The DeGiorgi-Nash-Moser estimate plays a crucial role in the study of quasilinear elliptic and parabolic equations. In the present paper we shall show that this fundamental estimate holds for solutions of a linear parabolic Volterra inte-grodiierential equation: @u @t = @ @x i a ij (x; t) @u @x j + Z t 0 @ @x i b ij (x; t;) @u @x j dd; where fa ij g and fb ij g are only assumed to be measurable...

Using a forward– backward stochastic differential equations (FBSDE) associated to a transmutation process driven by a finite sequence of Poisson processes, we obtain a probabilistic interpretation for a non-degenerate system of quasilinear parabolic partial differential equations (PDEs). The novetly is that the linear second order differential operator is different on each line of the system.