2005/12/13 by В. И. Богачев, Michael Röckner, Bogachev, Vladimir I. +3 · 1 citation
Computer Science · Mathematics · #35K10 #35K12 #47D07 #60J35 #60J60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.math/0512264
openalex publication_date 2005/12/13 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
Given a second order parabolic operator\n \n Lu(t,x)\n :=
frac
partial u(t,x)
partial t\n + aij(t,x)
partialxi
partialxju(t,x)\n + bi(t,x)
partialxiu(t,x),\n we consider the weak parabolic equation L*\μ=0 for Borel probability\nmeasures on (0,1)\×\ℝd. The equation is understood as the\nequality\n \n
int_(0,1)
times
mathbbRd Lu d
mu =0\n for all smooth functions u with compact support\nin~(0,1)\×\ℝd. This equation is satisfied for the transition\nprobabilities of the diffusion process associated with~L.\n We show that under broad assumptions \μ has the form \μ= varrho(t,x) dt\ndx, where the function x\↦ varrho(t,x) is Sobolev, |\∇x\n varrho(x,t)|2/ varrho(t,x) is Lebesgue integrable over\n[0,\τ]\×\ℝd, and varrho\∈ Lp([0,\τ]\×\ℝd)\nfor all p\∈ [1,+\∞) and \τ<1. Moreover, a sufficient condition for\nthe uniform boundedness of varrho on [0,\τ]\×\ℝd is given.\n