Appl Math Optim 37:295–353 (1998)
1998 Springer-Verlag New York Inc.
Exponential Integrability and Application to
and G. Kallianpur
Center for Stochastic Processes, Department of Statistics,
The University of North Carolina at Chapel Hill,
321 Phillips Hall, Chapel Hill, NC 27599-3260, USA
Abstract. We study the exponential integrability problem for I
)} < ∞, where I
) is a multiple Itˆo–Wiener integral on some Gaus-
sian space arising from the constructive quantum ﬁeld theory and from stochastic
quantization. We also study a class of singular inﬁnite-dimensional stochastic dif-
ferential equations whose drift coefﬁcients are measurable and unbounded. Using
a condition due to Kazamaki, we prove the existence of a weak solution assuming
some integrability conditions on the drift coefﬁcient. Then we apply the exponen-
tial integrability theorem and the existence theorem to study inﬁnite-dimensional
stochastic differential equations of stochastic quantization.
Key Words. P()
ﬁeld, Multiple integral, Stochastic quantization, Inﬁnite-
dimensional stochastic differential equation, Exponential integrability, Cameron–
AMS Classiﬁcation. Primary 60H10, 60H15, Secondary 34F05, 60J60, 60K40,
Let H be a separable Hilbert space. It is well known that there is a nuclear space (CNHS)
, continuously embedded in H: ⊂ H = H
is the dual of , and
a probability measure on (
, B), where B is the Borel σ-algebra on
(in the weak
This research was supported by the National Science Foundation and Air Force Ofﬁce of Scientiﬁc
Research Grant No. F49620 92 J 0154 and Army Research Ofﬁce Grant No. DAAL 03 92 G 0008.
Current address: Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA.