Summary
Let D be a bounded C 2 domain in ℝ d and let q be a bounded Borel function in D. For x∃D and z∃∂D suppose (X t) under the law P x;z is Brownian motion in D starting at x and conditioned to converge to z. Let Τ be the lifetime of (X t). We show that if the quantity \(E^{x;z} \left\{ {\exp \left[ {\int\limits_0^t {q(X_s )ds} } \right]} \right\}\) is finite for one x∃D and one z∃∂D, then this quantity remains bounded as x varies over D and z varies over ∂D. This may be considered one quantitative expression of the qualitative statement that no matter where Brownian motion in D eventually hits ∂D, it goes all over D before it gets there. We apply this result to show that if the equation 1/2δu+qu=0 admits a non-negative solution in D, which is strictly positive on a subset of ∂D of positive harmonic measure, then for any non-negative bounded Borel function f on ∂D it admits a unique bounded solution u satisfying u=f on ∂D, and this solution u is non-negative.
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Research supported in part by NSF grant MCS-8103473
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Falkner, N. Feynman-Kac functionals and positive solutions of 1/2Δu+qu=0. Z. Wahrscheinlichkeitstheorie verw Gebiete 65, 19–33 (1983). https://doi.org/10.1007/BF00534991
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DOI: https://doi.org/10.1007/BF00534991