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On the distribution density of the first exit point of a diffusion process from a small neighborhood of its initial position. (English. Russian original) Zbl 1002.60044

Theory Probab. Appl. 45, No. 3, 450-465 (2000); translation from Teor. Veroyatn. Primen. 45, No. 3, 536-554 (2000).
An asymptotic expansion up to \(o(R^2), R \to 0\), in powers of a scaling parameter \(R\) is obtained for the density indicated in the title. Actually this density is obtained by the Laplace transform in \(t \geq 0\) from the semi-Markov transition function and the result refers to multidimensional semi-Markov diffusion processes having transition functions generated by a (given) elliptic partial differential equation (e.p.d.e.) in a domain \(S \subset \mathbb{R}^d\) \((d \geq 2)\). The one-dimensional case was considered by the author [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 130, 190-205 (1983; Zbl 0534.60084)]. For scaling of spherical neighborhoods of the initial point the coefficients of expansion are given explicitely, “reflecting the probabilistic sense” of the coefficients of the 2nd order e.p.d.e.

MSC:

60G99 Stochastic processes
35J25 Boundary value problems for second-order elliptic equations
60J99 Markov processes

Citations:

Zbl 0534.60084
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