Reversible Diffusion
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov. Kolmogorov''s characterization of reversible diffusions Let B denote a d-dimensional standard Brownian motion; let b : Rd Rd be a Lipschitz continuous vector field. Let X : [0, + ) × Rd be an It diffusion defined on a probability space ( , , P) and solving…mehr

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a reversible diffusion is a specific example of a reversible stochastic process. Reversible diffusions have an elegant characterization due to the Russian mathematician Andrey Nikolaevich Kolmogorov. Kolmogorov''s characterization of reversible diffusions Let B denote a d-dimensional standard Brownian motion; let b : Rd Rd be a Lipschitz continuous vector field. Let X : [0, + ) × Rd be an It diffusion defined on a probability space ( , , P) and solving the It stochastic differential equation mathrm{d} X_{t} = b(X_{t}) , mathrm{d} t + mathrm{d} B_{t} with square-integrable initial condition, i.e. X0 L2( , , P; Rd). Then the following are equivalent: The process X is reversible with stationary distribution on Rd. There exists a scalar potential : Rd R such that b = , has Radon Nikodym derivative frac{mathrm{d} mu (x)}{mathrm{d} x} = exp left( - 2 Phi (x) right) and int_{mathbf{R}^{d}} exp left( - 2 Phi (x) right) , mathrm{d} x = 1.