[論文レビュー] Stochastic Flows on Non-compact Manifolds

Xue-Mei Li|arXiv (Cornell University)|May 31, 2021

Mathematical Dynamics and Fractals参考文献 36被引用数 27

ひとこと要約

本稿は、非コンパクト多様体上の SDE に対する解の流れの存在と性質を研究し、derivative flows を介して strong p-completeness を導入し、forms 上の heat semigroups に対する martingale-based representations を導出する。幾何学的・位相的な意味を導く。

ABSTRACT

I was asked to make my, by now quite old PhD thesis, available on the arxiv, for parts of it was never submitted for publication. The thesis offers a systematic study of stochastic differential equations (SDEs) on non-compact spaces. In particular we solve the open problem on strong completeness. An SDE is strongly complete if its solution can be chosen to depend continuously in space and in time for all time. The question is whether non-explosion, with possibly additional assumptions, implies strong completeness. Strong completeness of an SDE implies that its solution depends continuously on the initial condition, opening up possibility for numerical solutions, and the existence of a perfect Cocycle (a basic assumption on random dynamical systems). This was known only for compact manifolds and for linear state spaces, methods for either are not applicable to a general space. We also obtain existence of the global smooth solution flow of SDEs on $R^n$ (sometimes allowing substantial growth of the derivative of the coefficients, removing the global Lipschitz continuity condition). Non-explosion, the $C_0$-property, and the derivative flow are studied. We showed Bismut-Witten Laplacians are essentially self-adjojnt, paving the way for studying theirs semigroups acting on functions and on differential forms. We relate the Markovian semi-group on differential one forms with the semi-group $P_t$ on functions (inter-twining), find a method for proving path integration formulas for $dP_tf$, path integration formula for semi-group on differential forms, moment bounds for the derivative flows. Relation are obtained between intrinsic topological and geometrical properties of the manifold and that of SDEs. Information on the homotopy and cohomology of the manifolds are obtained from moment stability of the stochastic flows.

研究の動機と目的

• 非コンパクト多様体上の stochastic flows の存在と性質を調べる。
• derivative-flow 条件下で strong completeness と強い p-完備性を確立する。
• 拡散過程の挙動を基礎となる多様体の幾何学的/位相的特徴と結びつける。

提案手法

• derivative flow 法を導入し、解の流れと初期データに関するその微分を結合させる。
• strong p-completeness と非爆発を許す係数の成長条件を展開する。
• dP_t f = δP_t(df) の intertwinings を導出し、函数と微分形式の半群を研究する。
• 逐次重み付き熱半群 on forms に対して martingale-based integral representations を提供し、Bismut's formula を拡張する。
• 弱一様被覆技法を用いて無限遠での挙動と非爆発を分析する。
• 熱半群の Lp-有界性によるモーメント安定性と π1(M) の消滅・コホモロジーとの関連を確立する。

実験結果

リサーチクエスチョン

• RQ1 係数と derivative flow に関する条件の下で、非compact-manifold 上に globally defined な滑らかな解の流れが存在するにはどうすればよいか。
• RQ2 強い p-完備性は拡散の非爆発とモーメント安定性とどのように関連するか。
• RQ3 derivative-flow の成長制約が π1(M) や調和形式といったトポロジー的特徴に対して何を意味するか。
• RQ4 非 compact な多様体上で、函数および微分形式の半群に対する intertwinings と勾配公式を得られるか。
• RQ5 1-forms 以上の形式に対して、martingale 法を用いて forms の熱半群の積分表示を得られるか。

主な発見

• Strong 1-completeness and strong p-completeness can be achieved under integrability bounds on derivative flows.
• Non-explosion results for SDEs on open sets of R^n and for Brownian systems on submanifolds with bounded second fundamental forms.
• Intertwining property dP_t f = δP_t(df) holds under conditions on the derivative flow, enabling gradient representations.
• An Lp-boundedness and contractivity framework for heat semigroups on differential forms leads to cohomology vanishing results.
• A martingale-based integral representation formula for d(P_t f) extends Bismut’s formula to forms and higher-degree forms.
• Formulae for d(P_t φ) for q-forms (q ≥ 1) obtained via gradient SDEs, providing gradient of logarithmic heat kernels in certain settings.