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Additional info for A hierarchical Bayesian approach to modeling embryo implantation following in vitro fertilization (2

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This identity is a stochastic ‘fundamental theorem of calculus’ of sorts. Since things are random the difference between h0 (t) − h0 (0) and the integral of the infinitesimal rate cannot be identically zero. Instead it is a martingale. This is a process whose increments have mean-zero in a very strong sense, namely even when conditioned on the entire past. 22) in the stationary situation. Let Eνρ denote expectation of functions of the stationary process η(·) whose state η(t) is νρ -distributed at each time t.

Let Eνρ denote expectation of functions of the stationary process η(·) whose state η(t) is νρ -distributed at each time t. Normalize the height process h(·) at time zero so that h0 (0) = 0. Then h(·) is entirely determined by η(·). 23) where the particle flux is defined by: f (ρ) = ρ(1 − ρ). 24) (ii) Envelope property. Even though the flux f is nonlinear and therefore, as we see later, TASEP is governed by a nonlinear PDE, the height process has a valuable additivity property. 25) hi (0) = sup zi (0) for each site i ∈ Z.

1 20 Analysis and stochastics of growth processes and interface models gives this variational equality: (k) hi (t) = sup hk (0) + wi (t) . 18), and a general ‘hydrodynamic limit’ that describes the large scale evolution of the process. 27) in the form: t−1 h0 (t) = sup t−1 h[ty] (0) + t−1 w0 ([ty]) (t) . 28) y∈R Let t → ∞. s. by the law of large numbers. 18). With some work take the limit outside the supremum. Then we know t−1 h0 (t) converges. 23) arrive at: ([ty]) −f (ρ) = sup{ρy + g(−y)}. 29) y∈R This is a convex duality (Rockafellar 1970).

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A hierarchical Bayesian approach to modeling embryo implantation following in vitro fertilization (2 by Dukk V.


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