Neil Falkner

Embedding in Brownian motion with drift and the Azéma-Yor construction

We consider the embedding of a probability distribution in Brownian motion with drift. We first give a sufficient condition on the target measure, under which a variant of the Azema-Yor (1979a, Seminaire de Probabilites XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, pp. 90-115) construction for this problem works. A necessary and sufficient condition for embeddability by means of some stopping time, not necessarily finite, is also provided. This latter condition is then analyzed in some detail.

Stopping distributions for right processes

SummaryLetX be a transient right process for which semipolar sets are polar. We characterize the measures which can arise as the distribution ofXT withT a non-randomized stopping time.

Feynman-Kac functionals and positive solutions of 1/2Δu+qu=0

SummaryLet D be a bounded C2 domain in ℝ d and let q be a bounded Borel function in D. For x∃D and z∃∂D suppose (Xt) under the law Px;z is Brownian motion in D starting at x and conditioned to converge to z. Let Τ be the lifetime of (Xt). We show that if the quantity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Gqpe0-% rq0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadweadaahaa% WcbeqaaiaadIhacaGG7aGaamOEaaaakmaacmaabaGaciyzaiaacIha% caGGWbWaamWaaeaadaWdXbqaaiaadghacaGGOaGaamiwamaaBaaale% aacaWGZbaabeaakiaacMcacaWGKbGaam4CaaWcbaGaaGimaaqaaiaa% dshaa0Gaey4kIipaaOGaay5waiaaw2faaaGaay5Eaiaaw2haaaaa!4AF6!

The distribution of Brownian motion in Rn at a natural stopping time

E^{x;z} \left\{ {\exp \left[ {\int\limits_0^t {q(X_s )ds} } \right]} \right\}

Some Generalizations of the Riemann-Lebesgue Lemma

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.

Approximation of Debuts

SummaryLet X and

Review: Basic Real Analysis by Anthony W. Knapp; Advanced Real Analysis by Anthony W. Knapp.

\hat X