By Andreas Eberle
This booklet addresses either probabilists engaged on diffusion methods and analysts attracted to linear parabolic partial differential equations with singular coefficients. The valuable query mentioned is whether or not a given diffusion operator, i.e., a moment order linear differential operator with out zeroth order time period, that's a priori outlined on try out features over a few (finite or limitless dimensional) country area in simple terms, uniquely determines a strongly non-stop semigroup on a corresponding weighted Lp area. specific emphasis is put on phenomena inflicting non-uniqueness, in addition to at the relation among various notions of strong point showing in analytic and probabilistic contexts.
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Additional info for Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators
In fact, suppose first p > 1, and assume that, for some c, F~ is, for example, in Lq(c, Y0; pdx). 4) f~~ --~raldz p dy E Lq( c, Y0; p dx), < ~. and, in particular, CHAPTER 2. L P UNIQUENESS IN FINITE DIMENSIONS 44 Now fix 5 E (x0,y0). 4) still hold, if c is replaced by 5. From this and Fc E Lq(c, yo; pdx) one easily concludes t h a t Fa is in Lq(5, yo; pdx) as well. 4) does not necessarily hold. , pdx = //o pdx = oo for some (rasp. all) c E (xo,Yo). ) + b~xd , C~(xo,yo) ) on LP(xo,Yo ; pdx) is the generator of a C O semigroup for e v e r y absolutely continuous function a on (xo, Yo) such that a(x) > 0 for all x, and e v e r y function b E Lfoc(Xo, Y0; pdx) satisfying (A 1).
In Section a) above, we have shown that strong Markov uniqueness on L I(E; m ) , respectively Markov uniqueness on L2(E; m), implies uniqueness of the martingale problem among all sub-stationary, respectively reversible, time-homogeneous diffusion processes on E with initial distribution m. , does uniqueness of the martingale problem in an approriate sense imply Markov uniqueness, strong Markov uniqueness, or even L p uniqueness for some p E [1, ~ ) ? Appendix A S o m e s t a n d a r d t o o l s for p r o v i n g e x istence and uniqueness of C Osemigroups on Banach spaces We briefly review some standard techniques for proving existence and uniqueness of C O semigroups generated by linear operators on Banach spaces.
REMARK. Suppose (Tt)t>_o as in the theorem is a contraction semigroup, and let s > 0. Then the assertions in the theorem are also equivalent to: (iii') (Tt)t>_o is the only C O semigroup on B such that its generator extends (L, A), and Tt[l _< e Et. However, (Tt)t>o can be the only C O contraction semigroup with a generator that extends (L-, A), even if A is not a core for the generator of (Tt)t>_o. 2 a n d t h e r e m a r k . Obviously, (i) implies (ii). Suppose (ii) holds, and L is an extension of (L, A) which generates a C O semigroup.