existence of measurable choices and Markov selections. The book also includes an appendix with some estimates for singular integral operators and a bibliography. The book is written for mathematicians and researchers in the field of probability theory and stochastic processes. It is a valuable resource for those interested in the theory and applications of diffusion processes."Multidimensional Diffusion Processes" is a book by Daniel W. Stroock and S.R. Srinivasa Varadhan, first published in 1997. It is a reprint of the original edition and is part of the "Grundlehren der mathematischen Wissenschaften" series. The book is a comprehensive study of multidimensional diffusion processes, which are mathematical models used to describe the movement of particles or other entities in a continuous space. The book is divided into 12 chapters, each covering different aspects of diffusion processes. The first chapter introduces the necessary preliminary material, including extension theorems, martingales, and compactness. The second chapter discusses Markov processes, the regularity of their sample paths, and the Wiener measure. The third chapter covers parabolic partial differential equations, which are used to model diffusion processes. The fourth chapter introduces the stochastic calculus of diffusion theory, including Brownian motion, equivalence of certain martingales, Itô processes, and Itô's formula. The fifth chapter discusses stochastic differential equations, including existence and uniqueness, the Lipschitz condition, and the equivalence of different choices of the square root. The sixth chapter presents the martingale formulation of diffusion processes, including existence, uniqueness, the Cameron-Martin-Girsanov formula, and random time change. The seventh chapter discusses uniqueness in more detail, including local and global cases. The eighth chapter covers Itô's uniqueness and uniqueness to the martingale problem. The ninth chapter provides some estimates on the transition probability functions, while the tenth chapter discusses explosion, which refers to the possibility of a diffusion process reaching infinity in finite time. The eleventh chapter covers limit theorems, including the convergence of diffusion processes and Markov chains to diffusions. The twelfth chapter discusses the non-unique case, including the existence of measurable choices and Markov selections. The book also includes an appendix with some estimates for singular integral operators and a bibliography. The book is written for mathematicians and researchers in the field of probability theory and stochastic processes. It is a valuable resource for those interested in the theory and applications of diffusion processes.