TY - JOUR
T1 - On Lp-theory of stochastic partial differential equations in the whole spaces
AU - Krylov, N. V.
PY - 1996/3
Y1 - 1996/3
N2 - It is shown that equations like du = (aijuxixj + biuxi + cu + f) dt + (σikuxi + νku + gk) dwkt, t > 0, with variable random coefficients and with zero initial condition have unique solutions in the Sobolev spaces W2p, p ∈ [2, ∞), under natural ellipticity condition and under conditions that (i) a is uniformly continuous with respect to x, (ii) σ, ν have bounded first derivatives in x and all other coefficients are bounded, (iii) f ∈ Lp, g ∈ W1p. A corresponding result in the spaces of Bessel potentials Hnp is proved, which implies that better differentiability properties of the coefficients and free terms of the equations lead to the better regularity of solutions. Applications to equations with space-time white noise are given.
AB - It is shown that equations like du = (aijuxixj + biuxi + cu + f) dt + (σikuxi + νku + gk) dwkt, t > 0, with variable random coefficients and with zero initial condition have unique solutions in the Sobolev spaces W2p, p ∈ [2, ∞), under natural ellipticity condition and under conditions that (i) a is uniformly continuous with respect to x, (ii) σ, ν have bounded first derivatives in x and all other coefficients are bounded, (iii) f ∈ Lp, g ∈ W1p. A corresponding result in the spaces of Bessel potentials Hnp is proved, which implies that better differentiability properties of the coefficients and free terms of the equations lead to the better regularity of solutions. Applications to equations with space-time white noise are given.
KW - Bessel potentials
KW - Cylindrical white noise
KW - Stochastic equations
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U2 - 10.1137/S0036141094263317
DO - 10.1137/S0036141094263317
M3 - Article
AN - SCOPUS:0030528986
SN - 0036-1410
VL - 27
SP - 313
EP - 340
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
IS - 2
ER -