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By Seeger R. J.

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Markovian Gaussian stochastic processes . . . . . . . . . . . How to make a Markovian reservoir . . . . . . . . . . . . . 44 48 50 Ergodic properties: the chain . . . . . . . . . . . . . . . . . . 3 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . Strong Feller Property . . . . . . . . . . . . . . . . . . . Liapunov Function . . . . . . . . . . . . . . . . .

142) For example if can show that all (or a dense subset) of the points in a set F are accessible in time t, then we have Pt (x, F ) > 0 , (143) that is the probability to reach F from x in the time t is positive. 3. Let us consider the SDE dxt = b(xt ) dt + σdBt , (144) where b is such that there is a unique solution for all times. Assume further that σ : Rn → Rn is invertible. For any t > 0 and any x ∈ Rn , the support of the [0,t] diffusion Sx = {f ∈ C([0, t], Rn ) , f (0) = x} and, for all open set F , we have Pt (x, F ) > 0.

Note that if we combine this computation with Kolmogorov Continuity Theorem we obtain that the paths of Brownian motion are almost surely H¨older continuous with exponents α < 1/2 and almost never H¨older continuous with exponents α ≥ 1/2. If we consider the wave equation in Rd , then one obtains similar results (random fields indexed by Rd instead of ”stochastic processes”). 2 Markovian Gaussian stochastic processes In this section we describe a few facts on Gaussian stochastic process, in particular we describe a situation when Gaussian stochastic processes are Markovian (see [3]).

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A Critique of Recent Quantum Theories by Seeger R. J.


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