Summary
Using standard reflected Brownian motion (SRBM) and martingales we define (in the spirit of Stroock and Varadhan-see [S-V]) the probabilistic solution of the boundary value problem
whereD is a bounded domain withC 3 boundary andn is the inward unit oormal vector on ∂D. The assumptions forq, c andf are quite general.
The corresponding Dirichlet problem was studied by Chung, Rao, Zhao and others (see [C-R1] and [Z-M]) and the corresponding Neumann by Pei Hsu in [H2]. Here we show that the probabilistic solution of our problem exists, is unique (unless we hit an eigenvalue), continuous on\(\bar D\) and equivalent to the weak analytic solution. The method we use is to reduce the problem to an integral equation inD that involves the associated semigroup and, hence, to the study of the properties of this semigroup. In this way we do not have to assume that the spectrum is negative (almost every previous work on these probabilistic solutions makes this assumption). We construct the kernel of this semigroup and we prove certain estimates for it which help us to establish many other results, including the gauge theorem. We also show that, if the boundary functionc is continuous, our semigroup is a uniform limit of Neumann semigroups and, furthermore, that the Dirichlet semigroup is a uniform limit of semigroups of our type. Therefore the Dirichlet spectrum is a “monotone” limit of spectra of mixed problems (see Sect. 5B), a fact which is mentioned without proof in Vol 1, Ch. IV, Sect. 2 of theMethods of Mathematical Physics by Courant and Hilbert. This establishes the interrelation of the three boundary value problems. Finally, we add a drift term to our differential equation, which becomes
and we solve the third boundary value problem for this equation probabilistically, with the help of Girsanov's transformation.
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References
[A-S] Aizenman, M., Simon, B.: Brownian Motion and Harnack's inequality for Schrödinger operators. Commun. Pure Appl. Math.35, 209–271 (1982)
[C1] Chung, K.L.: Lectures from Markov processes to Brownian motion. Berlin Heidelberg New York: Springer 1980
[C2] Chung, K.L.: Doubly-Feller process with multiplicative functional. Seminar on Stochastic Processes. Boston: Birkhäuser 1985
[C-H] Chung, K.L., Hsu Pei: Gauge theorem for the Neumann problem. Seminar on Stochastic Processes. Boston: Birkhäuser 1984
[C-R1] Chung, K.L., Rao, K.M.: Feynmann-Kac functional and Schrödinger equation. Seminar on Stochastic Processes. Boston: Birkhäuser 1981
[C-R2] Chung, K.L., Rao, K.M.: General Gauge theorem for multiplicative functionals. Trans. Am. Math. Soc.306, 819–836 (1988)
[C-W] Chung, K.L., Williams, R.J.: Introduction to stochastic integration. Boston: Birkhäuser 1983
[C-Z] Chung, K.L., Zhao, Z.: From Brownian motion to Schrödinger equation (to appear)
[Co-H] Courant, R., Hilbert, D.: Methods of mathematical physics
[Cr-Z] Cranston, M., Zhao, Z.: Conditional transformation of drift formula and potential theory for 1/2 Δ+b(·)·Δ. Commun. Math. Phys.112, 613–625 (1987)
[D-M] Dellacherie, C., Meyer, P.A.: Probabilities and potential B. Mathematics Studies, 72. Amsterdam: North Holland 1982
[Fr] Freidlin, M.: Functional integration and partial differential equations. Study 109. Annals of Mathematics Studies. Princeton: Princeton University Press 1985
[G-T] Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Berlin Heidelberg New York: Springer 1977
[H1] Hsu, Pei: Reflecting Brownian motion, boundary local time and the Neumann problem. Thesis, June 1984, Stanford University
[H2] Hsu, Pei: Probabilistic approach to the Neumann problem. Commun. Pure Appl. Math.38, 445–472 (1985)
[H3] Hsu, Pei: On the Poisson kernel for the Neumann problem of Schrödinger operators. (preprint)
[I-W] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North Holland 1981
[I] Itô, S.: Fundamental solutions of parabolic differential equations and boundary value problems. Jap. J. Math.27, 55–102 (1957)
[K-S] Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Berlin Heidelberg New York: Springer 1987
[K-T] Karlin, S., Taylor, H.: A second course in stochastic processes. New York: Academic Press 1981
[K] Katznelson, Y.: An introduction to harmonic analysis. New York: Dover 1976
[Kh] Khas'minskii, R.Z.: On positive solutions of the equationAu+Vu=0. Theory Probab. Appl. (translated from the Russian)4, 309–318 (1959)
[L-S] Lions, P.L., Sznitman, A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math.37, 511–537 (1984)
[P-S] Port, S., Stone, C.: Brownian motion and classical potential theory. New York: Academic Press 1978
[R] Royden, H.: Real Analysis, Second edition. London: Macmillan 1968
[S.Y] Saisho, Y.: Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probab. Th. Rel. Fields74, 455–477 (1987)
[S-U] Sato, K., Ueno, T.: Multi-dimensional diffusion and Markov processes on the boundary. J. Math. Kyoto Univ.3–4, 529–605 (1965)
[S.B] Simon, B.: Schrödinger semigroups. AMS Bulletin7, 447–526 (1982)
[S.L] Simon, Leon: private communication
[S-V] Stroock, D.W., Varadhan, S.R.S.: Diffusion processes with boundary conditions. Commun. Pure. Appl. Math.24, 147–225 (1971)
[T] Trèves, F.: Topological vector spaces, distributions and kernels. New York: Academic Press 1967
[Ø] Øksendal, B.: Stochastic differential equations. Berlin Heidelberg New York: Springer 1985
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Papanicolaou, V.G. The probabilistic solution of the third boundary value problem for second order elliptic equations. Probab. Th. Rel. Fields 87, 27–77 (1990). https://doi.org/10.1007/BF01217746
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DOI: https://doi.org/10.1007/BF01217746