Diffusion equations and geometric inequalities

Let theta = (theta(0), theta(1)) be a fixed vector in R-2 with strictly positive components and suppose sigma(0), sigma(1) > 0. Set sigma(theta) = theta(0)sigma(0) + theta(1)sigma(1) and, if x(0), x(1) is an element of R-n, set x(theta) = theta(0)x(0) + theta(1)x(1). Moreover, for any j is an element of{0, 1, theta}, let c(j) : R-n --> R be a continuous, bounded function and denote by p(sigma j,cj)(t, x, y) the fundamental solution of the diffusion equation partial derivative v/partial derivative t = sigma(j)(2)/2 Delta v - 1/sigma(j)(2)c(j)(x)v, t > 0, x is an element of R-n. If 1/sigma(theta)c(theta)(x(theta)) less than or equal to theta(0)/sigma(0)c(0)(x(0)) + theta(1)/sigma(1)c(1)(x(1)), x(0), x(1) is an element of R-n then by applying the Girsanov transformation theorem of Wiener measure it is proved that sigma(theta)(n)p(sigma theta,c theta)(t, x(theta), y(theta)) greater than or equal to {sigma(0)(n)p(sigma 0,c0)(t, x(0), y(0))}(theta 0 sigma 0/sigma theta){sigma(1)(n)p(sigma 1,c1)(t, x(1), y(1))}(theta 1 sigma 1/sigma theta) for all x(0), x(0), y(0), y(1) is an element of 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.