Download Analytical methods for Markov equations by Lorenzi, Luca PDF

By Lorenzi, Luca

The moment variation of this booklet has a brand new name that extra properly displays the desk of contents. during the last few years, many new effects were confirmed within the box of partial differential equations. This variation takes these new effects into consideration, specifically the examine of nonautonomous operators with unbounded coefficients, which has obtained nice realization. also, this variation is the 1st to take advantage of a unified method of include the recent leads to a unique place.

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2 We note that the maximum principle implies that, if f ≥ 0 does not identically vanish, then the sequence {un } is positive and increasing. Therefore, u is positive and the whole sequence {un } converges to u. 1, corresponding, respectively, to the data f + and f −. 5). Indeed, if v is another positive solution, then the maximum principle yields v(t, x) ≥ un (t, x) for any t > 0, x ∈ Bn , n ∈ N and, eventually, v ≥ u. 1). 5), associated with the one-dimensional operator Au = u′′ + x3 u′ , admits a nontrivial bounded solution u satisfying u(0, ·) = 0.

Some estimates of the Green function g are provided in Chapter 15. It is interesting and important for many applications to study the behaviour of the function s → (G(t, s)f )(x) when t and x are fixed and f ∈ Cb (RN ). If f ∈ Cc2 (RN ), then this function is differentiable and (Ds G(t, s)f )(x) = −(G(t, s)A(s)f )(x). By a straightforward density argument, the continuity of the function s → (G(t, s)f )(x) can be guaranteed for any f ∈ C0 (RN ). Such a result can be extended to any f ∈ Cb (RN ) assuming the existence of a suitable family of Lyapunov functions, which allow to prove that the family of measures Introduction xxxv {p(t, s, x, dy) : (t, s, x) ∈ {(t, s) ∈ I × I : t ≥ s} × B r } is tight for any J ⊂ I and r > 0, where p(t, s, x, dy) = g(t, s, x, y)dy.

Clearly, g(t, ·, ·), g(t, x, ·) and g(t, ·, y) are measurable functions for any t > 0 and x, y ∈ RN since they are the pointwise limit of measurable functions. We now prove the regularity properties of g. Fix R, T > 0, x0 ∈ BR and let y0 ∈ RN be such that g(T, x0 , y0 ) < +∞; actually we have seen that this holds true for almost any y0 ∈ RN . 1), and hence gk (·, ·, y0 ) − gh (·, ·, y0 ) is as well. Moreover, gk (·, ·, y0 ) − gh (·, ·, y0 ) is positive and, for any fixed 0 < t0 < t1 < T , it satisfies the following Harnack inequality (see [161]): sup [t0 ,t1 ]×B R gk (T, ·, y0 ) − gh (T, ·, y0 ) gk (·, ·, y0 ) − gh (·, ·, y0 ) ≤C inf x∈B R ≤C gk (T, x0 , y0 ) − gh (T, x0 , y0 ) , where C > 0 is a constant, independent of h and k.

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