Feynman-Kac formula under a finite entropy condition

• 成果类型: Article
• 署名作者: Leonard, Christian
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01155-8
• 发表日期: 2022
• 页码: 1029-1091
• 关键词: diffusion-processes; REPRESENTATION; LAWS

摘要

Motivated by entropic optimal transport, we are interested in the Feynman-Kac formula associated to the parabolic equation (L + V)g = 0 with a final nonnegative boundary condition and a Markov generator L := partial derivative(t) + b.del + Delta(a)/2. It is well-known that when the drift b, the diffusion matrix a and the scalar potential V are regular enough and not growing too fast, the classical solution g of this PDE, is represented by the Feynman-Kac formula g(t)(x) = E-R[exp (integral([t, T]) V (s, X-s) ds) g(X-T) vertical bar X-t = x] where R is the Markov measure with generator L. We do not assume that g, b and V are regular, and only require that their growth is controlled by a finite entropy condition. These hypotheses are less restrictive than the standard assumptions of the theory of viscosity solutions, and allow for instance V to belong to some Kato class. We prove that g defined by the Feynman-Kac formula belongs to the domain of the extended generator L of the Markov measure R and satisfies the trajectorial identity: [(L + V)g](t, X-t) = 0, dtdP-a.e. where the path measure P is defined by P := f (X-0) exp (integral([0, T]) V (t, X-t) dt) g(X-T) R, with f : R-n -> [0, infinity) another non-negative function. We also show that the forward drift b(P) of P satisfies b(P) (t, X-t) = [b + a (del) over tilde log g](t, X-t), dtdP-a.e., where (del) over tilde is some extension of the standard derivative. Our probabilistic approach relies on stochastic derivatives, semimartingales, Girsanov's theorem and the Hamilton-Jacobi-Bellman equation satisfied by log g.