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From Brownian motion to Schrödinger's equation
Kai Lai Chung,Chung-hsin Chao +1 more
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In this paper, the case of one dimension is considered and the q-Green function is used to measure the number of vertices in the one-dimensional space of Brownian motion.Abstract:
1. Preparatory Material.- 2. Killed Brownian Motion.- 3. Schrodinger Operator.- 4. Stopped Feynman-Kac Functional.- 5. Conditional Brownian Motion and Conditional Gauge.- 6. Green Functions.- 7. Conditional Gauge and q-Green function.- 8. Various Related Developments.- 9. The Case of One Dimension.- References.read more
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Lévy Processes and Stochastic Calculus
TL;DR: In this paper, the authors present a general theory of Levy processes and a stochastic calculus for Levy processes in a direct and accessible way, including necessary and sufficient conditions for Levy process to have finite moments.
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Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
TL;DR: In this article, the authors provide an overview of the properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion.
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Censored stable processes
TL;DR: In this article, the authors presented several constructions of a censored stable process in an open set, i.e., a symmetric stable process which is not allowed to jump outside the boundary of the set.
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Estimates on Green functions and Poisson kernels for symmetric stable processes
Zhen-Qing Chen,Renming Song +1 more
TL;DR: In this article, a symmetric α-stable process X on Rn is a Levy process whose transition density p(t, x − y) relative to the Lebesgue measure is uniquely determined by its Fourier transform ∫ Rn e ix ·ξp(t, x )dx = e−t|ξ| α.
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Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators
TL;DR: In this article, a continuous transition density of the semigroup generated by the Kato class was constructed, where the transition density is comparable with that of the fractional Laplacian.