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A note on Lévy’s Brownian motion. (English) Zbl 0604.60080

This paper starts with the investigation of the canonical representation of the Lévy’s Brownian motion on a general plane curve C of \(C^ 3\)- class. It asserts, in particular, that the Brownian motion on a circle is a double Markov Gaussian process and its representation is obtained by solving a Riccati’s differential equation. The nature of the kernel function of the conditional expectation E[X(P)/X(A), \(A\in C]\) is discussed for some particular C, then the author proceeds to the determination of the kernel function for a general curve C. According to the main idea to find out the dependency of the Lévy’s Brownian motion, there naturally arises a variational problem on the conditional expectation.

MSC:

60J65 Brownian motion
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References:

[1] Processus stochastiques et Mouvment Brownein (1948)
[2] DOI: 10.1137/1108042 · Zbl 0124.08702
[3] Gaussian Processes, Kinokuniya Pub. Co (1976)
[4] Random functions: General theory with special reference to Laplacian random functions 1 pp 331– (1953) · Zbl 0052.14402
[5] Random functions: A Laplacian random function depending on a point of Hilbert space 2 pp 195– (1956) · Zbl 0070.14103
[6] Proceeding of the Third Berkeley Symposium on Math. Stat. and Prob 2 pp 133– (1956)
[7] Mem. Coll. Sci. Univ 33 pp 258– (1960)
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