Diffusion Equations and Geometric Inequalities

Abstract

Let θ =(θ₀, θ₁) be a fixed vector in R ² with strictly positive components and suppose σ₀, σ₁ > 0. Set σ_θ = θ₀ σ₀ + θ₁ σ₁ and, if x ₀, x ₁ ∈ R ⁿ, set x _θ = θ₀ x ₀ + θ₁ x ₁. Moreover, for any j ∈{0, 1, θ}, let c _j : R ⁿ → R be a continuous, bounded function and denote by p _{σ j} , c _j (t, x, y) the fundamental solution of the diffusion equation

$$\frac{{\partial \upsilon }}{{\partial t}} = \frac{{\sigma _j^2 }}{2}\Delta \upsilon - \frac{1}{{\sigma _j^2 }}cjx\upsilon ,t >0,x \in {\mathbf{R}}^n$$

If

$$\frac{1}{{\sigma \theta }}c_\theta \left( {x_\theta } \right) \leqslant \frac{{\theta _0 }}{{\sigma _0 }}\left( {x_0 } \right) + \frac{{\theta _1 }}{{\sigma 1}}c_1 \left( {x_1 } \right),x_0 ,x_1 \in {\mathbf{R}}^n$$

then by applying the Girsanov transformation theorem of Wiener measure it is proved that

σⁿ _θ p _{σ θ}, c _θ(t, x _θ, y _θ) ≥{σⁿ ₀ p _{σ 0}, c ₀(t, x ₀, y ₀)}^θ ₀ σ₀ / σ_θ{σⁿ ₁ p _{σ 1}, c ₁(t, x ₁, y ₁)}^θ ₁ σ₁ / σ_θ for all x ₀, x ₀, y ₀, y ₁ ∈ R ⁿ and t > 0. Finally, in the last section, another proof of this inequality is given more in line with earlier investigations in this field.