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Existence of Solutions To Weak Parabolic Equations For Measures
Author(s) -
Bogachev Vladimir I.,
Prato Giuseppe Da,
Röckner Michael
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611503014540
Subject(s) - mathematics , parabolic partial differential equation , mathematical analysis , partial differential equation
Let A = ( a ij ) be a Borel mapping on [0, 1] × R d with values in the space of non‐negative operators on R d and let b = ( b i ) be a Borel mapping on [0, 1] × R d with values in R d . Let L u ( t , x ) = ∂ t u ( t , x ) + a i j ( t , x ) ∂ x i∂ x ju ( t , x ) + b i ( t , x ) ∂ x iu ( t , x ) , u ∈ C 0 ∞ ( ( 0 , 1 ) × R d ) .Under broad assumptions on A and b , we construct a family μ = (μ t ) t ∈ [0, 1] of probability measures μ t on R d which solvesthe Cauchy problem L * μ = 0 with initial condition μ 0 = ν, where \nu is a probability measure on R d , in the following weak sense: ∫ 0 1 ∫R dL u ( t , x )μ t ( d x ) d t = 0 , u ∈ C 0 ∞ ( ( 0 , 1 ) × R d ) ,and lim t → 0∫R dζ ( x )μ t ( d x ) = ∫R dζ ( x ) ν ( d x ) , ζ ∈ C 0 ∞ ( R d ) .Such an equation is satisfied by transition probabilities of a diffusion process associated with A and b provided such a process exists. However, we do not assume the existence of a process and allow quite singular coefficients, in particular, b may be locally unbounded or A may be degenerate. An infinite‐dimensional analogue is discussed as well. Main methods are L p ‐analysis with respect to suitably chosen measures and reduction to the elliptic case (studied previously) by piecewise constant approximations in time. 2000 Mathematics Subject Classification 35K10, 35K12, 60J35, 60J60, 47D07.