I am currently looking at a $1$D-Burger's equation defined by \begin{equation} \label{ex burgers} \left\{ \begin{array}{ll} {} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \frac{\partial^2 V_m}{ \partial x^2} (t,x) - V_m(t,x) \frac{\partial V_m}{\partial x} (t,x), \quad \forall (t,x) \in (0,T] \times \mathbb{R}, \\ {} & {} \\ {} & V_m(0,x)= \int_{-\infty}^x \, m(dy), \quad \forall x \in \mathbb{R}, \end{array} \right. \end{equation} where $m$ is a probability measure on $\mathbb{R}$. It is known that $V_m(t,\cdot)$ is a cumulative distribution function and satisfies the following closed form solution via Cole-Hopf transformation (https://projecteuclid.org/euclid.aoap/1034968229, page $822$)

$$ V_m(t,x) = \frac{V_1(t,x,m)}{V_2(t,x,m)}, $$ where \begin{equation} V_1(t,x,m):= \int_{\mathbb{R}} \frac{x-y}{t} \exp \bigg\{ - \frac{1}{\sigma^2} \bigg[ \frac{(x-y)^2}{2t} + \int_{x}^y \int_{-\infty}^z m(d\theta) \,dz \bigg] \bigg\} \, dy \label{V1 } \end{equation} and \begin{equation} V_2(t,x,m):= \int_{\mathbb{R}} \exp \bigg\{ - \frac{1}{\sigma^2} \bigg[ \frac{(x-y)^2}{2t} + \int_{x}^y \int_{-\infty}^z m(d\theta) \,dz \bigg] \bigg\} \, dy. \label{V2 } \end{equation}

My question is how can we show the properties that $V_m(t,\cdot)$ is a cumulative distribution function, for any $t \in [0,T]$, i.e. that

$$ \lim_{x \to \infty} V_m(t,x) =1, \quad \quad \lim_{x \to -\infty} V_m(t,x) =0, \quad \quad V_m(t, \cdot) \text{ is increasing, } \quad \forall t,$$

based on the above representation formula of the solution. I believe that there are certain estimates that I am missing and I need those to prove properties of functions with a similar form.


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