By Dominique Bakry, Ivan Gentil, Michel Ledoux (auth.)
The current quantity is an intensive monograph at the analytic and geometric points of Markov diffusion operators. It makes a speciality of the geometric curvature houses of the underlying constitution with the intention to research convergence to equilibrium, spectral bounds, sensible inequalities resembling Poincaré, Sobolev or logarithmic Sobolev inequalities, and diverse bounds on strategies of evolution equations. whilst, it covers a wide type of evolution and partial differential equations.
The ebook is meant to function an advent to the topic and to be available for starting and complex scientists and non-specialists. concurrently, it covers a variety of effects and methods from the early advancements within the mid-eighties to the most recent achievements. As such, scholars and researchers attracted to the fashionable facets of Markov diffusion operators and semigroups and their connections to analytic sensible inequalities, probabilistic convergence to equilibrium and geometric curvature will locate it in particular beneficial. chosen chapters is usually used for complicated classes at the topic.
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Extra resources for Analysis and Geometry of Markov Diffusion Operators
Xt−tk , X0 ). 4) when the initial measure ν is μ since in this case the measure pt (x, dy)μ(dx) is symmetric in (x, y), and then, by an immediate induction, the measures ptk −tk−1 (yk−1 , dyk ) ptk−1 −tk−2 (yk−2 , dyk−1 ) · · · pt1 (y0 , dy1 ) μ(dy0 ) are invariant under the change (y0 , . . , yk ) → (yk , . . , y0 ). Therefore, for any t > 0, the law of the process (Xs )0≤s≤t is the same as the law of the process (Xt−s )0≤s≤t . Hence, the law of the Markov process is “reversible in time”. 1) at t = 0.
A . It is also generated by a partition (B1 , . . 2 Markov Semigroups, Invariant Measures and Kernels 15 sets with m-measure 0 have been removed. For f ∈ L1 (m), define P f as the conditional expectation E(Pf | F ) of Pf with respect to F . The operator P is bounded from L1 (m) into L∞ (m) with norm M and may be represented as j P f (x) = Qj f 1B (x) j j =1 where Qj f = 1 m(Bj ) Pf dm. Bj The Qj ’s, j = 1, . . , j , are linear operators, bounded on L1 (m), with norm bounded above by M. By duality, Qj f = E f (y) qj (y)dm(y) for some bounded (by M) measurable functions qj , j = 1, .
1) is usually called a diffusion process (more accurately, since the initial point x is a variable, it is a family of diffusion processes). 1, p.