The nonlocal boundary value problem for one-dimensional backward Kolmogorov equation and associated semigroup.

*(English)*Zbl 1434.60220Summary: This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) incurvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to thecase when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovianparticle leaves the boundary of the domain by jumps.

##### MSC:

60J60 | Diffusion processes |

35K20 | Initial-boundary value problems for second-order parabolic equations |

##### Keywords:

parabolic potential; boundary integral equation method; Feller semigroup; nonlocal boundary condition
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\textit{R. V. Shevchuk} et al., Carpathian Math. Publ. 11, No. 2, 463--474 (2019; Zbl 1434.60220)

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##### References:

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