{"product_id":"9789811938306","title":"SpringerBriefs in Probability and Mathematical Statistics","description":"\u003ch1\u003eSpringerBriefs in Probability and Mathematical Statistics\u003c\/h1\u003e \u003ch2\u003eLee, Haesung; Stannat, Wilhelm; Trutnau, Gerald\u003c\/h2\u003e \u003cp\u003e\u003c\/p\u003e\u003cdiv\u003eThis book provides analytic tools to describe local and global behavior of solutions to Itô-stochastic differential equations with non-degenerate Sobolev diffusion coefficients and locally integrable drift. Regularity theory of partial differential equations is applied to construct such solutions and to obtain strong Feller properties, irreducibility, Krylov-type estimates, moment inequalities, various types of non-explosion criteria, and long time behavior, e.g., transience, recurrence, and convergence to stationarity. \u003c\/div\u003e\u003cdiv\u003eThe approach is based on the realization of the transition semigroup associated with the solution of a stochastic differential equation as a strongly continuous semigroup in the \u003ci\u003eL\u003c\/i\u003e\u003csup\u003e\u003ci\u003ep\u003c\/i\u003e\u003c\/sup\u003e-space with respect to a weight that plays the role of a sub-stationary or stationary density. This way we obtain in particular a rigorous functional analytic description of the generator of the solution of a stochastic differential equation and its full domain. The existence of such a weight is shown under broad assumptions on the coefficients. A remarkable fact is that although the weight may not be unique, many important results are independent of it. \u003c\/div\u003e\u003cdiv\u003eGiven such a weight and semigroup, one can construct and further analyze in detail a weak solution to the stochastic differential equation combining variational techniques, regularity theory for partial differential equations, potential, and generalized Dirichlet form theory. \u003c\/div\u003e\u003cdiv\u003eUnder classical-like or various other criteria for non-explosion we obtain as one of our main applications the existence of a pathwise unique and strong solution with an infinite lifetime. These results substantially supplement the classical case of locally Lipschitz or monotone coefficients.\u003c\/div\u003e\u003cdiv\u003eWe further treat other types of uniqueness and non-uniqueness questions, such as uniqueness and non-uniqueness of the mentioned weights and uniqueness in law, in a certain sense, of the solution.\u003c\/div\u003e\u003cdiv\u003e\u003cbr\u003e\u003c\/div\u003e \u003ch3\u003eDetails\u003c\/h3\u003e \u003cp\u003ePublished by: Springer\u003c\/p\u003e \u003cp\u003ePublication Date: 2022-08-28\u003c\/p\u003e \u003cp\u003eFormat: Paperback\u003c\/p\u003e \u003cp\u003eISBN-13: 9789811938306\u003c\/p\u003e \u003cp\u003eDOI: 10.1007\/978-981-19-3831-3\u003c\/p\u003e \u003cp\u003eDimensions: 235cm x155cm\u003c\/p\u003e \u003cp\u003ePages: 126\u003c\/p\u003e ","brand":"Springer Nature Singapore","offers":[{"title":"Default Title","offer_id":45588240367756,"sku":"9789811938306","price":53.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0710\/9545\/1788\/files\/9789811938306.jpg?v=1775962898","url":"https:\/\/fh90cf-fv.myshopify.com\/products\/9789811938306","provider":"Late Knight Books and Services, LLC","version":"1.0","type":"link"}