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Brownian-time processes: The PDE connection and the half-derivative generator. (English) Zbl 1018.60066

Let \(B(t)\) be a one-dimensional Brownian motion starting at 0 and let \(X^x(t)\) be an independent \(\mathbb{R}^d\)-valued continuous Markov process started at \(x\), both defined on a probability space \((\Omega,{\mathcal F},\{{\mathcal F}_t\},\mathbb{P})\). The process \({\mathbf X}^x_B(t)=X^x(|B(t)|)\) is called a Brownian-time process. The authors study the connection between these processes and their exit distribution to fourth order parabolic and to second and fourth order elliptic partial differential equations. It is also shown that it is possible to assign a formal generator to these processes by giving such a generator in the half-derivative sense.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
60J45 Probabilistic potential theory
60J35 Transition functions, generators and resolvents
60J60 Diffusion processes
60J65 Brownian motion
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References:

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