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Freezing coefficients locally in the Keplerian viscous diffusion equation gives ${\Sigma }_{1,t}\simeq 3{\partial }_{\Sigma }(\bar{\nu }\Sigma ){\Sigma }_{1,rr}$. A radial Fourier mode grows at rate $-3{\partial }_{\Sigma }(\bar{\nu }\Sigma )k_r^2$. Negative response reverses ordinary smoothing and gives a backward heat equation, within the wavelength range where the thin disc transport closure applies.

Here $[M{]}_t=t$ and $d{A}_t=dt$, so the formula in part (b) givesIf $g$ vanishes on $\mathbb{R}\times [0,\infty )$, this integral is zero. Conversely suppose $A_t^f=0$ for all $t$ almost surely. Its integrand is continuous along each path; the fundamental theorem of calculus implies $g(B_t,t)=0$ for all $t$ on that event. For any fixed $t>0$, the normal distribution of $B_t$ has a strictly positive density at every real point. Since $g(\cdot ,t)$ is continuous, a nonzero value at any spatial point would give a nonzero value on an open interval of positive Brownian probability, a contradiction. Thus $g(x,t)=0$ for every $x$ and every positive $t$. Continuity extends this to $t=0$.

Consequently the exact condition isThese are space-time harmonic functions along Brownian motion, satisfying the backward heat equation. The second coordinate never takes negative values, so no equation is imposed for $y<0$, beyond compatibility with the assumed global $C^2$ regularity. The normalized finite-variation term is unique by part (a), so it cannot be removed by choosing a different decomposition.

For $0\le s\le t$ and $E\in {\mathcal{F}}_s$, the definition of the Radon-Nikodym derivative givesThe density is nonnegative and integrable; with the probability normalization used here its expectation is one. Henceso $Z$ is a martingale. When $Z_t=h(X_t,t)$, the Itô formula under Wiener measure givesThe first term is a continuous local martingale, by localization on compact space-time sets. Since $Z$ is a martingale, uniqueness of the continuous semimartingale decomposition makes ${\int }_0^{t}g(X_s,s)ds=0$ identically. The integrand is continuous in time along every continuous path, so $g(X_t,t)=0$ for every $t$, outside one common null event.

For each fixed $t>0$, $X_t$ has a Gaussian distribution with strictly positive density everywhere on $\mathbb{R}$. Since $g(\cdot ,t)$ is continuous, $g(X_t,t)=0$ almost surely implies $g(x,t)=0$ for every real $x$. ThereforeIf the stated smoothness includes time zero, continuity extends the equation there. This is the backward heat equation. The full-support argument is needed to pass from an identity along Brownian paths to an equation at every spatial point.

For $\bar{\nu }=C{r}^a{\Sigma }^b$, the response is $q=b+1$. A nonaccreting background has ${\bar{\nu }}_0{\Sigma }_0\propto r^{-1/2}$, while ${\Sigma }_0\propto r^{-p}$, soConsequently ${\bar{\nu }}_0\propto r^{p-1/2}$. On this steady background, introduce the square-root-radius diffusion transformSince $q$ is constant and ${\partial }_r=(2x{)}^{-1}{\partial }_x$, the linear equation becomesThus $A(x)\propto x^{2p-3}$, and the required choice isFor a Fourier mode $g\propto e^{\lambda t+i{k}_{x}x}$, $\lambda =-A{k}_x^2$. Positive kinematic viscosity makes $A$ have the sign of $q$, so $q<0$ gives growing modes whose rate increases with $k_x^2$. The formal equation is a backward heat equation and predicts arbitrarily rapid small-scale amplification. Physically the thin disc diffusion closure applies only to wavelengths sufficiently larger than the thickness and stress-relaxation scales; this formal limit identifies the need for a cutoff, rather than a finite fastest wavelength absent from the model. The boundary case $q=0$ has vanishing linear transport response.

The negative-diffusion criterion for viscous disk instability is ${\partial }_{\Sigma }(\bar{\nu }\Sigma )<0$. A surface density enhancement then transports angular momentum less effectively and grows. Freezing the Keplerian viscous diffusion equation locally gives growth proportional to the square of radial wavenumber, within the thin-disc transport range; wavelengths must still be sufficiently long compared with the disk scale height for that closure. The formal short-wave limit is a backward heat equation.