Nonprobabilistic Analogs of the Cauchy Process
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NONPROBABILISTIC ANALOGS OF THE CAUCHY PROCESS A. K. Nikolaev∗ and M. V. Platonova†
UDC 519.2
It is known that a solution to the Cauchy problem for the evolution equation, the right-hand side of which contains a convolution operator with generalized function |x|−2 , admits a probabilistic representation in the form of the expectation of the trajectory functional of the Cauchy process. Similar representations are constructed for evolution equations containing convolution operator with generalized function (−1)m |x|−2m−2 for arbitrary m ∈ N. Bibliography: 11 titles.
1. Introduction We consider the Cauchy problem for the equation ∂u = A0 u, u(0, x) = ϕ(x), ∂t where the operator A0 is given by the formula ϕ(x + y) − ϕ(x) dy A0 ϕ(x) = v.p. y2
(1)
R
and the function ϕ belongs to the Sobolev class W22 (R). We note that A0 is a pseudo-differential operator with symbol cos (p y) − 1 dy. a0 (p) = v.p. y2 R
It is known that a solution to the Cauchy problem (1) admits the probabilistic representation u(t, x) = E ϕ(x + ξ(t)),
(2)
where ξ(t) is the standard Cauchy process. In the present paper, we obtain similar probability representations for solution to the Cauchy problem ∂u = (−1)m Am u, u(0, x) = ϕ(x), ∂t where m ∈ N and the operator Am is defined as m ϕ(2j) (x)y 2j dy . Am ϕ(x) = v.p. ϕ(x + y) − ϕ(x) − (2j)! y 2m+2 j=1
R
For each m, the operator Am is also a pseudo-differential operator with symbol m dy (−1)j (p y)2j . cos (p y) − 1 − am (p) = v.p. 2m+2 (2j)! y R
j=1
We note that it is impossible to obtain exactly the probabilistic representation of the form (2) for m ≥ 1, since the fundamental solution of the corresponding equation is not the probabilistic ∗
St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
†
St.Petersburg Department of the Steklov Mathematical Institute; P. L. Chebyshev Laboratory in St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 183–194. Original article submitted October 26, 2018. 1072-3374/20/2511-0119 ©2020 Springer Science+Business Media, LLC 119
density. In this case, to obtain a probabilistic representation of the solution to the Cauchy problem, we use a method proposed in [3, 6–9] and based on the use of theory of generalized functions (see [1]). The cases of even and odd m, i.e., m = 2k and m = 2k − 1, are considered separately. The case m = 2k − 1 is more complicated and requires the use of complex-valued processes. 2. Basic notation and definitions In the present paper, the direct Fourier transform is defined as ∞ ϕ(x) eipx dx, ϕ(p) = −∞
and, accordingly, the inverse transformation is defined as ∞ 1 ϕ(p) e−ipx dp. ϕ(x) = 2π −∞
W2k (R)
the Sobolev space of functions defined on R and having quadratically Denote by summable generalized derivatives up to the order k inclusive (see [11, p. 146]). The standard norm in the space W2k (R) is defined by the formula k 2 ψ = |ψ (l) (x)|2 dx. k l=0 R
It is convenient
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