In recent years, curve evolution, applied to a single contour or to the level sets of an image via partial differential equations, has emerged as an important tool in image processing and computer vision. Curve evolution techniques have been utilized in problems such as image smoothing, segmentation, and shape analysis. We give a local stochastic interpretation of the basic curve smoothing equation, the so called geometric heat equation, and show that this evolution amounts to a tangential diffusion movement of the particles along the contour. Moreover, assuming that a priori information about the shapes of objects in an image is known, we present modifications of the geometric heat equation designed to preserve certain features in these shapes while removing noise. We also show how these new flows may be applied to smooth noisy curves without destroying their larger scale features, in contrast to the original geometric heat flow which tends to circularize any closed curve.