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Infinite dimensional diffusions in modelling and analysis. (English) Zbl 0956.60083
From the author’s introduction: There exists an important interplay between the theory of parabolic and elliptic equations and the theory of Markov processes. This was first noticed by L. Bachelier, A. Einstein, M. Smoluchowski and N. Kolmogorov. Probabilistic interpretations of solutions to PDEs were very fruitful for analysis and the analytic theory provided tools for finding formulae for probabilistic quantities. This intimate connection guided L. Gross and Yu. Daletskij in their approaches to the theory of partial differential equations on Hilbert and Banach spaces. The infinite-dimensional theory is developing rapidly. The probabilistic objects behind the new theory are the so-called infinite-dimensional diffusions, which have appeared in physics, population biology, chemistry and economics. Modelling questions stimulated the development of the theory. The aim of the paper is to present some recent and typical results from the field in a rather self-contained way.
MSC:
60J60 Diffusion processes
35K10 Second-order parabolic equations
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