## 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:

 [1] Allouba, H. (1999). Measure-valued Brownian-time processes: the PDE connection. Unpublished manuscript. [2] Burdzy, K. (1993). Some path properties of iterated Brownian motion. In Seminar on Stochastic Processes (E. \?Cinlar, K. L. Chung and M. J. Sharpe, eds.) 67-87. Birkhäuser, Boston. · Zbl 0789.60060 [3] Burdzy, K. (1994). Variation of iterated Brownian motion. In Workshopand Conference on Measure-Valued Processes, Stochastic PDEs and Interacting Particle Systems 35-53. Amer. Math. Soc., Providence, RI. · Zbl 0803.60077 [4] Burdzy, K. and Khoshnevisan, D. (1995). The level sets of iterated Brownian motion. Séminaire de Probabilités XXIX. Lecture Notes in Math. 1613 231-236. Springer, Berlin. · Zbl 0853.60061 [5] Burdzy, K. and Khoshnevisan, D. (1998). Brownian motion in a Brownian crack. Ann. Appl. Probab. 8 708-748. · Zbl 0937.60081 [6] Durrett, R. (1996). Stochastic Calculus. A Practical Introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL. · Zbl 0856.60002 [7] Elworthy, K. D. (1982). Stochastic Differential Equations on Manifolds. Cambridge Univ. Press. · Zbl 0514.58001 [8] Funaki, T. (1979). Probabilistic construction of the solution of some higher order parabolic differential equation. Proc. Japan Acad. Ser. A Math. Sci. 55 176-179. · Zbl 0433.35039 [9] Getoor, R. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75-90. JSTOR: · Zbl 0104.11203 [10] Hochberg, K. (1996). Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theoret. Probab. 9 511-532. · Zbl 0878.60050 [11] Kinateder, K., McDonald, P. and Miller, D. (1998). Exit time moments, boundary value problems, and the geometry of domains in Euclidean space. Probab. Theory Related Fields 111 469-487. · Zbl 0911.60073 [12] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press. · Zbl 0743.60052 [13] Le Gall, J.-F. (1993). Solutions positives de u = u2 dans le disque unité. C. R. Acad. Sci. Paris Sér. I 317 873-878. · Zbl 0791.60049 [14] Le Gall, J.-F. (1994). A path-valued Markov process and its connections with partial differential equations. In First European Congress of Mathematics 2 185-212. Birkhäuser, Boston. · Zbl 0812.60058 [15] Le Gall, J.-F. (1995). The Brownian snake and solutions of u = u2 in a domain. Probab. Theory Related Fields 102 393-432. · Zbl 0826.60062 [16] Lyons, T. and Zheng, W. (1990). On conditional diffusion processes. Proc. Roy. Soc. Edinburgh 115 243-255. · Zbl 0715.60097 [17] Zheng, W. (1995). Conditional propagation of chaos and a class of quasilinear PDE’s. Ann. Probab. 23 1389-1413. · Zbl 0836.60053
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