Formulae for the derivatives of heat semigroups

Abstract

We use a basic martingale method to show a differentiation formula for the derivatives $$d(P_{t}f)(x_0)(v_0)=\frac{1}{t}Ef(x_t){\int }_0^t⟨Y(x_s)(v_s),d{B}_t{⟩}_{R^m}.$$ These are proved first on $R^n$, then on manifolds. Afterwards for solutions of heat equations on differential forms, and a second order formula.

References in corpus (1)

• Stochastic Flows on Non-compact Manifolds

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