By Mark I. Freidlin, Jean-Francois Le Gall, Paul L. Hennequin

CONTENTS: M.I. Freidlin: Semi-linear PDE's and restrict theorems for giant deviations.- J.F. Le Gall: a few homes of planar Brownian movement.

**Read or Download Ecole d'Eté de Probabilités de Saint-Flour XX - 1990 PDF**

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**Extra resources for Ecole d'Eté de Probabilités de Saint-Flour XX - 1990**

**Sample text**

Vr2~TeCs)), This function increases monotonically; remaining positive and c(x) be ~(s), s > O, equation ~Cs) nically, x < 0}, Consider the function ~0 = o. its derivative (see Figure 5). 1 that lim ue(t,x) = I show that in the shaded domain above the curve lim uC(t,x) = 0 if x > @(t). This means that exactly the position of the front at time ~(t-s)} ^ t. t. Denote @. S) Figure 5. E uC(t-T_ X C )exp[l--[ c(X~,uC(t-s,X~ X t' T t [c "0 ))ds uC(t'x) = -< (iV sup g(x))ExZ{Tt

O Figure 8 shows that at time tI the new species K I, was switched on by K O. m The species K 2, switched on by was switched on at time K0 and K5, t2 by was switched on by KlO. K2, The species at time ~3" was t S. We see that, though the evolution is the result of rB_ndom mutation, the sequence of new species and the times when they appear ape not random at least in the main term. This model does not take into account competition between species. We can include the competition by changing the fitness coefficient after appeara_nce of each new species.

X e is the Markov process corresponding s of the heat process is (-~,~)xR r = ~. Let Define the functional (-~,~)×Co~ T = TF[t, ~] on to the operator F is the The second component eL. The phase space be a closed subset of ~. with the values in [O,m] by the formula T = rF(t,~) It is clear that F; s ~F TF(t,XC) = inf{s : ( t - s , ~ s) e F}. is the first time when the heat process touches is a Markov time with respect to the family of is the minimal measurable f o r any v-fields ~-field in the probability space, such that si S s• The f u n c t l o n a l s ~F we c a l l {~s' s Z 0}; Xe sI is ~ s Markov / u n c t l o n a l s .